A Combinatorial Model of Lagrangian Skeleta · A Combinatorial Model of Lagrangian Skeleta by Alex Zorn Doctor of Philosophy in Mathematics University of California, Berkeley Professor

A Combinatorial Model of Lagrangian Skeleta

by

Alex Zorn

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Mathematics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor David Nadler, ChairAssistant Professor Vivek Shende

Professor Marjorie Shapiro

Spring 2018


Copyright 2018by

Alex Zorn

1

Abstract


by

Alex Zorn

Doctor of Philosophy in Mathematics

University of California, Berkeley

Professor David Nadler, Chair

We investigate a collection of posets- combinatorial arboreal singularities- which are thestrata posets of the arboreal singularities constructed by David Nadler. Nadler demonstratedthat any Lagrangian skeleton admits a non-characteristic deformation into a skeleton withonly arboreal singularities, suggesting that arboreal singularities form the basis for a combi-natorial theory of Lagrangian skeleta.

In this document, we introduce a form of combinatorial data called a ‘cyclic structure’,which is essentially codimension-one data with compatibility conditions in codimensions twoand three. We develop a comprehensive theory of isomorphisms of combinatorial arborealsingularities and cyclic arboreal singularities (singularities equipped with a cyclic structure).We show that a cyclic structure determines (up to quasi-equivalence) a sheaf of dg-categorieson a combinatorial arboreal singularity, and investigate combinatorial properties of this sheaf.We describe a class of ‘arboreal moves’, which are local mutations of combinatorial arborealspaces preserving this sheaf of categories. Finally, we discuss how this combinatorial pictureis related to the geometric understanding of Lagrangian embeddings of arboreal singularities.

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Throughout my life, my grandfather, Jens Zorn, has served as a constant source ofinspiration and support, and has entertained and furthered my scientific curiosity. This

dissertation is dedicated to him.

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Contents

Contents ii

List of Figures iv

List of Tables vii

1 Introduction 11.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Summary of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Combinatorial Arboreal Singularities and Arboreal Spaces 182.1 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Some Examples: An and Sn singularities . . . . . . . . . . . . . . . . . . . . 25

3 Cyclic Structures 293.1 Pre-Cyclic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Directed Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Cyclic Structures on PA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5 Cyclic Structures on PS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6 Cyclic Structures on An Singularities . . . . . . . . . . . . . . . . . . . . . . 423.7 Cyclic Structures on Sn Singularities . . . . . . . . . . . . . . . . . . . . . . 46

4 The Combinatorics of Cyclic Structures 504.1 Isomorphisms of C-Arb Singularities . . . . . . . . . . . . . . . . . . . . . . 504.2 Reflection Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Isomorphisms of Cyclic C-Arb Singularities . . . . . . . . . . . . . . . . . . . 594.4 Reflection Isomorphisms for Directed Trees . . . . . . . . . . . . . . . . . . . 594.5 The Coherence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 A Sheaf of Categories 68

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5.1 Dg-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.2 Sheaves of dg-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Representations of Directed Trees . . . . . . . . . . . . . . . . . . . . . . . . 735.4 The Trivalent Vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.5 Reflection Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.6 The Cyclic Structure Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 805.7 Construction of the Reflection Functors . . . . . . . . . . . . . . . . . . . . . 81

6 Graded Structures and Graded Branes 996.1 Graded Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.3 The Representation Theory of Graded Branes . . . . . . . . . . . . . . . . . 1186.4 Branes on An Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.5 In Search of a ‘Fukaya Category’ . . . . . . . . . . . . . . . . . . . . . . . . 125

7 Arboreal Moves 1267.1 The H-to-I Move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.2 Definition of Arboreal Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.3 The Topology of Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.4 Classification of Good Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.5 Examples for An and Sn Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8 The Geometry of Arboreal Spaces 1488.1 Stratified Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488.2 Arboreal Singularities and Arboreal Spaces . . . . . . . . . . . . . . . . . . . 1508.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.4 The Geometry of Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1648.5 The Geometry of Moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

A Cyclic Orders on Sets 175

B Infinity-Categories and Sheaves 177B.1 Quasi-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.2 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180B.3 Morphisms Between Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 182B.4 Pullback Functors and Homotopy . . . . . . . . . . . . . . . . . . . . . . . . 185

Bibliography 188

iv

List of Figures

1.1 The Geometry of the Neighborhood of a Codimension-One Cell . . . . . . . . . 21.2 The A3 (left) and S4 (right) trees. . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The H-to-I move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 The An Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Illustrations of PA2 (left) and PA3 (right) . . . . . . . . . . . . . . . . . . . . . . 71.6 A diagram of the sheaf Qµ on PA2 . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 The sheaf Qµ after applying a quasi-equivalence of global sections . . . . . . . . 101.8 Pre-Branes on PA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.9 The H-to-I move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.10 Total Space of the H-to-I move . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.11 A Generalized Cyclic Interval I(v, w,D). . . . . . . . . . . . . . . . . . . . . . . 141.12 The unique (up to equivalence) partitions of cyclic sets yielding moves of type A3

and A4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1 A Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 A Correspondence with Edges/Terminal Vertices Labelled . . . . . . . . . . . . 242.3 The An Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Hasse diagram for PA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Hasse diagram for PA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6 Illustrations of PA2 (left) and PA3 (right) . . . . . . . . . . . . . . . . . . . . . . 272.7 Another Visualization of PA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.8 The Sn Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 The Geometry of the Neighborhood of a Codimension-One Cell . . . . . . . . . 293.2 A Directed Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 A Codimension-One Cell, and the three Top Cells in the Induced Cyclic Order . 323.4 An Illustration of PA3\{{e0, e1}, {e2, e3}}. . . . . . . . . . . . . . . . . . . . . . 363.5 A Completed Arrow Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Translating Between Two Descriptions of Codimension-One Cells in PS4 . . . . . 393.7 An Illustration of PS4\{vi, ei}c. . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8 Constructing Pinwheel Orientations . . . . . . . . . . . . . . . . . . . . . . . . . 413.9 Constructing Pinwheel Orientations . . . . . . . . . . . . . . . . . . . . . . . . . 42

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3.10 A Directed Tree (A5, µ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.11 The Cyclic Order ∆(Oµ) on KA5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1 A tree T and the chains in KT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 An illustration of the relation ∼C : The subtree {v} is a witness to the relation

C1 ∼0C C3, while {v, w} is a witness to the relation C2 ∼0

C C3 . . . . . . . . . . . 534.3 Cases B1 and B2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4 Case B3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.5 (An, µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.6 The Construction of q and p1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.7 The construction in Case 2b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.1 The directed tree (A2, µ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Geometric Picture of the Indecomposable Representations . . . . . . . . . . . . 76

6.1 Illustration of Mondromy Axiom (M2) . . . . . . . . . . . . . . . . . . . . . . . 1006.2 Illustration of Monodromy Axiom (M3) . . . . . . . . . . . . . . . . . . . . . . . 1006.3 Arrow Diagram Setup With Labelled Cells . . . . . . . . . . . . . . . . . . . . . 1016.4 Arrow Diagrams Giving m(ϕ

{e0}t,x ϕ

{e2}z,t ϕ

{e1}y,z ϕ

{e3}x,y ) = −2 (left) or 0 (right) . . . . . 102

6.5 A Pinwheel Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.6 A Neighborhood of {v1} in the Pinwheel Orientation . . . . . . . . . . . . . . . 1036.7 Cells Near {v3} In PS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.8 Hasse diagram for PA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.9 Hasse diagram for PA3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.10 Illustration of the Graded Structure Gµ . . . . . . . . . . . . . . . . . . . . . . . 1066.11 Pre-Branes on PA2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.12 An Illustration of L = PA3\{{e0, e1}, {e2, e3}} with two cyclic structures- On the

left, L is not a Brane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.1 The H-to-I move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.2 Total Space of the H-to-I move . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.3 The H-to-I move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.4 An Invalid Move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.5 A Second Invalid ‘H-to-I-esque’ Move . . . . . . . . . . . . . . . . . . . . . . . . 1287.6 A Generalized Cyclic Interval I(v, w,D). . . . . . . . . . . . . . . . . . . . . . . 1337.7 Illustrating The Edge Expansion expe . . . . . . . . . . . . . . . . . . . . . . . . 1357.8 Illustrating the Edge Cut cute. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.9 Possible Partitions of KA3- Only the Fourth Gives a Move . . . . . . . . . . . . 1457.10 The Partition Givng The Unique A4 Move Up To Equivalence/Reversal . . . . . 1457.11 A Topological Picture of The A4 Move . . . . . . . . . . . . . . . . . . . . . . . 1467.12 The Four Distinct Partitions of KS4 Giving Moves . . . . . . . . . . . . . . . . . 147

8.1 Embeddings Which Are Not Germ Equivalent . . . . . . . . . . . . . . . . . . . 155

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8.2 The embeddings i+ (left) and i− (right) of LA2 . . . . . . . . . . . . . . . . . . . 1578.3 A Lagrangian Embedding of LA3 . . . . . . . . . . . . . . . . . . . . . . . . . . 1588.4 Another Lagrangian Embedding of LA3 . . . . . . . . . . . . . . . . . . . . . . . 1598.5 A Cross-Section of the Embedding at ξy = t0 . . . . . . . . . . . . . . . . . . . . 1598.6 The planes LT (v) and LT (w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1618.7 The planes Hw1 ,Hw2 in Lv(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1618.8 A Disallowed Embedding of LA2 × R. . . . . . . . . . . . . . . . . . . . . . . . . 1628.9 An Embedding of LA3 Realizing a Non-Cyclic Structure . . . . . . . . . . . . . . 1638.10 The non-brane in (PA3 ,O) relative to two Lagrangian embeddings . . . . . . . . 1658.11 A Movie Realizing the ‘H-to-I’ Move . . . . . . . . . . . . . . . . . . . . . . . . 1678.12 Realizing the PA4 move with a movie . . . . . . . . . . . . . . . . . . . . . . . . 1678.13 Labelled Cells In the PA4 Movie . . . . . . . . . . . . . . . . . . . . . . . . . . . 1688.14 The Partition Corresponding to the PA3 Move . . . . . . . . . . . . . . . . . . . 1698.15 One Movie Realizing a PS4 Move . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.16 A Second PS4 Movie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.17 A movie realizing a new move? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1708.18 Horizontal Cross-Sections of the LA3 × R Movie . . . . . . . . . . . . . . . . . . 1718.19 Two New Moves? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1728.20 Diagramming the Type I Noncompact Move of Type A4 . . . . . . . . . . . . . 1748.21 Diagramming the Type I Noncompact Move of type S4 . . . . . . . . . . . . . . 174

B.1 A Commutative Diagram Giving Γ(Z; i∗Q)→ Γ(P ;Q) . . . . . . . . . . . . . . 186

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List of Tables

3.1 A Dictionary for Cyclic Structures on PA3 . . . . . . . . . . . . . . . . . . . . . 37

5.1 The Restrictions of Indecomposable Elements of Rep(A2, µ). . . . . . . . . . . . 75

6.1 Non-Branes In (PA3 ,Oµ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.2 Branes in PA3 , Four Ways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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Acknowledgments

First and foremost, this dissertation would not have been possible without the incredibleamount of aid and encouragement from my advisor, David Nadler. He has been with thisproject every step of the way- giving advice on directions to pursue, helping me understandtechnical details, and clarifying key concepts. Perhaps most importantly he has always be-lieved in the value of my work, and his faith in my ideas and abilities has kept me motivatedduring the most stressful and frustrating parts of the writing process. I really cannot over-state his strength as a mentor during my six years at Berkeley.

I also owe thanks to useful conversations with Ben Gammage, Vivek Shende, and LauraStarkston.

I owe a great deal to Zvedelina Stankova, who I have worked with as a GSI on multi-ple occasions, and who gave me the opportunity to lead sessions with the Berkeley MathCircle. My growth as a math educator while at Berkeley has been in no small part due to her.

Finally, I want to thank the people at the UC Berkeley math department who have helpedmake the department a great community in which to study math, including Judie Filomeo,Barb Waller, and Vicky Lee.

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Chapter 1

Introduction

This dissertation is part of an ongoing program of finding combinatorial models for sym-plectic topology, and combinatorial computations of associated symplectic invariants. In[15], Kontsevich conjectures roughly that the Fukaya category of a symplectic manifold hasa combinatorial description in terms of a sheaf of differential graded categories along a La-grangian skeleton. David Nadler has made substantial progress toward a resolution of thisconjecture with the introduction of arboreal singularities in [20] and [24]. In this work wepush Nadler’s program further by undertaking a detailed combinatorial analysis of arborealsingularities.

The category of microlocal sheaves associated to a Lagrangian skeleton, developed byKashiwara-Schapira [13], is intimately connected to the Fukaya category by the work ofNadler and Zaslow in [25] and [22]. In [20] and [24], Nadler introduces a collection of singularstratified spaces termed ‘arboreal singularities’, described using combinatorial data (namely,trees), and shows that these spaces form a fundamental set of combinatorial building blocksfor singular Lagrangian skeleta, in the sense that arbitrary Lagrangian singularities can benon-characteristically deformed into a skeleton with only arboreal singularities. In addition,[20] includes a computation of the category of microlocal sheaves on the skeleton.

In recent years, the combinatorial description provided by arboreal singularities has seenapplications to mirror symmetry in [10], [19], [23], and the combinatorics of Legendrianknots due to Shende, Treumann, Zaslow, and others in [31], [30], [29], [26]. There is alsowork in the direction of understanding the structure of Weinstein manifolds [9] via arborealLagrangian skeleta by Starkston in [33].

The main goals of this work are: (i) to describe a local combinatorial invariant (termeda ‘cyclic structure’) which is meant to capture the local symplectic geometry of an arborealskeleton, (ii) to describe a sheaf of dg-categories on an arboreal poset associated to such aninvariant, and (iii) to describe a class of combinatorial ‘moves’ which are meant to capturelocal modifications to an arboreal skeleta which do not affect the local symplectic geometry.

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This work can be thought of a higher-dimensional version of the theory of ribbon graphs,which label cells in the moduli space of Riemann surfaces [11] [18]. The sheaf of dg-categoriesshould be thought of as an extension of Sibilla, Treumann, and Zaslow’s construction on rib-bon graphs in [32]. The combinatorial moves generalize the classic H-to-I mutation of ribbongraphs, also known as the 2-2 Pachner move on the dual graph to a triangulation of a surface[27].

The combinatorial moves discussed here also appear in connection with cluster structuresassociated to Legendrian knots, for example in [31].

1.1 Main Results

For a tree T , we let LT denote the arboreal singularity associated to T , and PT denotethe strata poset. Any codimension-one strata in LT is in the closure of exactly three topdimensional strata, as pictured in 1.1. It follows that any codimension-one element of PThas exactly three maximal elements greater than it.

Figure 1.1: The Geometry of the Neighborhood of a Codimension-One Cell

More generally, we let a C-Arb space denote a poset P such that, for any x ∈ P , theposet N(x) = {y ∈ P | y ≥ x} is isomorphic to a C-Arb singularity. (We remark that C-Arbsingularities are themselves C-Arb spaces). Then C-Arb spaces also have the property thatevery codimension-one element has exactly three elements greater than it.

3

Definition 1.1.1. (Definition 3.1.1) A pre-cyclic structure O on a C-Arb space P isa choice, for every codimension-one element x, a cyclic ordering Ox on the three maximalelements greater than x.

Among pre-cyclic structures, there is a collection of preferred structures which we termcyclic structures. These can be specified with a construction involving directed trees. Adirected tree is a pair (T, µ), where µ is an orientation of the edges of T . To an orientationµ of T , one can define (Definition 3.2.3) a pre-cyclic structure Oµ on PT . Then:

Definition 1.1.2. (Definition 3.2.4) A pre-cyclic structure O on a C-Arb space P is acyclic structure if, for every x ∈ P, there exists a directed tree (T, µ) and an isomorphismϕ : N(x)

∼→ PT such that ϕ∗Oµ = i∗O, where i : N(x) ↪→ P is the inclusion.

A cyclic C-Arb space is a pair (P ,O), where O is a cyclic structure on P .

The first main result we prove shows that cyclic structures are distinguished among pre-cyclic structures by local coherence conditions. Namely:

Definition 1.1.3. (Definition 3.3.1) Fix a tree R. A pre-cyclic structure O on a C-Arbspace P is R−coherent if for every embedding i : PR ↪→ P, i∗O is a cyclic structure.

Theorem 1.1.1. The Coherence Theorem (Theorem 3.3.1): A pre-cyclic structure is acyclic structure iff it is A3−coherent and S4−coherent.

A3 is the tree with three vertices connected in a line, and S4 is the tree with three verticesconnected to a central vertex (Figure 1.2). One way to interpret the Coherence Theorem isthat cyclic structure can be thought of as codimension-one data which satisfies codimension-two and codimension-three compatibility conditions.

Figure 1.2: The A3 (left) and S4 (right) trees.

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Our next result concerns a sheaf of dg-categories Qµ on the poset PT associated toa directed tree (T, µ). This is a sheaf meant to model a category of microlocal sheaveson the arboreal singularity LT , as computed in [20]. In particular, the global sectionsΓ(Qµ;PT ) ∼= Rep(T, µ)- the derived dg category of finite-dimensional representations ofthe directed tree (quiver) (T, µ).

We show that a cyclic structure is enough to determine the sheaf up to quasi-equivalence.Precisely:

Theorem 1.1.2. The Cyclic Structure Theorem (Theorem 5.6.1): Let (T, µ) and

(T ′, µ′) be directed trees, and ϕ : PT∼↪→ PT ′ be a poset isomorphism. Then ϕ∗Qµ′ is quasi-

equivalent to Qµ if and only if ϕ∗Oµ′ = Oµ.

Following this, we take the first steps toward constructing a canonical A∞−categoryassociated to a cyclic arboreal space (P ,O), which does not depend in isomorphisms withdirected trees. As in [32], this category is meant to model the Fukaya category on a symplec-tic neighborhood of the space. In this document, we construct a set Brgr(P , G) of gradedbranes associated to an arboreal space P equipped with a graded structure G. Locally,graded structures up to isomorphism are equivalent to cyclic structures. We also specify asubcollection of objects Rep(T, µ) termed rank-one objects, and prove the following:

Theorem 1.1.3. (Propositions 6.3.2,6.3.3) Given a directed tree (T, µ) there is a canonicalbijection: {

Rank one objectsin Rep(T, µ)

}/homotopy←→ Brgr(PT , Gµ)

Where Gµ is the graded structure on PT associated to µ.

Rep(T, µ) is the triangulated hull of the full subcategory of rank-one objects. It is there-fore our hope that one could define a sheaf of A∞−categories whose objects are gradedbranes, such that locally one gets a category whose triangulated hull is quasi-equivalent toa category of representations of directed trees. An advantage of this perspective is that thiscategory would manifestly respect all symmetries, for example those observed by Nadler in[21].

Our last main result concerns a generalization of the ‘H-to-I move’ (Figure 1.3) to higher-dimensional arboreal spaces. Namely, we construct a certain classification of combinatoriallocal ‘mutations’ of cyclic C-Arb spaces such that the total space of the mutation looks likean arboreal singularity PT . The ‘H-to-I’ move is such a move where T = A3.

5

Figure 1.3: The H-to-I move

More concretely, for a tree T , let KT = E(T )t Vt(T ), where E(T ) and Vt(T ) denote theset of edges of T and the set ofterminal vertices of T , respectively. We show that moves areclassified by a directed tree (T, µ) along with a partition KT = K1 tK2 of KT into subsetswhich are good with respect to µ. We give an explicit combinatorial classification of goodsubsets:

Theorem 1.1.4. (Theorem 7.4.1) A subset K of KT is good if and only if it is not ageneralized cyclic interval (definition 7.4.1).

As a consequence, we are able to describe a move with total space PA4 , as well as fourinequivalent moves with total space PS4 .

1.2 Summary of the Paper

We provide a brief summary of each chapter, outlining the main ideas and methods.

Summary of Chapter 2: Combinatorial Arboreal Singularities andArboreal Spaces

In this chapter we introduce an abstract construction of posets arising from certain simply-behaved categories. We then apply this construction to the category of trees and treecorrespondences in order to obtain our main objects of interest: Combinatorial ArborealSingularities.

Briefly, a tree is a nonempty finite acyclic graph. A correspondence of trees (definition2.2.2) is a diagram:

p = (Rq� S

i↪→ T )

6

Where i is the inclusion of a subtree, and q is a quotient map (a surjective map with connectedfibers). If one has another correspondence

q = (R′q′

� S ′i↪→ R)

One can form the composition:

p′ ◦ p = (R′q′◦q�− q−1(S ′)

j↪→ T )

The elements of a C-Arb singularity PT are equivalence classes of correspondences p = (Rq�

Si↪→ T ), denoted [p]. They are equipped with a partial order in which [p] ≥ [p′] iff there

exists q such that p = p′ ◦ q. There is a unique minimal element 0 represented by:

p0 = (T∼← T

∼→ T )

Maximal elements are in one-to-one correspondence with subtrees S of T via:

S ⇔ (?� S ↪→ T )

Where ? is the tree with one vertex (corollary 2.3.1).

This construction ensures that neighborhoods of points in C-Arb singularities are againC-Arb singularities. Explicitly, if we let:

p = (Rq� S

i↪→ T )

Then N([p]) = {[q] ∈ PT | [q] ≥ [p]} ∼= PR.

An important property of PT which we will exploit repeatedly is that it is atomistic(lemma 2.3.2), meaning that each element can be written as the least upper bound of a set of

atoms−minimal elements of PT\{0}. Such elements are of the form p = (Rq� S

i↪→ T ) with

|V (R)| = |V (T )|−1. They come in two forms: deleting a terminal vertex of T or contractingan edge of T . For this reason we will often make use of the notation KT = Vt(T ) t E(T ),where Vt(T ) denotes the set of terminal vertices of T and E(T ) the set of edges.

As a consequence of PT being atomistic, we have an embedding PT ↪→ P(KT ) (thepower set of KT ).

Example 1.2.1. Let An denote the tree with n vertices connected in a straight line (Figure1.4). We label the vertices 1 to n from left to right.

Then |KAn| = n+ 1, and one can show (lemma 2.4.1) PAn ∼= {X ⊆ KAn | |K| ≤ n− 1}.In other words, PAn is isomorphic to the cone over the n− 2−skeleton of the combinatorialn−simplex. We illustrate PA2 and PA3 in Figure 1.5.

7

1 2 3 n

Figure 1.4: The An Tree

Figure 1.5: Illustrations of PA2 (left) and PA3 (right)

In Chapter 1 we give a similar description of the image of PSn in P(KSn), where Sn isthe tree with a central vertex connected to n− 1 exterior vertices.

Summary of Chapter 3: Cyclic Structures

In this chapter we give the definition of a pre-cyclic structure and cyclic structure on a C-Arb space P , as well as the construction of the cyclic structure Oµ on PT associated to anorientation µ of a tree T .

We provide an in-depth discussion of cyclic structures on PA3 and PS4 , since according tothe Coherence Theorem (theorem 1.1.1), these cyclic structures encode all the compatibilityrequirements distinguishing cyclic structures among pre-cyclic structures. As a part of thisdiscussion, we construct a family pre-cyclic structures on PS4 which are A3−coherent, butnot cyclic structures, showing that the S4−coherence is a necessary condition.

We end the chapter by generalizing these observations to describe cyclic structures onPAn and PSn . For instance, we show the following result:

Proposition 1.2.1. (Proposition 4.4) Cyclic structures on PAn are in natural bijection withcyclic orders on KAn.

8

Summary of Chapter 4: The Combinatorics of Cyclic Structures

In this section we undergo a detailed analysis of isomorphisms of C-Arb singularities, and ofcyclic C-Arb singularities.

Note that we have an embedding PT ↪→ P(KT ). So for any ϕ ∈ Aut(PT ), we getϕ ∈ S(KT )- the symmetric group on KT . Recall KT = E(T ) t Vt(T ).

The key constructions of this chapter are the reflection isomorphisms: Let v ∈ V (T )be a vertex of degree 2. If e, e′ are the two edges in T incident to v, we construct a posetisomorphism:

ωv ∈ Aut(PT )

Such that ωv is the transposition (ee′). Similarly, if v ∈ Vt(T ), and e is the unique edgeincident to v, we construct ωv ∈ Aut(PT ) such that ωv is the transposition (ve).

Our first main result in this section is that these reflection isomorphisms generate allnontrivial automorphisms of PT . In more detail: For any tree isomorphism σ : T

∼→ T ′, oneobtains iσ : PT

∼→ PT ′ . Then:

Proposition 1.2.2. (Corollary 4.2.2) If ϕ : PT∼→ PT ′ is a poset isomorphism, there exists

a tree isomorphism σ : T∼→ T ′ such that i−1

σ ◦ ϕ ∈ Aut(PT ) is a product of reflectionisomorphisms.

We then extend these results to cyclic C-Arb singularities. If (T, µ) is a directed tree andv ∈ V (T ) is a source or sink, we define rvµ to be the orientation µ with the edges incidentto v reversed. We prove:

Lemma 1.2.1. (Lemma 4.4.2) If v is a source or sink of degree ≤ 2, then ωv : (PT ,Oµ)∼→

(PT ,Orvµ) is an isomorphism of cyclic C-Arb singularities. In particular:

ω∗vOrvµ = Oµ

We call such isomorphisms directed reflection isomorphisms. We also prove a similarclassification result to proposition 1.2.2. As in that discussion, an isomorphism of σ directedtrees induces an isomorphism iσ of cyclic C-Arb singularities. Then:

Theorem 1.2.1. (Theorem 4.4.1) If (T, µ) and (T ′, µ′) are directed trees, and ϕ : (PT ,Oµ)∼→

(PT ′ ,Oµ′) is an isomorphism of cyclic C-Arb singularities, then there is an orientation λ onT and an isomorphism of directed trees σ : (T, λ)

∼→ (T ′, µ′) such that i−1σ ◦ ϕ : (PT ,Oµ) →

(PT ,Oλ) is a product of directed reflection isomorphisms.

9

As a first application of our understanding of isomorphisms of C-Arb singularities, weprove the Coherence Theorem (theorem 1.1.1).

Summary of Chapter 5: A Sheaf of Categories

Our main justification for our definition of cyclic structures is that a cyclic structure deter-mines a sheaf of dg-categories on a C-Arb singularity up to quasi-equivalence. That is thecontent of the Cyclic Strucure Theorem (theorem 1.1.2). In this chapter we construct thesheaf of dg-categories Qµ associated to an orientation µ and prove this theorem.

To a directed tree (T, µ) one can associate the ‘dg derived category’ Rep(T, µ). In thiswork, we consider Rep(T, µ) to be the (formal) pretriangulated hull of the dg categoryRep0(T, µ), which is a k−linear category with one object Pv for each vertex v.

If p = (Rq� S

i↪→ T ), the stalk Qµ,p of the sheaf Qµ at the point [p] ∈ PT is Rep(R, λ),

where λ = p∗µ is the pullback of the orientation to R. The restriction map cp : Rep(T, µ)→Rep(R, λ) is characterized by its behavior on Rep0(T, µ):

c0p : Rep0(T, µ)→ Rep0(R, λ)

c0p(Pv) =

{Pq(v) v ∈ V (S)0 Else

(We must also characterize the behavior of c0p on morphisms- see definition 5.3.2 for more

details). This data defines a sheaf of dg-categories on the poset.

The proof of the Cyclic Structure Theorem (theorem 1.1.2), completed in this chapter,consists of two parts. The first assertion follows from a detailed analysis of the sheaf Qµ onPA2 . This sheaf can be viewed diagrammatically as in Figure 1.6. In that figure, V and Ware finite-dimensional chain complexes and f is an injective morphism.

One sees that there is a cyclic rotation symmetry to this sheaf: Applying the quasi-

equivalence (Vf→ W ) 7→ (W

i→ Cone(f)) to the stalk at the vertex, one gets an equivalentpresentation as shown in Figure 1.7:

And one notices that the restriction functors have been rotated clockwise. We prove:

Proposition 1.2.3. (Proposition 5.4.1) Let (A2, µ) be a directed tree. If ϕ ∈ Aut(PA2) andϕ∗Qµ is quasi-equivalent to Qµ, then ϕ∗Oµ = Oµ.

It follows easily that if (T, µ) and (T ′, µ′) are directed trees, and ϕ : PT∼→ PT ′ be a poset

isomorphism such that ϕ∗Qµ′ is quasi-equivalent to Qµ. Then we must have ϕ∗Oµ′ = Oµ.

10

(Vf→ W ) V

W

W/f(V )

Figure 1.6: A diagram of the sheaf Qµ on PA2

(Wi→ Cone(f)) W

Cone(f)q.i.−→ W/f(V )

Cone(f)/i(W ) ∼= V [1]

Figure 1.7: The sheaf Qµ after applying a quasi-equivalence of global sections

This is half of the Cyclic Structure Theorem. The bulk of the remainder of the chapter isdevoted to showing the following:

Proposition 1.2.4. (Proposition 5.5.1) Let (T, µ) be a directed tree, v ∈ V (T ) be a sourceor sink of degree ≤ 2, and ωv : (PT ,Oµ)

∼→ (PT ,Orvµ) be the directed reflection isomorphism.Then there exists a quasi-equivalence:

Qµ∼→ ω∗vQrvµ

Combining proposition 1.2.3 with proposition 1.2.4 yields the Cyclic Structure Theorem.

Proving proposition 1.2.4 requires the construction of a morphism:

Γ+v : Qµ → ω∗vQrvµ

When v is a source of degree ≤ 2, and likewise:

Γ−v : Qµ → ω∗vQrvµ

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When v is a sink. These morphisms are essentially the BGP reflection functors of [2], a clas-sical construction which gives quasi-equivalences of the categories Rep(T, µ) and Rep(T, rvµ)whenever v is a source or sink. Our proof extends these functors to give quasi-equivalencesof sheaves of dg-catgories, meaning they must be compatible with the restriction functors.

Additionally, we must show these functors are inverses, which formally involves somehigher coherence data. These constructions involve some tedious verifications of commuta-tive diagrams, which we meticulously present in Chapter 5.

Summary of Chapter 6: Graded Structures and Graded Branes

While we have shown that one can construct a sheaf of dg-categories up to quasi-equivalenceon a cyclic C-Arb singularity, we have not given a canonical construction- i.e. one thatdoesn’t require the choice of an isomorphism with (PT ,Oµ). Such a construction is impor-tant if one wants to consider sheaves on general C-Arb spaces- since it gives an explicit recipefor gluing together local models.

We remark that such a canonical construction exists for An singularities, appearing in [21].

This chapter does not solve the general problem, but presents some ideas in this direc-tion. We construct a set of objects, called graded branes, which are meant to serve asthe objects in this theoretical canonical category. The use of the term brane is meant to besuggestive: We expect the correct answer to this problem is an A∞−category which modelsthe Fukaya category of a symplectic neighborhood of an arboreal singularity.

To begin, we introduce the notion of a graded structure G on a C-Arb space P (defini-tion 6.1.1). To a graded structure, we associate a pre-cyclic structure OG (definition 6.1.2).We are able to show that OG is in fact a cyclic structure (proposition 6.1.1) by using theCoherence Theorem (theorem 1.1.1). We also prove that, locally, each cyclic structure is in-duced by a unique (up to isomorphism) graded structure. In particular, for any orientationµ of T there is a canonical graded structure Gµ such that OGµ = Oµ.

We then define, for a C-Arb space, a set of pre-branes Brpre(P), which are essentiallysubsets representing top-dimensional cycles over Z/2Z (definition 6.2.1). As an example,pre-branes on A2 are illustrated in Figure 1.8.

The set of pre-branes forms a vector space over Z/2Z. When P = PT we have a ‘standardbasis’ of the form {Lv}v∈V (T ) (though note- this basis is not invariant under automorphismsof PT ).

12

(1)

(2)

(12)

Figure 1.8: Pre-Branes on PA2

For a cyclic C-Arb space (P ,O), one has a subset of branes Br(P ,O) ⊂ Brpre(P) givenby a coherence condition in codimension-two (definition 6.2.3). We give a nice classificationresult:

Proposition 1.2.5. (Proposition 6.2.1) Given a directed tree (T, µ), we can construct abijection:

Br(PT ,Oµ) \ {0} ←→{

Subtrees of T isomorphicto An for some n.

}Where 0 is the ‘zero brane’ corresponding to the empty set.

Furthermore, to a graded structureG one can construct the set of graded branes Brgr(P , G).There is a ‘forgetful map’

Brgr(P , G)→ Br(P ,OG)

Whose nonempty fibers have a natural free Z−action. When P = PT , the forgetful map issurjective, and Z acts transitively on its fibers.

Our main result in this chapter is a representation-theoretic interpretation of branes.Namely, we say X ∈ Rep(T, µ) is rank-one if, for any maximal correspondence p = (? �

Si↪→ T ), cpX ∈ C(k) has one-dimensional cohomology. (C(k) ∼= Rep(?, ·) is the dg category

of finite-dimensional chain complexes over k). Then one has a bijection (propositions 6.3.2,6.3.3): {

Rank one objectsin Rep(T, µ)

}/homotopy←→ Brgr(PT , Gµ)

The rest of the chapter gives a more in-depth discussion of branes and graded structuresin the case T = An. In particular, we provide a description of branes and graded branes that

13

doesn’t explicitly reference the tree (like proposition 1.2.5 does).

Summary of Chapter 7: Arboreal Moves

In the theory of trivalent ribbon graphs there is a ‘move’ or ‘mutation’ that preserves theassociated Riemann surface (Figure 7.1).


This move can be viewed as a transformation of cyclic C-Arb spaces. Viewing this trans-formation as a continuous deformation over time, the total space of this move is the A3−singularity (Figure 7.2).

Figure 1.10: Total Space of the H-to-I move

In this section, our goal will be to generalize this observation to describe a wide class ofmoves of C-Arb spaces whose total space is a C-Arb singularity.

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The process goes as follows: If T is a tree, for a subset K of KT , we define:

UK :=⋃k∈K

N(〈k〉)

And we say that K is good with respect to an orientation µ on T if the restrictionΓ(Qµ;PT ) → Γ(Qµ;UK) is a quasi-equivalence of dg-categories. Then a partition: KT =K1 t K2 into good sets determines an arboreal move of type T between cyclic C-Arbspaces:

(UK1 , i∗1Oµ)→ (UK2 , i

∗2Oµ)

The bulk of this chapter is focused on a classification of good subsets. Our main tool willbe the use of combinatorial homotopies (specifically, retracts) to conclude that certain mapsare quasi-equivalences without explicitly computing a limit of dg-categories. We introducethe following notion:

Definition 1.2.1. (Definition 7.4.1) Let (T, µ) be a directed tree. A generalized cyclicinterval is a subset of KT of the form I(v, w,D), where:

• v, w ∈ V (T )

• D is a subset of the terminal vertices Vt(T ) of T .

• I(v, w,D) ⊂ KT consists of all elements of D, all edges between a point in D and theshortest path joining v and w, and all edges on the shortest path joining v and w whichare pointing from v to w.

v′

v w

Figure 1.11: A Generalized Cyclic Interval I(v, w,D).

An example is given in Figure 1.11. Elements of I(v, w,D) are thick blue edges andblue terminal vertices, while elements not in I(v, w,D) are thin gray lines and gray terminalvertices. D is the set of blue terminal vertices. The result is:

(Theorem 1.1.4) A subset K of KT is good if and only if it is not a generalizedcyclic interval.

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We conclude this chapter with an explicit discussion on moves of type An and Sn. Ex-plicitly, we have the following nice characterization. Recall (proposition 4.4) that giving acyclic structure O on PAn is equivalent to giving a cyclic order on KAn . Then:

Proposition 1.2.6. (Proposition 7.5.1) K ⊂ KAn is a generalized cyclic interval with respectto an orientation µ iff it is a cyclic interval in the cyclic order determined by Oµ.

As a result, moves of type An are equivalent to giving a partition of a cyclically orderedset of n + 1 elements into two subsets, neither of which are cyclic intervals. For example,when n = 3, the unique valid partition of 4 cyclically ordered elements (two opposite pairs)gives the H-to-I move.

Figure 1.12: The unique (up to equivalence) partitions of cyclic sets yielding moves of typeA3 and A4

Summary of Chapter 8: The Geometry of Arboreal Spaces

In this chapter, we bravely leave the combinatorial realm and attempt some honest symplec-tic geometry.

Of central importance is the notion of a stratified space. Our definition resembles thatin [17].

Definition 1.2.2. (Definition 8.1.1) A stratified space is a (Hausdorff, locally compact,second countable) topological space X which is can be written as a disjoint union: X =tα∈AXα, where the Xα are called strata. In addition we have:

(i) Each Xα is equipped with the structure of a smooth manifold of some dimension, whichis compatible with the subspace topology on Xα.

(ii) For α 6= β ∈ A, if Xβ intersects Xα, then Xβ ⊂ Xα and dim(Xβ) < dim(Xα).

We then describe a construction given in [20] of the arboreal singularity LT associatedto a tree T . Each stratified space X has an associated poset S (X) whose elements are the

16

strata of X. We have S (LT ) ∼= PT - in other words, the strata of LT are written LT (p), for

correspondences p = (Rq� S

i

↪→ T ).

Following this, we define an arboreal space as a stratified space which is locally isomor-phic to a stratified space of the form LT × Rk. As shown in (ref), LT itself satisfies the

definition: For any correspondence p = (Rq� S

i↪→ T ) there is an open neighborhood of

LT (p) isomorphic to LR × R|T |−|R|.

We then discuss Lagrangian embeddings of arboreal spaces: Topological embeddings i :L ↪→ (M,ω), where:

• (M,ω) is a symplectic manifold.

• i a smooth on strata and maps each strata to an isotropic submanifold.

• Certain regularity conditions on the relative positions of the strata (see definition 8.3.1).

We define the notion of a pre-cyclic structure on an arboreal space mimicking the defini-tion for a C-Arb space, and show that a Lagrangian embedding induces a canonical pre-cyclicstructure. In particular, for a tree T there is a natural bijection {Pre-cyclic structures on LT} ↔{Pre-cyclic structures on PT}.

What we term our ‘main conjecture’ for Lagrangian embeddings is a bijection of theform: {

‘Regular’ Lagrangianembeddings of LT

}/equivalence←→ {Cyclic Structures on PT}

While we give a plausible notion of ‘equivalence’, we are not able to give a satisfactory notionof ‘regular’- though, we do explain why such a notion is necessary. Furthermore, we have amap:

χT : Sh(PT ; dgCat)→ Sh(LT ; dgCat)

Yielding what we expect to be a commutative (up to quasi-equivalence) diagram:

i : LT ↪→ (M,ω) (PT ,O)

Shi(LT )(M) QχT

Where i is a ‘regular’ Lagrangian embedding, Shi(LT )(M) is the category of sheaves onM with

microsupport along i(LT ), and Q = ϕ∗Qµ for some isomorphism ϕ : (PT ,O)∼→ (PT ,Oµ).

17

Lastly, we give a definition of ‘arboreal moves’ of cyclic arboreal spaces (arboreal spacesequipped with a cyclic structure). Here, the conjectures are even less precisely formulated-we are content to give some examples of geometric realizations of some of the combinatorialmoves described in Chapter 7, as well as some geometric moves which are not as neatlycaptured by our formalism from that chapter.

18

Chapter 2

Combinatorial Arboreal Singularitiesand Arboreal Spaces

2.1 Posets

References for posets and the poset topology are [4], [35].

Definition 2.1.1. A poset P is a set equipped with a partial order ≤. Throughout this paperwe will view a poset as a T0 topological space, where U ⊂ P is open if it is upward closed:for every x ≤ y ∈ P, if x ∈ U then y ∈ U . A open embedding is a continuous mapP1 ↪→ P2, which is injective and has open image. For x ∈ P, we have the neighborhoodof x: N(x) := {y ∈ P | x ≤ y}, which is the intersection of all open sets containing x.

We let Pos denote the category of posets whose morphisms are open embeddings.

Definition 2.1.2. Let P be a poset. P is graded if there exists a dimension functiondim : P → Z≥0 such that:

(G1) Any minimal element x has dim(x) = 0.

(G2) If x < y, dim(x) < dim(y).

(G3) If x < y and there does not exist z with x < z < y, then dim(y) = dim(x) + 1

P is co-graded if there exists a codimension function codim : P → Z≥0 which is a dimensionfunction for Pop: the poset P with the opposite order relation.

Note that, for a graded (resp. co-graded) poset, the dimension function (resp. codimen-sion function) is uniquely determined. If P is graded, we let Cj(P) = {x ∈ P | dim(x) = j}.

19

We sometimes refer to these elements as j−cells. Similarly, if P is co-graded we let Cj(P) ={x ∈ P | codim(x) = j}.

Definition 2.1.3. Let P be a poset with a least element 0 (that is, P = N(0)). Then anatom of P is a minimal element of P\{0}. In the case P is graded, atoms are the same asone-cells. P is atomistic if each element is the least upper bound of a set of atoms.

Note that if P is atomistic, then there is an injection P ↪→ P(C1(P))- where P(X)denotes the power set of X- which sends x ∈ P to {y ∈ C1(P) | y ≤ x}.

We continue by reviewing a construction from [20], from an abstract perspective.

First, let C be any category1

Definition 2.1.4. For an object A ∈ Ob(C), let C/A denote the slice category: Objects of C/Aare arrows f : B → A, and a morphism f → g is a map h satisfying f = gh.

Definition 2.1.5. A category is thin if there is at most one morphism between any twoobjects. For any thin category D, we can construct a poset Pos(D) whose objects are iso-morphism classes of objects in D. If X, Y ∈ Ob(D), and [X] and [Y ] are the correspondingequivalence classes, we set [X] ≤ [Y ] if and only if there exists a map from Y to X in D.

Now, let C be a category in which every morphism is monic. For A ∈ Ob(C), C/A is athin category. Write PA = Pos(C/A). The following facts are readily verified: (We use [f ] todenote the equivalence class of a morphism f ∈ CA)

Lemma 2.1.1.

(i) PA has a unique minimal element, represented by the identity morphism idA : A→ Ain CA.

(ii) If f : A → B is a morphism, the map if : PA → PB defined by if ([g]) = [f ◦ g] iswell-defined, and is an open embedding of posets. Furthermore, the image of PA underif is the open neighborhood N([f ]) ⊂ PB.

1For this exposition our categories will be assumed to be essentially small- that is, the collection ofisomorphism classes of objects forms a set, and morphisms between any two objects form a set.

20

�

Part (ii) of the lemma implies that the assignment A 7→ PA, f 7→ if defines a functorC → Pos.

Definition 2.1.6. A combinatorially graded category2 C is a category in which everymorphism is monic, equipped with a functor | · | : C → N, where the category N has objectsthe positive integers, and a single morphism from m to n if and only if m ≤ n. We requirethat | · | satisfies the following properties:

(i) If X, Y ∈ Ob(C) with |X| = |Y |, any morphism X → Y is an isomorphism.

(ii) If X, Y ∈ Ob(C), if there exists a morphism f : X → Y and |X| < |Y |, there existsZ ∈ Ob(C) with |Z| = |Y | − 1, and morphisms g : X → Z, h : Z → Y , with f = hg.

(iii) For any X ∈ Ob(C), there exists Y ∈ Ob(C) with |Y | = 1, and a morphism f : Y → X.

Lemma 2.1.2. If C is a combinatorially graded category, then for any A ∈ Ob(C), PA isgraded and co-graded. In particular, for f : B → A, dim([f ]) = |A| − |B| and codim([f ]) =|B| − 1 are dimension/codimension functions.

Proof. First we show dim is a dimension function.

For G1: we know PA has a unique minimal element represented by idA, and dim([idA]) =|A| − |A| = 0, as desired.

For G2: If [f ] < [g], with f : X → A and g : Y → A, there exists h : Y → Xsuch that g = f ◦ h. Since | · | is a functor, that implies |Y | ≤ |X|. If |Y | = |X|, thenh is an isomorphism by (i), which would make [f ] = [g], hence |Y | < |X|. So we havedim([f ]) = |A| − |X| < |A| − |Y | = dim([g]).

For G3: Suppose [f ] < [g], with f : X → A and g : Y → A, and there does not existz with [f ] < [z] < [g]. Then there exists h : Y → X such that g = f ◦ h, and |Y | < |X|.By (ii) there exists Z with |Z| = |X| − 1, and morphisms j : Y → Z, k : Z → X, withh = k ◦ j. So we have [f ] ≤ [f ◦ k] ≤ [f ◦ k ◦ j] = [g]. By assumption we must have[g] = [f ◦ k] or [f ] = [f ◦ k]. The latter is impossible, as |Z| = |X| − 1, so [g] = [f ◦ k], hence|Y | = |Z| = |X| − 1, and dim([g]) = dim([f ]) + 1.

2Terminology is of my own creation, though most likely this notion exists elsewhere in the literature

21

To show that codim is a codimension function, the proofs of G2 and G3 are identical.To prove G1, note that by (iii) any minimal element in PopA must be of the form f : B → Awith |B| = 1. Hence codim([f ]) = |B| − 1 = 0 as desired.

In this paper, we will use the above formalism for the categories of trees and directedtrees to construct our central objects of interest: Combinatorial arboreal singularities.

2.2 Trees

Material in this section can also be found in [20].

To fix notation, for a set X, we let P(X) denote the power set of X. We review somebasic definitions of graph theory (with conventions adopted to our purposes- we only con-sider finite nonempty graphs with no loops or multi-edges):

Definition 2.2.1. A graph G is the data of a finite nonempty set V (G) (whose elements arecalled vertices), along with a set of edges E(G) ⊆P(V (G)) consisting of subsets of V (G)of size 2. A path in G is a sequence (v0, v1, . . . , vn), where vi ∈ V (G), and {vi, vi+1} ∈ E(G)for all 0 ≤ i < n. A graph is connected if every pair of vertices can be joined by a path. Atree is a connected graph with no cycles, where a cycle is a path (v0, v1, . . . , vn = v0), wherev0, . . . , vn−1 are all distinct vertices, and n > 2. If G1 and G2 are graphs, a map of graphsf : G1 → G2 is the data of a function (which we also call f) f : V (G1) → V (G2) such thatif {v, w} ∈ E(G1), either f(v) = f(w) or {f(v), f(w)} ∈ E(G2).

If G and H are graphs, H is called a subgraph of G if V (H) ⊂ V (G) and E(H) ⊂ E(G).A subtree is a subgraph which is a tree. Note that if S is a subtree of T , S is determined byV (S) ⊆ V (T )- because of this we will occasionally refer to a subset of V (T ) as a subtree of T .

If T1, T2 are trees a map q : T1 → T2 is a quotient map if it is surjective and, forv ∈ V (T2), q−1({v}) is a connected subtree of T1.

Definition 2.2.2. The category Tree is the category whose objects are trees and whosemorphisms are correspondences of trees. A correspondence of trees is a diagram:

p = (Rq� S

i↪→ T )

Where S is a subtree of T , and q is a quotient map. If we have a pair of correspondences:

p = (Rq� S

i↪→ T ) p′ = (R′

q′

� S ′i′

↪→ R)

22

The composition is given by:

p′ ◦ p = (R′q′◦q�− q−1(S ′)

j↪→ T )

A correspondence is pictured in Figure 2.1. The subtree S consists of the fillled-in ver-tices and solid lines, while the edges contracted by q are the thick edges- we will use thisconvention to illustrate correspondences throughout this work.

Figure 2.1: A Correspondence

It is easy to verify that the composition of correspondences is associative. In light of theabove definition, we will sometimes use the notation p : R→ T to denote a correspondenceof the form (R � S ↪→ T ).

Lemma 2.2.1. Each morphism in Tree is monic.

Proof. Suppose we have morphisms:

p = (Rq� S

i↪→ T ) q1 = (X

w1

� Y1

j1↪→ R) q2 = (X

w2

� Y2

j2↪→ R)

With p◦q1 = p◦q2. Then q−1(Y1) = q−1(Y2). Since q is surjective, this implies Y1 = Y2. Callthis subtree Y . We have w1 ◦ q = w2 ◦ q as maps q−1(Y )→ Y → X. Since q is surjective wehave w1 = w2.

Definition 2.2.3. We can now apply the construction from the beginning of the section toa tree T to obtain a poset PT . A poset which is isomorphic to PT for some tree T is acombinatorial arboreal singularity, or C-Arb singularity.

Definition 2.2.4. A combinatorial arboreal space, or C-Arb space, is a cogradedposet P such that, for all x ∈ P, N(x) is a C-Arb singularity.

23

2.3 Basic Properties

Let us examine some properties of PT . We use the notation from definition 2.1.5- Wheneverp : R→ T is a correspondence, [p] will denote the element of PT .

First we introduce some terminology that will be crucial in what follows:

Definition 2.3.1. For a tree T , let |T | = #(V (T )). If v ∈ V (T ), the degree of v is thenumber of edges incident to v (equivalently, the number of vertices w with {v, w} ∈ E(T )).A vertex is called terminal if its degree is one. We let Vt(T ) denote the set of terminalvertices. Note that T\{v} is a subtree of T if and only if v is a terminal vertex of T .

Lemma 2.3.1. The category Tree, with | · | as defined above, is a combinatorially gradedcategory. In particular:

(i) If p : R→ T , then |R| ≤ |T | with equality if and only if p is an isomorphism.

(ii) Let p : R → T be any correspondence with |R| < |T |. Then there exists a correspon-dence q : R′ → T with |R′| = |T | − 1, and [p] ≥ [q].

(iii) Let ? denote the tree with a single vertex. Then for any tree T there exists a corre-spondence p : ?→ T .

Proof.

(i) If p = (Rq� S

i↪→ T ) is a correspondence, then it is clear |R| ≤ |S| ≤ |T |. If |R| = |T |,

then S = T and q must be an isomorphism, meaning p is an isomorphism with inverse

(Tq−1

� R=↪→ R).

(ii) Let p = {Rq� S

i↪→ T} be any correspondence with |R| < |T |. If |S| < |T |, then there

is a terminal vertex v ∈ Vt(T ) which is not in V (S), and we can take q = {T\{v} ∼←T\{v} ↪→ T}. If |S| = |T |, we have |R| < |S|, and so there are two neighboring verticesv, w ∈ V (S) such that q(v) = q(w). Then we can set q = {T/(v ∼ w) � T

=→ T}.

(iii) Given any vertex v ∈ V (T ) we can set p = {? ∼← {v} ↪→ T}.

Corollary 2.3.1. For any tree T , PT is a graded and co-graded poset. For p : R → T ,dim([p]) = |T | − |R|, and codim([p]) = |R| − 1. In particular:

24

(i) [p] is maximal iff |R| = 1: Such correspondences are in bijection with subtrees S of T ,via S ⇔ (?� S ↪→ T ), where ? is the tree with a single vertex.

(ii) [p] is a one-cell iff |R| = |T | − 1. Such correspondences arise from contracting an edgeor deleting a terminal vertex, so we have a bijection C1(PT ) ∼= E(T ) t Vt(T ).

Corollary 2.3.2. If PT ∼= PT ′, then T and T ′ have the same number of vertices.

In fact, if PT ∼= PT ′ , we must have T ∼= T ′. This is not trivial to prove, however, and isestablished in Chapter 4.

We have the following important lemma:

Lemma 2.3.2. PT is atomistic.

Proof. We want to show that a correspondence p = (Rq� S

i↪→ T ) is the least upper bound

of the set {x ∈ C1(PT ) | x ≤ [p]}. For the rest of this proof, denote this set by X.

Use the identification C1(PT ) ∼= E(T ) t Vt(T ). If e ∈ E(T ), we see that e ∈ X if andonly if e is disjoint from S (i.e. none of its endpoints are in S) or is in S and contracted byq. Similarly, if v ∈ Vt(T ), v ∈ X if and only if v is not in S. As an example, consider thecorrespondince in Figure 2.2, (which uses the same convention of Figure 2.1) with the edgesand terminal vertices labelled:

v2

v3

v1

v4

e4

e3

e7

e1 e2

e9e5

e6

e8

Figure 2.2: A Correspondence with Edges/Terminal Vertices Labelled

The set X is {v1, v4} ∪ {e1, e5, e6, e8}.

We note that v ∈ V (S) if and only if v cannot be connected by a path to a terminal

vertex w ∈ X such that each edge in the path is in X. So if q = (R′q′

� S ′i↪→ T ) is another

correspondence such that x ≤ [q] for each x ∈ X, we must have S ′ ⊂ S.

25

Secondly, suppose v ∈ V (S ′), w ∈ V (S), and v and w are connected by an edge e ∈ X,so e is contracted by q. Then e ≤ [q], meaning e is either disjoint from S ′ or contracted byq′, and the assumption that v ∈ V (S ′) means that the latter must hold.

These two statements imply that [q] ≥ [p]. So [p] is the least upper bound of X asclaimed.

Notation

Following the lemma, we introduce the following notation, which we use throughout thispaper:

• For a tree T , let KT = E(T ) t Vt(T ). For k ∈ KT , we let 〈k〉 ∈ C1(PT ) denote thecorresponding one-cell.

• For a correspondence p, let Xp = {k ∈ KT | 〈k〉 ≤ [p]}. By Lemma 2.5, this gives aninjection from PT into the power set P(KT ). (Though this is not an open embedding).

• It follows that there is an embedding Aut(PT ) ↪→ S(KT ), where S(KT ) denotes thesymmetric group on KT . For ϕ ∈ Aut(PT ), let ϕ denote its image in S(KT ).

• More generally, if ϕ : PT∼→ PT ′ is a poset isomorphism, let ϕ : KT

∼→ KT ′ denote thecorresponding bijection of sets.

2.4 Some Examples: An and Sn singularities

Description of PAn

Let An denote the tree with n vertices connected in a straight line (Figure 2.3). We labelthe vertices 1 to n from left to right.

1 2 3 n

Figure 2.3: The An Tree

We’ll examine the Hasse diagrams for PA2 and PA3 . The notation used in this section todenote a correspondence is as follows: We list the vertices contained in S, grouping the fibersof q in parentheses. So PA2 has four elements, represented by the following correspondences:

26

(1) = (?∼← {1} ↪→ A2)

(2) = (?∼← {2} ↪→ A2)

(12) = (?� A2∼→ A2)

(1)(2) = (A2∼← A2

∼→ A2)

And has Hasse diagram illustrated in Figure 2.4.

(1) (2) (12)

(1)(2)

Figure 2.4: Hasse diagram for PA2

PA3 has eleven elements, including, for example:

(12)(3) = (A2

q� A3

∼→ A3)

(23) = (?� {2, 3} ↪→ A3)

Hasse diagram is illustrated in Figure 2.5.

(1)(2) (1)(23) (12)(3) (2)(3)

(1) (12) (2) (123) (23) (3)

(1)(2)(3)


Writing [n] = {1, 2, . . . , n}, we observe that PA2 is equivalent to the order poset of{S ⊆ [3] | |S| ≤ 1} and PA3 is equivalent to the order poset of {S ⊆ [4] | |S| ≤ 2}. Thispattern generalizes. We have KAn = E(An) t Vt(An) has n+ 1 elements. Then:

Lemma 2.4.1. The image of PAn in P(KAn) is {X ⊆ KAn | |X| ≤ n− 1}.

27

Proof. Given a subset X of KAn , the proof of Lemma 2.3.2 suggests how to construct p:We have v ∈ V (S) if and only v cannot be connected to a terminal vertex which is in Xby a path, all of whose edges are in X. Then q contracts all the edges in V (S) which are in X.

This will only fail if the subtree S determined by this procedure is empty. This occurswhen X contains both the terminal vertices and all but one edge, or when X contains atleast one one terminal vertex and all edges. These cases occur exactly when |X| = n orn+ 1.

Another way to phrase this result is that PAn\{0} is the face poset of the n−1−skeletonof a n+ 1−simplex (see [4]). This leads to the visualizations of PA2 and PA3 in Figure 2.6.

Figure 2.6: Illustrations of PA2 (left) and PA3 (right)

Another useful visualization of PA3 is as a plane with two half-planes glued along thelines x = 0 and y = 0, see Figure 2.7.

Figure 2.7: Another Visualization of PA3

28

Remark 2.4.1. Observe that Aut(P(An)) ∼= S(KAn). In general we will have more auto-morphisms of the poset PT than of the tree T - this is investigated more thoroughly in sections4-6.

Description of PSn

The Sn tree is a single vertex connected to n− 1 other vertices (Figure 2.8).

01

23

n− 1

Figure 2.8: The Sn Tree

We take n ≥ 4. The set KSn has size |K| = 2(n − 1): there are n − 1 edges, and n − 1terminal vertices. For a terminal vertex v, let e(v) denote the edge connected to v. Wedefine an equivalence relation ∼ on KSn whose equivalence classes are {v, e(v)}.

Lemma 2.4.2. The image of PSn in P(KSn) is:

{X ⊆ KSn | Xc intersects each equivalence class of ∼, or is a single equivalence class of ∼}

Proof. It is straightforward to describe all possible correspondences p = (Rq� S

i↪→ T ).

There are two cases: if S does not contain the central vertex, then it must be a single ter-minal vertex v. Then we see X = {v, e(v)}c, i.e. Xc is a single equivalence class of ∼.

If S does contain the central vertex, then S also contains some (possibly empty) subsetW ⊂ Vt(T ). Then q is formed by contracting edges of the form e(w) for w ∈ W0 ⊂ W .Then we see X = {Vt(T )\W} t {e(w) | w ∈ W0}- in other words, X is arbitrary with therestriction that it intersects each equivalence class at most once, i.e. Xc insersects eachequivalence class of ∼.

From the description, we see that Aut(P(Sn)) consists of ϕ ∈ S(K) which descend to anautomorphism of K/ ∼. Explicitly, Aut(PSn) ∼= (Z/2Z)n−1 oSn−1.

Analogously to the illustration in Figure 2.7, we can picture PSn as Rn−1 with half-spacesglued along the hyperplanes xi = 0.

29

Chapter 3

Cyclic Structures

3.1 Pre-Cyclic Structures

The notion of a cyclic structure is combinatorial data meant to capture the local geometryof a Lagrangian embedding of an arboreal singularity. This is discussed more thoroughlyin Chapter 8, here we provide a brief non-rigorous explanation of the geometric motivationbehind the definition.

The C-Arb singularity PT\{0} is the face poset of a compact regular cell complex (see [4]for definitions), as proven in [20]. Therefore PT is the strata poset of the cone over a compactregular cell complex- the resulting stratified space is the arboreal singularity LT . sectionIfx ∈ PT is a codimension-one cell, we have N(x) ∼= PA2 . In LT , this corresponds to the factthat codimension-one strata have neighborhoods homeomorphic to Rn−1 × Cone({a, b, c})(refer to the Hasse diagram- Figure 2.4). This is illustrated in Figure 3.1.

Figure 3.1: The Geometry of the Neighborhood of a Codimension-One Cell

If we have an embedding of L = Rn−1 × Cone({a, b, c}) into a 2n−dimensional symplec-

30

tic manifold (M,ω) such that each strata is smoothly embedded and the top-dimensionalstrata are Lagrangian submanifolds, then under some extra assumptions, we can obtain anembedding of L into the symplectic normal bundle of the ‘central’ strata Rn−1. This is a2-dimensional vector bundle which has a symplectic form, i.e. an orientation, and so inducesa cyclic order on the set {a, b, c}. (See appendix A for a discussion on cyclic orders on sets).

All this leads to the following definition:

Definition 3.1.1. Let P be a C-Arb space. A pre-cyclic structure O on P is a choice,for each codimension-one cell x ∈ P, a cyclic order Ox on the three top-cells containing x.

There are no compatibility conditions in this definition, hence if a C-Arb space has kcodimension-one cells, there are 2k possible pre-cyclic structures.

Definition 3.1.2. If i : P ↪→ P ′ is an open embedding of C-Arb spaces, and O is a pre-cyclicstructure on P ′, let i∗O denote the pullback of O, where if x ∈ C1(P) and a, b, c > x weset:

(i∗O)x(a, b, c)⇔ Oi(x)(i(a), i(b), i(c))

A cyclic structure is a pre-cyclic structure satisfying certain compatibility conditions,which we will explain in the next section using directed tree.

3.2 Directed Trees

Definition 3.2.1. A directed tree is a pair (T, µ), where T is a tree and µ is an ori-entation on T . Explicitly, µ is the data, for each edge e ∈ E(T ), a head h(e) and a tailt(e) in V (T ), such that e = {h(e), t(e)}. We say e “points from t(e) to h(e)”. The cate-gory DirTree has objects directed trees and morphisms correspondences of directed tree. Acorrespondence of directed trees is a correspondence

p = ((R, λ)q� (S, ν)

i↪→ (T, µ))

Where each map preserves orientations. To be more explicit for q: whenever two verticesv1, v2 ∈ V (R) are connected by an edge, there is a unique pair of vertices w1, w2 ∈ V (S)connected by an edge such that q(w1) = v1 and q(w2) = v2. If that edge in S points from w1

to w2, then the edge in R points from v1 to v2.

31

Figure 3.2: A Directed Correspondence

Figure 3.2 illustrates a directed version of the correspondence in Figure 2.1 in Chapter2. We also let (?, ·) denote the directed tree with a single vertex and the trivial orientation.

The orientation µ of T determines the orientations ν, λ on S,R respectively. If p : R→ Tis a correspondence and µ is an orientation of T , we let p∗µ denote the induced orientationon R.

Another way of saying that is the forgetful functor F : DirTree → Tree is a discretefibration:

Definition 3.2.2. Let F : D → C be a functor. F is a discrete fibration if for everyc ∈ C, d ∈ D, and g : c→ F (d), there is a unique morphism f s.t. F (f) = g.

This notion is useful for the following reason:

Lemma 3.2.1. Suppose F : D → C is a discrete fibration. If C is a combinatorially gradedcategory, then the functor | · | ◦ F makes D a combinatorially graded category. Additionally,for any object A ∈ Ob(D), F induces an isomorphism of slice categories DA → CF (A). Hence

we get a canonical isomorphism PA∼→ PF (A).

Proof. The existance/uniqueness of lifts automatically implies that F induces an isomor-phism of slice categories. C being a combinatorially graded category is a local property, inthe sense that all relevant properties can be stated as properties of its slice categories, henceD is a combinatorially graded category as well.

The poset P(T,µ) has objects which are directed correspondences p : (R, λ) → (T, µ) upto equivalences. By Lemma 3.1, P(T,µ) is canonically isomorphic to PT .

Definition 3.2.3. Given a directed tree (T, µ), there is a functorial pre-cyclic structure Oµon PT ∼= P(T,µ). Namely, every codimension-one cell is represented by a unique correspon-dence of the form p : (A2, λ0)→ (T, µ), where λ0 is the orientation on A2 in which the edge

32

points from vertex 1 to vertex 2. The cyclic order O[p]µ on the three top-cells containing [p]

is given by the successor function:1

s([p ◦ 〈1〉]) = [p ◦ 〈e〉]s([p ◦ 〈e〉]) = [p ◦ 〈2〉]s([p ◦ 〈2〉]) = [p ◦ 〈1〉]

Recall (Corollary 2.3.1) that maximal cells in PT are in bijection with subtrees S of

T . Similarly, codimension-one cells determined by correspondences p = (A2

q� S

i↪→ T )

are specified by the fibers q−1(1), q−1(2)- two subtrees of T separated by a single edge.The three top-cells greater than this correspondence are represented by the three subtreesq−1(1), q−1(2), q−1(A2).

In determining the cyclic order on these top-cells, only the orientation of the edge sep-arating the two subtrees matters. See Figure 3.3 illustrating a codimension-one cell for thedirected tree in Figure 3.2.

Figure 3.3: A Codimension-One Cell, and the three Top Cells in the Induced Cyclic Order

The meaning of ‘functorial’ in the above definition is as follows: let i : P ↪→ P ′ be anembedding of C-Arb singularities. Then given a pre-cyclic structure O on P ′, one can pullit back to obtain a pre-cyclic structure i∗O on P . Then we have:

Lemma 3.2.2. Suppose we have a correspondence of directed trees p : (R, λ) → (T, µ),yielding an open embedding ip : PR ↪→ PT . Then i∗pOµ = Oλ.

1Refer to the notation introduced in Chapter 2

33

Proof. Consider a codimension-one cell represented by q : (A2, λ0) → (R, λ), where λ0 is as

discussed above. The cyclic order Oip([q])µ = O[p◦q]

µ is given by the successor function:

s([(p ◦ q) ◦ 〈1〉]) = [(p ◦ q) ◦ 〈e〉]s([(p ◦ q) ◦ 〈e〉]) = [(p ◦ q) ◦ 〈2〉]s([(p ◦ q) ◦ 〈2〉]) = [(p ◦ q) ◦ 〈1〉]

Since [(p◦q)◦ 〈k〉] = ip([q◦ 〈k〉]) for k ∈ KA2 , the cyclic order i∗pO[q]µ is given by the successor

function:

s([q ◦ 〈1〉]) = [q ◦ 〈e〉]s([q ◦ 〈e〉]) = [q ◦ 〈2〉]s([q ◦ 〈2〉]) = [q ◦ 〈1〉]

Which is equal to O[q]λ .

This leads to the following definitions:

Definition 3.2.4. Let P be a C-Arb space, and O a pre-cyclic structure on P. O is acyclic structure if, for all x ∈ P, there exists a directed tree (T, µ) and an isomorphismϕ : N(x)

∼→ PT such that ϕ∗Oµ = i∗O, where i : N(x) ↪→ P is the inclusion.

Definition 3.2.5. A cyclic C-Arb singularity (resp. cyclic C-Arb space) is a pair (P ,O),where P is a C-Arb singularity (resp. space) and O is a cyclic structure on P.

Lemma 3.2.3.

(i) If i : P ↪→ P ′ is an open embedding of C-Arb spaces and O is a cyclic structure on P ′,i∗O is a cyclic structure on P.

(ii) If (T, µ) is a directed tree, Oµ is a cyclic structure on PT .

Proof. (i) Is clear from the local nature of the definition of cyclic structures. (ii) Followsfrom Lemma 3.2.2.

Remark 3.2.1. It is not the case (in general) that every cyclic structure on PT is of theform Oµ, where µ is an orientation of T . Instead, every cyclic structure is of the form ϕ∗Oµ,where µ is an orientation of T and ϕ ∈ Aut(PT ). This follows trivially from the definition,once we show that there cannot exist an isomorphism PT

∼→ PT ′ when T 6∼= T ′, which weshow in section 4.3.

34

3.3 Coherence

Definition 3.2.4 is simple but somewhat unilluminating. In fact, cyclic structures can alsobe defined as pre-cyclic structures satisfying two local ‘coherence conditions’. We provide apartial discussion of this point in the rest of this section, and a proof appears in Chapter 4.

Definition 3.3.1. Fix a tree R. O is R-coherent if for every embedding i : PR ↪→ P, i∗Ois a cyclic structure.

It is clear from the definition that a cyclic structure O on a C-Arb space P is R−coherentfor every tree R. Our main result regarding coherence is the following:

Theorem 3.3.1. A pre-cyclic structure is a cyclic structure if and only if it is A3−coherentand S4−coherent.

Recall that A3 is the tree consisting of three vertices connected in a line, while S4 is thetree with three vertices all connected to a central vertex.

In the following two sections, we will undergo a careful investigation of cyclic structures onPA3 and PS4 , and we will also look at A3−coherent pre-cyclic structures on PS4 which are notcyclic structures. In the following sections we will examine cyclic structures on PAn and PSn .

A complete proof of Theorem 3.3.1 will involve techniques developed in Chapter 4.

3.4 Cyclic Structures on PA3

In this section we will carefully describe cyclic structures on PA3 , and give both a geometricand combinatorial way of understanding these objects.

To begin, write K = KA3 = Vt(A3) t E(A3) = {1, e1, e2, 3}. Here 1, 3 are terminal ver-tices, and e1, e2 the internal edges. For simplicity of notation we will write 1 = e0 and 3 = e3.Identify PA3 with its image in P(K), which is {S ⊆ K | |S| ≤ 2}.

If c is a cyclic order on K, there is a corresponding pre-cyclic structure Oc on PA3 asfollows:

35

Definition 3.4.1. (Definition of Oc) A codimension-one cell is given by S ⊆ K with |S| = 1.If we write K\S = {x, y, z}, the three top-cells containing S are S ∪ {x}, S ∪ {y}, S ∪ {z}.We define:

OSc {S ∪ {x}, S ∪ {y}, S ∪ {z}} ⇔ c(x, y, z)

.

We have:

Proposition 3.4.1. The construction c 7→ Oc gives a bijection between cyclic orders on Kand cyclic structures on PA3.

Proof. If ϕ : PA3

∼→ PT is a poset isomorphism, then T ∼= A3- this follows from Corollary2.3.2. Hence any cyclic structure is of the form ϕ∗Oµ, where µ is an orientation of A3 andϕ ∈ Aut(PA3).

Observe that the association c 7→ Oc is functorial, in the sense that if ϕ ∈ Aut(PA3)induces ϕ ∈ S(KA3), then ϕ∗Oc = Oϕ∗c. (The pullback ϕ∗c is the cyclic order defined by(ϕ∗c)(x, y, z)⇔ c(ϕ(x), ϕ(y), ϕ(z))).

An easy computation (the results of which are shown in Table 3.1 later in this section)verifies that for an orientation µ, there exists a cyclic order c on K with Oµ = Oc. Thenϕ∗Oµ = Oϕ∗c, so each cyclic structure corresponds to a cyclic order.

On the other hand, Aut(PA3)∼= S(K) acts transitively on cyclic orders on K, so if one

cyclic order yields a cyclic structure, all of them do. Hence each cyclic order corresponds toa cyclic structure.

As a result, we see there are 6 = 3! distinct cyclic structures on PA3 .

We now present a visual way of diagramming pre-cyclic structures on PA3 . Recall fromChapter 1 that PA3 can be visualized as a plane with two half-planes glued on- see Figure2.7 Removing the top-cells {e0, e1} and {e2, e3} from PA3 , we can represent the remainingcells as a stratification of R2, illustrated in Figure 3.4:

In the figure, each of the four one-cells are pictured, along with two of the top-cells con-taining it. Suppose we have a pre-cyclic structure O on PA3 . For a one-cell {e}, let {e, x},{e, y} denote the top cells containing {e} represented in the figure. One of these top-cellsmust be the successor of the other in the cyclic order O{e}. We can represent this by drawingan arrow from the predecessor to its successor, crossing {e}.

36

∅ {e0}

{e2}

{e1}

{e3}

{e0, e2}{e1, e2}

{e1, e3} {e0, e3}

Figure 3.4: An Illustration of PA3\{{e0, e1}, {e2, e3}}.

For example, suppose the cyclic order O{e0} is induced by the total order {e0, e1} <{e0, e2} < {e0, e3}. Then, since {e0, e3} is the successor of {e0, e2}, we draw an arrow point-ing from the top-right quadrant to the bottom-right quadrant, crossing {e0}. A completeddiagram might look like Figure 3.5.

∅ {e0}

{e2}

{e1}

{e3}

{e0, e2}{e1, e2}

{e1, e3} {e0, e3}

Figure 3.5: A Completed Arrow Diagram

One can check that the corresponding pre-cyclic structure is not a cyclic structure, byobserving that it does not arise from a cyclic order.

Table 3.1 provides a translation between the three ways we have seen of describing cyclicstructures on PA3 : Using an orientation on A3, using a cyclic order on K, and using an‘arrow diagram’. Note that there are only four orientations of A3 while there are six cyclicorders of K.

37

Directed Tree Cyclic Order Arrow Diagram

e0

e1

e2

e3

∅ {e0}

{e2}

{e1}

{e3}

e0

e3

e2

e1

∅ {e0}

{e2}

{e1}

{e3}

e0

e1

e3

e2

∅ {e0}

{e2}

{e1}

{e3}

e0

e2

e3

e1

∅ {e0}

{e2}

{e1}

{e3}

N/A e0

e2

e1

e3

∅ {e0}

{e2}

{e1}

{e3}

N/A e0

e3

e1

e2

∅ {e0}

{e2}

{e1}

{e3}

Table 3.1: A Dictionary for Cyclic Structures on PA3

38

Definition 3.4.2. An arrow diagram for PA3 is coherent if arrows on opposite one-cellspoint in the same direction (the first four in the table), and spiral if the arrows form aspiral.

We see an arrow diagram represents a cyclic structure if and only if it is coherent or spiral.However, as the automorphism group acts transitively on cyclic structures, the distinctionbetween coherent and spiral structures involves some symmetry breaking. As a result, thisdistinction between is not really fundamental. In particular, while we drew this arrow dia-gram by removing cells {e0, e1} and {e2, e3}, we could have also removed cells {e0, e2} and{e1, e3}, or cells {e0, e3} and {e1, e2}. For each of these presentations, four of the cyclicstructures will yield coherent arrow diagrams, and the other two will yield spiral diagrams,but which ones yield which is altered.

To be more explicit: The spiral structures occur when the top-cells removed correspondto pairs of one-cells which are opposites in the cyclic order.

We will use understand this idea to cyclic structures on PS4 via arrow diagrams, as wellas A3−coherent pre-cyclic structures on PS4 which are not cyclic structures.

3.5 Cyclic Structures on PS4

Now, consider PS4 . Write K = KS4 = E(S4) t Vt(S4) = {e1, e2, e3, v1, v2, v3}. The image ofPS4 in P(K) is {S ⊆ K | {vi, ei} * S, 1 ≤ i ≤ 3} ∪ {{vi, ei}c | 1 ≤ i ≤ 3}. For the rest ofthis section we identify PS4 with its image in P(K).

Definition 3.5.1. A polarization is a function ρ : K → {+,−} such that ρ(vi) = −ρ(ei)for i = 1, 2, 3.

Definition 3.5.2. Given a polarization ρ, the pre-cyclic structure Oρ is defined as follows:A codimension-one cell is given by a set X with |X| = 2, such that there is a unique i where{ei, vi} ∩X = ∅. The three top-cells containing X are X ∪ {ei}, X ∪ {vi}, and {ei, vi}c. Ifρ(ei) = +, the cyclic order is induced by the total order:

X ∪ {vi} < X ∪ {ei} < {ei, vi}c

Otherwise, it is the opposite.

39

Codimension-one cells can be also visualized as pairs of subtrees separated by a sin-gle edge. When T = S3, one of these subtrees is a single terminal vertex {v}, while theother subtree, call it S contains the central vertex 0. The three top-cells greater than thiscodimension-one cell are indexed by the subtrees {v}, S, S ∪ {v}. If the original correspon-dence is described by the set X, then the subtree S yields the set X ∪ {vi}, S ∪ {v} yieldsthe set X ∪ {ei}, and {v} the set {ei, vi}c.

As an example, when S = {0, v2} and v = v1, the translation between these two de-scriptions of codimension-one cells is illustrated in Figure 3.6. In the figure, [p] is thecodimension-one cell, and [pi], 1 ≤ i ≤ 3 are the three top-cells greater than it.

e2

e1

e3

0 v1

v2

v3

Xp = {e2, v3}

v = v1; S = {0, v2}

e2

e1

e3

0 v1

v2

v3

e2

e1

e3

0 v1

v2

v3

e2

e1

e3

0 v1

v2

v3

Xp1= {e2, v3, v1} = Xp ∪ {v1}Subtree: S = {0, v2}

Xp2= {e2, v3, e1} = Xp ∪ {e1}

Subtree: S ∪ {v} = {0, v1, v2}Xp3

= {e2, e3, v2, v3} = {e1, v1}c

Subtree: {v} = {v1}

Figure 3.6: Translating Between Two Descriptions of Codimension-One Cells in PS4 .

Definition 3.5.3. For an orientation µ of S3, we can define a polarization ρ = ρ(µ) whereρ(ei) = + if edge ei points toward vertex 0, and − otherwise. One can easily check thatOµ = Oρ.

Proposition 3.5.1. The association ρ 7→ Oρ gives a bijection between polarizations andcyclic structures on OS4. Therefore, does the association µ 7→ Oµ, where µ is an orientationof S4.

Proof. Suppose ϕ : PS4

∼→ PT is an isomorphism. By Corollary 2.3.2, T must have 4 vertices.By observation, PS4 and PA4 are not isomorphic (one has five one-cells and the other has

40

six, for example), hence T ∼= S4. Hence any cyclic structure is of the form ϕ∗Oµ, where µ isan orientation of S4 and ϕ ∈ Aut(PS4).

The association ρ 7→ Oρ is functorial, meaning that if ϕ ∈ Aut(PS4) induces ϕ ∈ S(KS4),ϕ∗Oρ = Oϕ∗ρ. Here (ϕ∗ρ)(x) = ρ(ϕ(x)), for x ∈ K.

As explained in definition 3.5.3, for any orientation µ, there is a polarization ρ = ρ(µ)with Oµ = Oρ. Then ϕ∗Oµ = Oϕ∗ρ, so each cyclic structure corresponds to a polarization.

On the other hand, for any polarization ρ there exists µ with Oρ = Oµ, so each polariza-tion corresponds to a cyclic structure.

We can present pre-cyclic structures on PS4 using arrow diagrams as well. If we removethe three top-cells {vi, ei}c, the remaining cells can be visualized as a stratification of R3

sliced by the three coordinate planes {xi = 0}, i = 1, 2, 3. Identify the one-cell correspondingto the positive xi−axis with {vi} and the negative xi−axis with {ei}. This is illustrated inFigure 3.7.

{e1}

{v1}

{e2}

{v2}

{e3}

{v3}

Figure 3.7: An Illustration of PS4\{vi, ei}c.

As in PA3 , we can specify an arbitrary pre-cyclic structure on PS4 by drawing an arrowcrossing each of the twelve two-cells in the picture. Such a pre-cyclic structure is A3−coherentif its restriction to each neighborhood isomorphic to PA3 is a cyclic structure. There are sixsuch neighborhoods, each corresponding to a one-cell, i.e. a positive or negative axis. Re-stricting to the neighborhood of a one-cell yields a geometric picture of a PA3 as in theprevious section. So we see that, in order for a pre-cyclic structure to be A3−coherent, thearrows on the four two-cells attached to a one-cell must be oriented coherently or spirally

41

(see Definition 3.4.2). This is illustrated in Figure 3.8- the one-cell in question is bold, andtwo possible options for the arrows around that one-cell are shown:

Figure 3.8: Constructing Pinwheel Orientations

One possibility is that the arrows on each hyperplane are oriented the same way (i.e. allpointing toward the same half-space). There are 8 = 23 such pre-cyclic structures, and onecan check that these are exactly the cyclic structures. Explicitly, we can write this structureas Oµ or as Oρ, where the following are equivalent:

• The arrows on the hyperplane {xi = 0} all point toward the half-space xi > 0 (i.e.toward {vi})

• The edge ei is pointing away from the center in µ.

42

• ρ(vi) = +.

However, this does not exhaust all possibilities for A3−coherent pre-cyclic structures. Togive an example: Fix an axis, and draw arrows on the eight two-cells attached to that axisso that they point cyclically about that axis (Figure 3.9).

Figure 3.9: Constructing Pinwheel Orientations

For the remaining hyperplane, orient the arrows pointing toward the same half-space.There are 3 · 2 · 2 = 12 of these pinwheel orientations. One can easily show that these are allthe possible A3−coherent pre-cyclic structures which are not coherent. This shows that theS4−coherence condition is necessary.

3.6 Cyclic Structures on An Singularities

An singularities and cyclic structures on them have a beautiful description in terms of cyclicorders on finite sets. This cyclic symmetry also appears in the representation theory of An,as observed in [21]. We will investigate this connection more thoroughly later.

Recall that from 2.4.1, PAn ∼= {X ∈ P(KAn) | |X| ≤ n − 1}. Throughout this sectionwe identify these posets. Note that the codimension of a subset X is |Xc| − 2.

Definition 3.6.1. Let ∆inj denote the category of finite sets whose morphisms are injectivemaps, and ArbA the full subcategory of Pos whose objects are PAn, n ≥ 2.

43

Definition 3.6.2. Define the functor ∆ : ArbA → ∆inj as follows:

∆(An) = KAn

For an embedding i : PAm ↪→ PAn, since open embeddings preserve codimension, the imageof the minimal point i(∅) = X is a subset of KAn of size n−m. We will define a functorialinjection:

∆(i) : KAm ↪→ KAn

Whose image is Xc, defined by:

i({k}) = X ∪ {∆(i)(k)}

More generally, for Y ⊆ KAn , we have:

i(Y ) = X ∪∆(i)(Y )

Definition 3.6.3. Let S be a finite set. A pre-cyclic order on S is a ternary relation csatisfying all the properties of a cyclic order except transitivity.

See appendix A for the axioms of cyclic orders. An equivalent way of thinking of pre-cyclic orders is that they are a choice of cyclic order on each 3-element subset of S- withoutthe transitivity axiom there is no relationship among these choices.

Definition 3.6.4. Let Arbpre−cycA denote the category whose objects are pairs (PAn ,O), whereO is a pre-cyclic structure on PAn, and whose morphisms (PAm ,O) → (PAn ,O′) are em-beddings i : PAm ↪→ PAn such that i∗O′ = O. Let ArbcycA denote the full subcategory ofArbpre−cycA where O is a cyclic structure.

Similarly, let ∆pre−cycinj denote the category whose objects are finite sets equipped with a

pre-cyclic order and whose morphisms are injections preserving the ternary relation. Let∆cycinj denote the full subcategory for which the pre-cyclic order is a cyclic order.

Given a pre-cyclic structure O on PAn , one can construct a pre-cyclic order c = ∆(O)on KAn as follows: Each codimension-one cell in PAn is represented by X ⊆ KAn with|X| = n−2. Write KAn\X = {x, y, z}. The three top-cells containing X are X∪{x}, X∪{y},and X ∪ {z}. Then we write:

OX(X ∪ {x}, X ∪ {y}, X ∪ {z})⇔ c(x, y, z)

It is clear that this construction yields a bijection between pre-cyclic structures on PAnand pre-cyclic orders in KAn .

44

Lemma 3.6.1. The construction described above is functorial, i.e., it allows us to extendthe functor ∆ : ArbA → ∆inj to a functor ∆ : Arbpre−cycA → ∆pre−cyc

inj .

Proof. Let i : PAm ↪→ PAn , O be a pre-cyclic structure on PAn , and c = ∆(O). Wewant to show ∆(i)∗c = ∆(i∗O). So let x, y, z ∈ KAm , X = KAm\{x, y, z}. Then i(X) =KAn\{∆(i)(x),∆(i)(y),∆(i)(z)}, so:

(∆(i)∗c)(x, y, z)⇔ c(∆(i)(x),∆(i)(y),∆(i)(z))

⇔ Oi(X)(i(X) ∪ {∆(i)(x)}, i(X) ∪ {∆(i)(y)}, i(X) ∪ {∆(i)(z)})⇔ (i∗O)X(X ∪ {x}, X ∪ {y}, X ∪ {z})⇔ ∆(i∗O)(x, y, z)

We are now able to prove the key proposition of this section:

Proposition 3.6.1. Let O be a pre-cyclic structure on PAn. The following are equivalent:

1. O is a cyclic structure.

2. O is A3−coherent.

3. ∆(O) is a cyclic order on KAn.

Proof. (1)⇒ (2) is clear.

To show (2) ⇒ (3), we exploit the functor ∆ : Arbpre−cycA → ∆pre−cycinj . Let c = ∆(O).

If Y is a 4-element subset of KAn , we can find an embedding i : PA3 ↪→ PAn such thati(∅) = Y c, meaning the image of ∆(i) is Y .

A3 coherence implies the pullback i∗O is a cyclic structure. By functoriality, ∆(i∗O) =∆(i)∗c. Proposition 3.4.1 then implies ∆(i)∗c is a cyclic order. It follows that c is a cyclicorder when restricted to any four-element subset of KAn , which is enough to ensure transi-tivity, so c is a cyclic order.

To show (3)⇒ (1), suppose ∆(O) = c is a cyclic order. If µ is any orientation on An, letcµ = ∆(Oµ). By the implication (1) ⇒ (3) already proved, cµ is a cyclic order, hence thereexists ϕ ∈ Aut(PAn) such that ϕ∗cµ = c. (Recall ϕ is the permutation of KT induced by anautomorphism of PT -, which in this case equals ∆(ϕ)). By the functoriality of ∆, O = ϕ∗Oµis a cyclic structure.

45

To finish this section we will more explicitly describe the relationship between cyclic or-ders and orientations of An.

Proposition 3.6.2. Let µ be an orientation on An. By proposition 4.4, ∆(Oµ) := cµ is acyclic order on KAn. cµ can be described using a successor function s as follows:

• If v ∈ Vt(An), begin travelling from v toward the opposite endpoint. s(v) is the firstedge encountered pointing in the direction of travel. If no such edge is encountered,s(v) is the opposite endpoint.

• If e ∈ E(An), begin travelling toward an endpoint of An in the direction indicated bye. s(e) is the next edge encountered pointing in the direction of travel. If no such edgeis encountered before hitting an endpoint, s(e) is the endpoint.

In particular, write KAn = {e0, e1, . . . , en−1, en}, where ei denotes the edge joining vertex ito vertex i+ 1 for 1 ≤ i < n, e0 is the vertex 1, and en is the vertex n. If µ is the orientationin which all edges point from vertex i to vertex i + 1, ∆(Oµ) is the cyclic order induced bythe total order e0 < e1 · · · en−1 < en.

Proof. In a cyclic order c, y is the successor of x if and only if c(x, y, z) whenever z 6= x, y.

We use 〈1〉, 〈2〉, 〈e〉 to denote the one-cells of PA2 .

Suppose ei ∈ KAn with 0 ≤ i < n, and suppose either i = 0 or ei is pointing to the right.(The other case can be handled by symmetry). Let j is the smallest index greater than i ofan edge ej pointing from vertex j to vertex j + 1 (or j = n if no such vertex does), we will

show s(ei) = ej. Let p = (A2

q� S

i↪→ T ) be a correspondence with Xp = KAn\{ei, ej, ek}.

Write λ = p∗µ. We can assume WLOG that the subtree q−1({1}) has vertices of smallerindex than q−1({2}). There are three cases:

Case 1: If k < i < j, then Xp◦〈1〉 = X ∪ {ek}, Xp◦〈e〉 = X ∪ {ei}, Xp◦〈2〉 = X ∪ {ej}, andthe edge points from vertex 1 to vertex 2 in λ.

Case 2: If i < k < j, then Xp◦〈1〉 = X ∪ {ei}, Xp◦〈e〉 = X ∪ {ek}, Xp◦〈2〉 = X ∪ {ej}, andthe edge points from vertex 2 to vertex 1 in λ.

Case 3: If i < j < k, then Xp◦〈1〉 = X ∪ {ei}, Xp◦〈e〉 = X ∪ {ej}, Xp◦〈2〉 = X ∪ {ek}, andthe edge points from vertex 1 to vertex 2 in λ.

46

Verification that c(ei, ej, ek) in each case, using the definitions of Oµ (definition 3.2.3)and ∆(Oµ), is straightforward and left to the reader.

Example 3.6.1. As an example, consider the directed A5 pictured in Figure 3.10, with theedges/terminal vertices labelled as elements of KA5:

e0 e5e1 e2 e3 e4

Figure 3.10: A Directed Tree (A5, µ).

The successor of e0 is e1, since it points to the right. The successor of e1 is e4, whichis the next edge that points to the right. The successor of e4 is e5, since there are no moreedges pointing to the right. Continuing this, we get the cyclic order in Figure 3.11.

e0

e1e4

e5

e3 e2

Figure 3.11: The Cyclic Order ∆(Oµ) on KA5 .

3.7 Cyclic Structures on Sn Singularities

In this section we explain cyclic structures on PSn . The exposition is analogous to the pre-vious section.

We identify PSn with its image in P(KSn), which is (lemma 2.4.2):

{X ⊆ KSn | Xc intersects each equivalence class of ∼, or is a single equivalence class of ∼}

When Xc intersects each equivalence class of ∼, the codimension of X is the number ofequivalence classes completely contained in Xc. Hence for an open embedding i : Sm ↪→ Sn,the image i(∅) = X is a set such that Xc contains m− 1 equivalence classses in KSn . Thatinspires the following definition:

47

Definition 3.7.1. Let �inj denote the category whose objects are finite sets equipped witha relation ∼ whose equivalence classes are of size 2. A morphism S → T is a pair (i,X)where i : S ↪→ T is an injective map of sets satisfying i(x) ∼ i(y) when x ∼ y, and X ⊆ T\Sintersects each equivalence class of T\S exactly once. The composition (i,X) ◦ (j, Y ) = (i ◦j,X ∪ j(Y )).

Definition 3.7.2. Let ArbS denote the full subcategory of Pos whose objects are PSn, n ≥ 4.

Definition 3.7.3. Define a functor � : ArbS → �inj as follows. �(PSn) = KSn. Leti : PSm ↪→ PSn be an embedding. If i(∅) = X, then Xc contains m− 1 equivalence classes inKSn, and intersects the rest once. We will construct:

�(i) = (j, Y )

Where the image of j is the m−1 equivalence classes contained in Xc, and Y = Xc\j(KSm)-it is defined by:

i({k}) = X ∪ {j(k)}

Definition 3.7.4. Let S ∈ Ob(�inj). A weak polarization ρ is a function which assignsto every x ∈ S and every set Y ⊂ S which contains exactly one element of each equiva-lence class besides [x], a sign ρY (x) ∈ {+,−}. Furthermore if x ∼ y but x 6= y we requireρY (x) = −ρY (y). A polarization is a weak polarization which does not depend on the ar-gument Y .

We let �w−polinj and �pol

inj denote the categories consisting of pairs (S, ρ) where S ∈ Ob(�inj)and ρ is a weak polarization (resp. polarization). Note that when (i,X) : S → T is a mor-phism and ρ a weak polarization on T , the pullback i∗ρ is defined by i∗ρY (v) = ρY ∪X(i(v)).

We let Arbpre−cycS and ArbcycS denote the category of pairs (PSn ,O), where O is a pre-cyclic structure (resp. cyclic structure).

Given a pre-cyclic structure O on PSn , we can construct a weak polarization �(O) asfollows: Each codimension-one cell in PSn is given by a subset X of KSn which intersectseach equivalence class of KSn once, except for one. Call the remaining class {x, y}. Thethree top-cells containing X are given by X ∪{x}, X ∪{y}, and {x, y}c. (See the discussionof PS4 , including Figure 3.6). If:

OX(X ∪ {x}, X ∪ {y}, {x, y}c)

48

We set �(O)X(x) = −, �(O)X(y) = +.

It is clear from this discussion that this establishes a bijection between weak polarizationsand pre-cyclic structures. In fact, we have:

Lemma 3.7.1. The above construction gives a functor � : Arbpre−cycS → �w−polinj .

Proof. Let O be a pre-cyclic structure on PSn and i : PSm ↪→ PSn be an embedding, with�(i) = (j, Y ). We want to show �(i∗O) = j∗�(O). Let X be a subset of KSm whichintersects each equivalence class of KSn once, except for one: Call it {x, y}. Then:

j∗�(O)X(x) = − ⇔ �(O)X∪Y (j(x)) = −⇔ Oj(X)∪Y (j(X) ∪ Y ∪ {j(x)}, j(X) ∪ Y ∪ {j(y)}, {j(x), j(y)}c)⇔ (i∗O)X(X ∪ {x}, X ∪ {y}, {x, y}c)⇔ �(i∗O)X(x) = −

Note that this convention agrees with our previous convention for polarizations on PS4 .In particular, we know from proposition 3.5.1 that a pre-cyclic structure O on PS4 is a cyclicstructure if and only if �(O) is a polarization. This idea generalizes to give a description ofcyclic structures on PSn , as the following proposition shows:

Proposition 3.7.1. Let O be a pre-cyclic structure on PSn, where n ≥ 4. The following areequivalent:

(1) O is a cyclic structure.

(2) O is S4−coherent.

(3) �(O) is a polarization on KSn.

(4) O = Oµ for some orientation µ of Sn.

Proof. (1)⇒ (2) is clear.

For (2)⇒ (3), let ρ = �(O), where O is a S4−coherent pre-cyclic structure on PSn . Pickan equivalence class {a, b} ⊆ KSn . Let X and Y be two subsets of KSn which intersect eachequivalence class of KSn once except {a, b}- we want to show ρX(a) = ρY (a). Assume thatX and Y differ on a single equivalence class, that is, that there is a class {x, y} such thatx ∈ X and y ∈ Y , but X\{x} = Y \{y}. Pick a third equivalence class {u, v}, with u ∈ X, Y .

49

We get a map (j, Z) : KS3 → KSn in �inj, where j sends v1, e1, v2, e2, v3, e3 to a, b, x, y, u, v,respectively, and Z = X\{x, u} = Y \{y, u}. This map defines i : PS4 ↪→ PSn , with �(i) =(j, Z). Since i∗O is a cyclic structure, by functoriality j∗ρ = �(i∗O) is a polarization. So:

ρX(a) = j∗ρ{v2,v3}(v1) = j∗ρ{e2,v3}(v1) = ρY (a)

We assumed X and Y differed only on a single equivalence class, but we can get betweenany two sets by repeatedly making changes on a single equivalence class.

For (3)⇒ (4), let ρ be a polarization. We define µ to be the orientation where the edgeei points toward vertex 0 if ρ(ei) = +, and toward vertex vi if ρ(ei) = −. It is easy to checkthat Oρ = Oµ.

Finally, (4)⇒ (1) follows immediately from the definition.

Remark 3.7.1. An interesting result is that, in this case, all of the cyclic structures comefrom orientations. This is true in general when there are no vertices of degree 2 in the tree(we will not explicitly prove this, but it will follow in a straightforward way from generalcombinatorial descriptions of cyclic structures in the following section).

50

Chapter 4

The Combinatorics of CyclicStructures

4.1 Isomorphisms of C-Arb Singularities

In this section, we will prove a handful of combinatorial results involving arboreal singulari-ties. An important component of this is the analysis of Aut(PT ), the automorphism groupof a C-Arb singularity, as well as isomorphisms PT

∼→ PT ′ . In this second case it turns outwe must have T ∼= T ′, but this is not immediately obvious.

Our first goal will be understanding how combinatorial properties of PT relate to prop-erties of the underlying tree T .

To begin, let T be a tree and KT = E(T ) t Vt(T ). As PT is atomistic, for ϕ ∈ Aut(PT )we can associate ϕ ∈ S(KT ). We use notation from Chapter 2.

Definition 4.1.1. Let x, y ∈ KT . A path connecting x and y is a path {v0, v1, . . . , vk} suchthat v0 ∈ x if x ∈ E(T ) or v0 = x if x ∈ Vt(T ), and likewise for vk and y.

Lemma 4.1.1. Let T be a tree, and x 6= y ∈ KT . Then there exists a correspondencep : R → T with Xp = {x, y}c if and only if x and y can be connected by a path where eachvertex has degree ≤ 2.

Proof. If there is such a path between x and y, let S be the subtree defined by that path.Then if p = (? � S ↪→ T ), Xp = {x, y}c. On the other hand, suppose there is a vertex vbetween x and y having degree > 2, and let p = (R � S ↪→ T ), with x, y /∈ Xp. Then xand y are either edges having a vertex in common with S or vertices in S, hence v ∈ V (S).

51

Since deg(v) > 2, S has a boundary vertex which is not in (or equal to) x or y, and hencethere is at least one more element in Xc

p .

Definition 4.1.2. Let x, y ∈ KT . Define x ∼ y if x = y or there exists p : R→ T such thatXp = {x, y}c. By the lemma, ∼ is an equivalence relation, we call the equivalence classeschains. An element ϕ ∈ Aut(PT ) stabilizes chains if x ∼ ϕ(x) for all x ∈ KT .

Chains are essentially the connected components of T once the vertices of degree > 2are removed. This is illustrated in Figure 4.1. We see that for each chain C there are atmost two vertices with degree > 2 that are incident to an edge in C- such vertices are calledboundary vertices of C.

v2

v1 v3

v4

e1

e4 e5

e6

e2

e3 e7

C1 = {v1, e1}

C2 = {v2, e2, e3}

C3 = {e4, e5}

C4 = {v3, e6}

C5 = {v4, e7}

Figure 4.1: A tree T and the chains in KT

The below definitions involve only the combinatorics of PT . We will see (lemma 4.1.2)that each of these notions has an interpretation in terms of the tree T (the names are meantto be suggestive).

Definition 4.1.3.

• For a tree T , let CT denote the set of chains.

• If C ∈ CT , we define ∼0C to be the relation on CT\{C} where C1 ∼0

C C2 if and onlyif there exists a maximal correspondence p : ? → T such that (Xp)

c intersects each ofC1, C2, C. We let ∼C denote the equivalence relation on CT\{C} generated by ∼0

C.

• If C,C1, C2 ∈ CT , C separates C1 and C2 if C,C1, C2 are distinct and C1 6∼C C2.

52

• Distinct chains C1, C2 ∈ CT are adjacent if they are not separated by any chain C.

• A node is a maximal collection of pairwise adjacent chains. A boundary node of achain is a node containing that chain.

• For a node N ⊂ CT , the node relation ∼N on CT is defined by C1 ∼N C2 if C1 = C2

or if C1, C2 are distinct, not both in N , and are not separated by any C ∈ N .

• A correspondence p is disjoint from a node N if the set of chains intersecting (Xp)c

is contained in a single equivalence class of ∼N . p intersects N if it is not disjointfrom N .

This terminology, which depends entirely on the combinatorics of the poset, is related tothe combinatorics of the tree T as follows:

Lemma 4.1.2.

(i) C separates C1 and C2 iff any path in T connecting x ∈ C1 to y ∈ C2 contains bothvertices of an edge in C. Equivalently, C1 ∼C C2 iff C1 and C2 are on the same sideof C.

(ii) C1 and C2 adjacent iff there is vertex of degree > 3 incident to an edge in each of them.

(iii) If v is a vertex of degree greater than 2, the subset Nv of chains containing an edgeincident to v is a node, and this describes all nodes in P(CT ).

(iv) For a node Nv, we can describe the node relation: C1 ∼Nv C2 iff some x ∈ C1 can beconnected to some y ∈ C2 by a path avoiding v.

(v) A correspondence p = (Rq� S

i↪→ T ) intersects a node Nv iff v ∈ V (S). In this case,

(Xp)c intersects at least one chain in each equivalence class of Nv.

Proof. (i)

Suppose p = (? � S ↪→ T ) satisfies (Xp)c ∩ C 6= ∅, and C has two boundary vertices.

Then S can only contain one of the boundary vertices. This means that if C1 ∼0C C2 then

C1, C2 are on the same side of C. Since ‘on the same side of’ is an equivalence relation, weare done if we can show that C1 ∼C C2 whenever C1 and C2 are on the same side of C.

For a subtree S of T , we say S is a witness to the relation C1 ∼C C2 if, lettingp = (?� S ↪→ T ), (Xp)

c intersects each of C1, C2, C.

53

First assume that C1, C2 are on the same side of C, and C1 is not in between C and C2,nor C2 in between C and C1. Then one can find a subtree S such that each of C,C1, C2 con-tains an edge incident to only one vertex in S, so this S is a witness to the relation C1 ∼0

C C2.

Suppose on the other hand that C1 is between C and C2. Letting v ∈ V (T ) be theboundary vertex of C on the same side as C1, C2, we can find an edge e incident to v whichis not between C and C1. If we write C3 for the chain containing e, by the argument in theprevious paragraph we see C1 ∼0

C C3 and C2 ∼0C C3, hence C1 ∼C C2.

Figure 4.2 for an illustrates this argument.

v

C3

w

C2

C C1

Figure 4.2: An illustration of the relation ∼C : The subtree {v} is a witness to the relationC1 ∼0

C C3, while {v, w} is a witness to the relation C2 ∼0C C3

(ii), (iii)

Both of these are clear from the definitions and part (i).

(iv)

If C1 and C2 are separated by a chain C ∈ Nv, then any path connecting C1 to C2 mustpass through C, and hence through v. On the other hand, if neither C1, C2 ∈ Nv and theyare not separated by a chain in Nv, the shortest path connecting any x ∈ C1 to y ∈ C2 passesthrough at most one boundary vertex of a chain in Nv, and hence not through v.

If C1 ∈ Nv but C2 /∈ Nv, and they are not separated by a chain in Nv, then C1 has aboundary vertex which is not equal to v which can be joined to C2 by a path avoiding v.

(v)

If v ∈ V (S), then each component of Nv has a chain which either has a boundaryvertex of T which is also a boundary vertex of S, or has an edge in T only one of whosevertices are in S. So we see (Xp)

c intersects at least one chain in each equivalence class of Nv.

54

On the other hand, if v /∈ V (S), then S is completely disjoint from all the chains in allbut one of the equivalence classes of Nv, hence p is disjoint from Nv.

The conclusion from the above is that we can reconstruct the tree from the poset struc-ture. To be more precise, let T, T ′ be trees, and σ : T

∼→ T ′ be a tree isomorphism. We

can view σ as an isomorphism in Tree, namely (Tσ−1

� T ′i↪→ T ′). One has an induced

isomorphism iσ : PT∼→ PT ′ . Then:

Proposition 4.1.1. Let ϕ : PT∼→ PT ′ be an isomorphism of C-Arb singularities. Then

there exists σ : T∼→ T ′ such that i−1

σ ◦ ϕ ∈ Aut(PT ) stabilizes chains. If T 6∼= An, then σ isunique.

Proof. Since chains are defined purely in terms of the poset structure, it is clear that ϕ sendschains in KT to chains in KT ′ , i.e. determines a map CT → CT ′ . If such a σ exists, we musthave iσ = ϕ as maps CT → CT ′ . When T 6∼= An, there are no nontrivial automorphisms ofT which act trivially on CT , meaning σ is unique. (When T ∼= An, n ≥ 2 there is only onechain, but there is a nontrivial automorphism of T ).

To complete the proof, we need to show that there exists such a σ. However, by lemma4.1.2, if C1, C2 ∈ CT are chains sharing a boundary vertex, the same must be true of ϕ(C1),ϕ(C2), since this property can be discerned from the poset structure. Since this relationshipcompletely determines the tree structure, the isomorphism σ must exist.

4.2 Reflection Isomorphisms

We now tackle the converse: Suppose ω ∈ S(KT ) stabilizes chains. Does there necessarilyexist ϕ ∈ Aut(PT ) such that ϕ = ω? The answer, as we will see in this section, is yes. Ourstrategy will be to construct reflection isomorphisms, which are automorphisms of PTwhich act as transpositions of adjacent elements in KT .

Lemma 4.2.1. Let v ∈ V (T ) be a vertex of degree 2, and e1, e2 the edges in E(T ) incidentto v. Then there is a ωv ∈ Aut(PT ) such that ωv is the transposition (e1e2) switching e1

and e2. Additionally, if v ∈ V (T ) is a vertex of degree 1, and e is the edge incident to v,then there is a ωv ∈ Aut(PT ) such that ωv is the transposition (ve) switching v and e. Theseisomorphisms are called reflection isomorphisms.

We will take the rest of this section to prove this lemma. It is enough to show that, forany correspondence p : R → T , the set that results from taking Xp and switching e1, e2 (or

55

v and e) is of the form Xq for some other correspondence q : R′ → T . In our approach, wewill explicitly construct the correspondence q. To begin, we have the following definition:

Definition 4.2.1. A pointed tree is a pair (T, v), where either (i) : v ∈ V (T ) is a vertexof degree ≤ 2, or (ii) : v = 0, a symbol we introduce to mean the ‘null vertex’. Let (T, v) be

a pointed tree, and p = (Rq� S

i↪→ T ) a correspondence. If v ∈ V (S) and q−1({q(v)}) = v,

write p∗v := q(v), and in all other cases p∗v = 0. A correspondence of pointed treesp : (R,w)→ (T, v) is a correspondence satisfying w = p∗v.

For a correspondence of pointed trees p : (R,w) → (T, v), with p = (Rq� S

i↪→ T ), we

define the reflected correspondence Rp : (R,w) → (T, v) as follows. First, in the followingcases, we define Rp = p:

(A1) If w, v are both in V (T ), or w = v = 0.

(A2) If v /∈ V (S) and none of the vertices adjacent to v are in V (S)1 .

(A3) If deg(v) = 2, v ∈ V (S), and for both vertices v1, v2 adjacent to v, vi ∈ V (S) andq(vi) = q(v).

In the following cases, we modify p:

(B1) If v /∈ V (S), but there is u ∈ V (S) adjacent to v, define Rp = (Rq′

� S ∪ {v} i↪→ T ),

where q′ = q except q′(v) = q(u).

(B2) If v ∈ V (S), there is exactly one u ∈ V (S) adjacent to v, and q(v) = q(u). Then

Rp = (Rq� S\{v} i

↪→ T ).

(B3) If deg(v) = 2, suppose v is incident to e1 = {v, v1} and e2 = {v, v2}. Supposev, v1, v2 ∈ V (S) and q(v) = q(v1) 6= q(v2). If we write q(v1) = x, q(v2) = y, then

Rp = (Rq′

� Si↪→ T ), where q′ = q except q′(v) = y.

Cases B1 (Right Hand Side) and B2 (Left Hand Side) are illustrated in Figure 4.3. CaseB3 is illustrated in Figure 4.4. These figures should be interpreted as in Chapter 2 (see, forexample, Figure 2.1): Thick lines are those contracted by q, and dotted lines/empty circlesare not contained in S.

A crucial fact is the following:

1u ∈ V (T ) is adjacent to v if {u, v} ∈ E(T )

56

v vT

R

q R

Figure 4.3: Cases B1 and B2

T

R

q

v v

R

Figure 4.4: Case B3

Lemma 4.2.2. If we let Tree• denote the category of pointed trees, R defines a functorTree• → Tree•, which acts at the identity on objects and satisfies R2 = Id. In particular,R(p ◦ q) = Rp ◦ Rq.

Proof. This proof amounts to checking a handful of cases. For each nontrivial case we il-lustrate p, q, Rp, and Rq in a neighborhood of v, from which one can easily verify the

identitites. Writing p = (Rq� S

i↪→ T ), we can assume v, or at least one of its neighbors,

is in S, otherwise Rp = p, Rq = q, and R(p ◦ q) = p ◦ q. For similar reasons, if we write

q = (R′q′

� S ′i↪→ R), we can assume v, or at least one of its neighbors, is in q−1(S ′).

First assume v has one neighbor v1 in S. If either v /∈ V (S) (RHS) or v ∈ V (S) withq(v) = q(v1) (LHS), we have:

T

R

q

R′q′

R

R

v v

If v ∈ V (S) with q(v) 6= q(v1), we have q(v) = p∗v := w. If either w /∈ V (S ′) (RHS) orw ∈ V (S ′) with q′(w) = q′(q(v1)) (LHS) we have:

57

T

R

q

R′q′

R

R

v

w

v

w

Now assume v has two neighbors, v1 and v2 in S. If q(v) = q(v1) = q(v2) we see Rp = p,Rq = q, and R(p◦q) = p◦q. Now suppose q(v) = q(v1) 6= q(v2). Write q(v1) = x, q(v2) = y.If x, y ∈ V (S ′), and q′(x) = q′(y), we have:

T

R

q

R′q′

R

R

v v

If q′(x) 6= q′(y), we have:

T

R

q

R′q′

R

R

v v

If x ∈ V (S ′) but y /∈ V (S ′) (LHS) or the reverse (RHS) we have:

T

R

q

R′q′

R

R

v v

Finally, assume q(v) 6= q(v1) 6= q(v2), so Rp = p. Write w = q(v) = p∗v, w1 = q(v1),w2 = q(v2). If w,w1 ∈ V (S ′) w2 /∈ V (S ′), and q′(w) 6= q′(w1), Rq = q and R(p ◦ q) = p ◦ q.If q′(w) = q′(w1) (LHS), or only w1 ∈ V (S ′) (RHS), we have:

58

T

R

q

R′q′

R

R

v

w

v

w

If w,w1, w2 ∈ V (S ′) with q(w) = q(w1) 6= q(w2) we have:

T

R

q

R′q′

R

R

v

w

v

w

If p : R → T , and v ∈ V (T ) has degree ≤ 2 or v = 0, p lifts to a correspondence ofpointed trees (R, p∗v) → (T, v). We use the notation rvp to denote the correspondence oftrees R → T underlying the correspondence of pointed trees Rp. If we write w = p∗v, wecan interpret the lemma as saying rv(p ◦ q) = rvp ◦ rwq.

By the functoriality of R, the map ωv : [p] 7→ [rvp] is an automorphism of PT . Further-more, it is easy to see from the description that if deg(v) = 2 and e1, e2 are the two edgesindicent to v, ωv(〈e1〉) = 〈e2〉, ωv(〈e2〉) = 〈e1〉, and for each other k ∈ KT , ωv(〈k〉) = 〈k〉,and a similar statement holds when deg(v) = 1. This completes the proof of lemma 4.2.1.

Corollary 4.2.1. Suppose ω ∈ S(KT ) stabilizes chains. Then there exists ϕ ∈ Aut(PT )such that ω = ϕ, and ϕ is a product of reflection isomorphisms.

Proof. Any such ω will act as a permutation on each chain. Any permutation can be gen-erated by transpositions of adjacent elements, which can be realized by reflection isomor-phisms.

Corollary 4.2.2. If ϕ : PT∼→ PT ′ is a poset isomorphism, there exists a tree isomorphism

σ : T∼→ T ′ such that i−1

σ ◦ ϕ ∈ Aut(PT ) is a product of reflection isomorphisms.

Proof. Follows immediately from proposition 4.1.1 and corollary 4.2.1.

59

4.3 Isomorphisms of Cyclic C-Arb Singularities

In this section, we investigate isomorphisms of the form ϕ : (PT ,Oµ)∼→ (PT ′ ,Oµ′), where

(T, µ) and (T ′, µ′) are directed trees. That is, we require that ϕ∗Oµ′ = Oµ.

By proposition 4.1.1, there exists an isomorphism σ : T → T ′ such that i−1σ ◦ϕ stabilizes

chains. Writing ψ = i−1σ ◦ϕ, we get ψ : (PT ,Oµ)→ (PT ,Oi∗σµ) is an isomorphism. Hence we

can restrict our attention to the case when T = T ′ and our isomorphism ϕ stabilizes chains.

As in the undirected case, we will see that any isomorphism of this kind can be written asa product of directed reflection isomorphisms. To explain this, we first need to extendthe notion of reflection isomorphisms to the directed case.

4.4 Reflection Isomorphisms for Directed Trees

Definition 4.4.1. Let (T, µ) be a directed tree. We let V+(T ), respectively V−(T ), denotethe set consisting of all vertices v ∈ V (T ) of degree ≤ 2 which are sources (all incident edgespoint away from v), respectively sinks (all incident edges point toward v). Additionally, welet each set contain the null vertex 0.

A + pointed directed tree is a triple (T, µ, v) with v ∈ V+(T, µ). A - pointeddirected tree is a triple (T, µ, v) with v ∈ V−(T, µ). A correspondence of ± pointeddirected trees is an orientation-preserving correspondence of ± pointed trees.

We let DirTree+• , resp. DirTree−• , denote the category of + pointed directed trees (resp.

- pointed directed trees).

Definition 4.4.2. For a directed tree (T, µ), and v ∈ V (T ), define rvµ to be the orientationon T with all arrows incident to v reversed. We also define r0µ = µ.

Now we have:

Lemma 4.4.1. If p : (R, λ, w) → (T, µ, v) is a correspondence of pointed directed trees, sois Rp : (R, rwλ,w) → (T, rvµ, v). This gives functor R± : DirTree±• → DirTree∓• , whichon objects acts as R±(T, µ, v) = (T, rvµ, v).

Proof. We need to prove that (rvp)∗(rvµ) = rwλ.

60

First assume w ∈ V (R), w = q(v) and q−1({w}) = {v}. Then any edge incident to w inR inherits its orientation from an edge incident to v in T , and rvp = p, so the statement holds.

If not, then w = 0 so rwλ = λ. Furthermore, if each edge incident to v in S is contractedby q, none of the edges in R inherits a direction from any of the edges whose direction weswitched- explicitly p∗µ = p∗(rvµ) = (rvp)∗(rvµ), so the statement holds.

The last case left is illustrated in the figure below:

T

R

q

v v

R

We can now prove:

Lemma 4.4.2. ωv ∈ Aut(PT ) satisfies ω∗vOrvµ = Oµ.

Proof. Consider a codimension-one cell in PT represented by p = (A2

q� S

i↪→ T ), and let

w = p∗v. The functoriality of R gives the following commutative diagram:

PT PT

PA2 PA2

ωv

ipωw

irvp

Which reduces the proof to the case T = A2. If w = 0, there is nothing to prove. Ifw ∈ V (A2), then ωw is a transposition on KA2 , but for any orientation µ on A2, Oµ andOrwµ give opposite cyclic orders, which shows ω∗wOrwµ = Oµ.

The central result concerning directed reflection isomorphisms is as follows:

Theorem 4.4.1. If (T, µ) and (T ′, µ′) are directed trees, and ϕ : (PT ,Oµ)∼→ (PT ′ ,Oµ′) is

an isomorphism of cyclic C-Arb singularities, then there is an orientation λ on T and anisomorphism of directed trees σ : (T, λ)

∼→ (T ′, µ′) such that i−1σ ◦ϕ : (PT ,Oµ)→ (PT ,Oλ) is

a product of directed reflection isomorphisms.

61

The argument is different depending on whether T ∼= An or not. In the next two partswe prove this theorem, first for An trees and then non-An trees.

Isomorphisms of Cyclic Singularities of An Type

In this part we prove theorem 4.4.1 in the case where T ∼= An, which has some importantdifferences from the general case. By the discussion at the beginning of the section, it isenough to consider T = T ′ = An. Furthermore, if µ and µ′ are any two distinct orientationson T , there is a sequence of reflections which transform µ into µ′. Hence we can take µ = µ′

to be the orientation in which all edges point to the right (i.e. from i to i+ 1).

1 2 3 n

Figure 4.5: (An, µ)

Proposition 4.4.1. If ϕ ∈ Aut(PAn) satisfies ϕ∗Oµ = Oµ, then ϕ can be written as aproduct of reflection functors. Explicitly, the group Aut(PAn ,Oµ) of such automorphisms isisomorphic to Z/(n + 1)Z, and a generator can be written ω = ω1ω2 · · ·ωn, where ωi is thereflection isomorphism at vertex i.

Proof. By proposition , the automorphisms of PAn preserving Oµ are the same as the auto-morphisms of KAn preserving the cyclic order induced by the total order e0 < e1 < · · · < en.A generator for this group is θ(ei) = ei+1, with index addition taken mod n + 1. We willprove that the ω written in the statement of the proposition is well-defined, and ω = θ.

To show ω is well-defined, note that n is a sink in An, so ωn : (PAn ,Oµ) → (PAn ,Ornµ)is well-defined. Assume ωk · · ·ωn is well-defined for some 1 < k ≤ n, and write µ(k) :=rk · · · rnµ. Then the orientation of every edge has been flipped either twice or not at all,except the edge ek−1, so vertex k− 1 is a sink in µ(k), meaning ωk−1ωk · · ·ωn is well-defined.In µ(1), every edge has been flipped twice so we have µ(1) = µ, so ω ∈ Aut(PAn ,Oµ).

We know ωi is the transposition (ei−1ei), for 1 ≤ i ≤ n. It follows that ω(ei) =ω1 · · ·ωn(ei) = ei+1 as desired.

Isomorphisms of Cyclic Singularities of Non-An Type

In this section we establish theorem 4.4.1 for trees not isomorphic to An. In some ways thissituation is simpler; as we will see there is more rigidity imposed by the cyclic structure here.

62

Proposition 4.4.2. Let T be a tree not isomorphic to An, and µ, µ′ be two orientations onT . If ϕ ∈ Aut(PT ) satisfies ϕ∗Oµ = Oµ′, and ϕ stabilizes chains, then ϕ is a product ofreflection isomorphisms.

Proof. Note first that reflection isomorphisms stabilize all chains, since the induced bijectionon K is the transposition of two edges which are connected by a vertex of degree 2, or anedge connected to a terminal vertex, and in each case both of these are contained in thesame chain. The result follows from the following two observations:

(i) Let C be a chain with two boundary vertices v and w. The number of edges in Cpointing from v to w in µ is the same as in µ′.

(ii) If µ = µ′, then ϕ is the identity.

(i) implies that we can obtain the orientation µ′ from µ by a sequence of reflections, and (ii)implies that the product of the corresponding reflection isomorphisms must equal ϕ.

As a preliminary step, suppose ϕ ∈ Aut(PT ) stabilizes chains, and suppose p = (R �

S ↪→ T ) is a correspondence. Write ϕ([p]) = [p′], where p′ = (R′q′

� S ′i↪→ T ). Recall (lemma

4.1.2) that if v ∈ V (T ) is a vertex of degree > 2, v ∈ V (S) iff p intersects the node Nv. Sinceϕ stabilizes chains, it also preserves the node relation ∼Nv , so p intersects the node Nv iff p′

does, in other words v ∈ V (S) iff v ∈ V (S ′).

Secondly, we use the notation 〈1〉, 〈2〉, 〈e〉 to denote the correspondences ? → A2. We

remark that if p = (A2

q� S

i↪→ T ) is a correspondence, and α is the edge in T joining the

fibers q−1(1) and q−1(2), we have Xp◦〈e〉 = Xp ∪ {α}.

We now proceed to the proofs of assertions (i) and (ii).

Proof of (i): Let α be in edge in a chain C with boundary vertices v and w. Writeϕ(α) = β ∈ C. We will show β must point the same direction as α. WLOG α points fromv to w.

Let p = (A2

q� S ↪→ T ) be any correspondence with v ∈ q−1(1), w ∈ q−1(2), and α the

edge joining the fibers q−1(1) and q−1(2). Our assumption is the edge in λ = p∗µ points

from 1 to 2. Write p′ = (A2

q′

� S ′i′

↪→ T ), with ϕ([p]) = [p′], such that p′∗µ′ = λ. Since ϕpreserves the cyclic structure, there are three options:

(A)

ϕ([p ◦ 〈1〉]) = [p′ ◦ 〈1〉]ϕ([p ◦ 〈e〉]) = [p′ ◦ 〈e〉]ϕ([p ◦ 〈2〉]) = [p′ ◦ 〈2〉]

; (B)

ϕ([p ◦ 〈1〉]) = [p′ ◦ 〈e〉]ϕ([p ◦ 〈e〉]) = [p′ ◦ 〈2〉]ϕ([p ◦ 〈2〉]) = [p′ ◦ 〈1〉]

;

63

(C)

ϕ([p ◦ 〈1〉]) = [p′ ◦ 〈2〉]ϕ([p ◦ 〈e〉]) = [p′ ◦ 〈1〉]ϕ([p ◦ 〈2〉]) = [p′ ◦ 〈e〉]

By our preliminary step, we know ϕ([p ◦ 〈2〉]) intersects Nv but not Nw, and ϕ([p ◦ 〈1〉])intersects Nw but not Nv. Since v, w ∈ V (S ′), [p′ ◦ 〈e〉] intersects both Nv and Nw, whichmeans option (A) is the correct one. That implies q′(v) = 1 and q′(w) = 2. Additionally,Xp′◦〈e〉 = ϕ(Xp◦〈e〉) = Xp′ ∪ {β}, meaning β is the edge joining the fibers q′−1(1) and q′−1(2).Therefore, β also points from v to w.

Proof of (ii): First, let C be a chain with two boundary vertices v, w. We have shownϕ must permute the edges between v and w pointing from v to w, we will show this permu-tation must be the identity. Pick two edges α, β between v and w, both pointing from v tow, with α closer to v than β. It is enough to prove that ϕ(α) = α′ is also closer to v thanϕ(β) = β′.

Construct the correspondence p = (A2

q� S ↪→ T ), where q−1(1) consists of the sub-

tree connecting v to the closer vertex of α, and q−1(2) is the subtree connecting α to β.By our assumption, p∗µ = λ is the orientation where the edge points from 1 to 2. Let

p′ = (A2

q′

� S ′ ↪→ T ) be a correspondence representing ϕ([p]), with p′∗µ′ = λ.

As before, we have options (A), (B), (C). The fact that p ◦ 〈1〉 does not intersect Nv

eliminates option (B).

We have Xp◦〈e〉 = Xp ∪ {α}, and Xp◦〈2〉 = Xp ∪ {β}. If option (C) were true, then wewould have Xp′◦〈e〉 = Xp ∪{β′}, meaning β′ is the edge joining the fibers q′−1(1) and q′−1(2).Furthermore, we would have p′ ◦ 〈1〉 intersects Nv, meaning v ∈ q′−1(2). This would meanβ′ points from w toward v, contradicting (i).

This leaves option (A), which means α′ is the edge joining the fibers q′−1(1) and q′−1(2),and v ∈ q′−1(1), so α′ is closer to v than β′.

Lastly, suppose C is a chain with a single boundary vertex v. WLOG we can assume alledges point away from v in µ, since any situation can be turned into this one by a series ofreflection isomorphisms. Then the same argument as before shows that if α is closer to vthan β, the same must be true of ϕ(α) and ϕ(β) (and this even works if β is a boundaryvertex).

This establishes theorem 4.4.1 for T 6∼= An.

64

4.5 The Coherence Theorem

We finish this section with a proof of theorem 3.3.1.

Theorem 3.3.1: Let T be a tree, and O a pre-cyclic structure on PT . If O isA3−coherent and S4−coherent, then O is a cyclic structure.

Proof. Step 1: Use A3−coherence to construct an orientation µ on T .

To begin, take vertices v and w of degree 6= 2, such that v and w can be connected by apath in which each intermediate vertex has degree 2. The path connecting these vertices isa subtree S isomorphic to An, and one has a correspondence:

p = (Anq� S

i↪→ T )

Where q is an isomorphism, q(v) = 1 and q(w) = n.

To fix terminology, call such a subtree a chain subtree, and the associated correspon-dence a chain correspondence. Notice that for each edge e there is a unique (up to

equivalence) chain correspondence p = (Anq� S

i↪→ T ) with e ∈ E(S).

Now, let u be a vertex between v and w, and write q(u) = j, where j is an integerbetween 1 and n. Then p gives a correspondence of pointed trees (An, j)→ (T, u), so by thefunctoriality of the reflection isomorphisms there is a commutative diagram

PT PT

PAn PAn

ωu

ipωj

ip

Write KAn = {e0, e1, . . . , en}, where ei is the edge joining vertex i to i+ 1 for 1 ≤ i < n,and e0, en are the terminal vertices. Taking transpositions centered on internal vertices allowsus to obtain any permutation of KAn fixing Vt(An) = {e0, en}. As a result, if η ∈ S(KAn)is any automorphism fixing {e0, en}, we can find η ∈ Aut(PAn) and ω ∈ Aut(PT ) such thatthere is a commutative diagram:

PT PT

PAn PAn

ω

ipη

ip

65

We know O is A3−coherent, so by proposition 4.4, i∗pO is a cyclic structure, and c =∆(ip∗O) is a cyclic order on KAn . There exists a bijection η ∈ S(KAn) fixing e0 and en suchthat c is induced by the total order e0 < η(e1) < · · · < η(ek) < en < η(en−1) < η(en−2) <· · · < η(ek+1).

For this η, pick η and ω as in the above commutative diagram.

If we let λ denote the orientation of An where e1, . . . , ek are pointing to the right andek+1, . . . , en−1 are pointing to the left, by proposition 3.6.2 the cyclic order ∆(Oλ) is in-duced by the total order e0 < e1 < · · · < ek < en < en−1 < · · · < ek+1. So we haveOλ = η∗i∗pO = i∗pω

∗O. Since ω is an automorphism of PT , O is a cyclic structure iff ω∗O is,so we can replace O with ω∗O. So we can assume i∗pO = Oλ for some orientation λ.

Repeating this process for each chain correspondence, we can assume (after poten-tially applying an automorphism of PT ) that for each chain correspondence p : An → T ,p∗O = Oλ(p) for some orientation λ(p). There is a unique orientation µ on T such thatp : (An, λ(p))→ (T, µ) is an orientation-preserving correspondence. This tells us i∗pO = i∗pOµfor any chain correspondence p.

Step 2: Our second step is to use S4−coherence to show O = Oµ.

Pick a correspondence p = (A2

q� S

i↪→ T ). We are done if we can show i∗pO = i∗pOµ.

Step 2a: We consider a special case. If p factors through a chain correspondence, thenwe are done. If not, assume there exists a chain subtree R such that q−1(2) ⊆ R, and q−1(1)contains one of the endpoints v of R with the degree of v greater than 2.

We will construct a correspondence q = (Snq′

� Wi′

↪→ T ) with [p] ≥ [q]:

• W = S ∪ n(v), where n(v) denotes the set of neighbors of the vertex v ∈ V (T ).

• We define the fiber of the central vertex, q′−1(0), to be R ∩ q−1(1).

• Define the fiber q′−1(1) to be q−1(2).

• The rest of the fibers are the connected components of W\V (since v has degree > 2there are at least two such connected components)

Let p0 : A2 → Sn denote the correspondence satisfying q ◦ p0 = p.

The pullback i∗qO is an S4−coherent cyclic structure on Sn. By proposition 3.7.1 itis a cyclic structure and equal to Oµ′ for some orientation µ′ on Sn- so i∗pO = i∗p0Oµ′ . We

66

are done, then (with this case), if we can show q∗µ agrees with µ′ on the edge joining 0 and 1.

To accomplish this, let p1 : A2 → Sn denote the correspondence whose fiber over 1 is thecentral vertex 0, and whose fiber over 2 is the vertex 1.

Since q ◦ p1 factors though the chain correspondence (An � R ↪→ T ), i∗q◦p1O = i∗q◦p1Oµ =i∗p1Oq∗µ. We also have i∗q◦p1O = i∗p1iq∗O = ip∗1Oµ′ . That implies q∗µ agrees with µ′ on theedge joining 0 and 1.

This strategy is illustrated in Figure 4.6. The figure shows the construction of q and p1

from p, and the edge e is the one whose direction is crucial.

e

v

p

e

v

q

e

v

q ◦ p1

Figure 4.6: The Construction of q and p1

Step 2b: Finally, we consider the general case of a correspondence p = (A2

q� S

i↪→ T ).

The edge connecting q−1(1) and q−1(2) is contained in a chain subtree R, call the end-points of R v and w. We can assume v ∈ q−1(1) and w ∈ q−1(2), and both v and w havedegree greater than 2 (if not, we are in one of the previous cases).

We can now repeat essentially the same argument as the previous case. Explicitly: we

again construct a correspondence q = (Snq′

� Wi′

↪→ T ):

• W = S ∪ n(v), where n(v) denotes the set of neighbors of the vertex v ∈ V (T ).

• We define the fiber of the central vertex, q′−1(0), to be R ∩ q−1(1).

• Define the fiber q′−1(1) to be q−1(2).

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• The rest of the fibers are the connected components of W\V (since v has degree > 2there are at least two such connected components)

Let p0 : A2 → Sn denote the correspondence satisfying q ◦ p0 = p.

i∗qO = Oµ′ for some orientation µ′ on Sn, we are done if we can show q∗µ agrees with µ′

on the edge joining 0 and 1. As before, let p1 : A2 → Sn denote the correspondence whosefiber over 1 is the central vertex 0, and whose fiber over 2 is the vertex 1:

The difference in this case is that q ◦ p1 does not necessarily factor through a chaincorrespondence- however, it is of the form covered in the previous case, so we again havei∗q◦p1O = iq◦p∗Oµ. See Figure 4.7 for an illustration of this case.

e

v

p

e

v

q

e

v

q ◦ p1

Figure 4.7: The construction in Case 2b

The rest of the argument is identical.

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Chapter 5

A Sheaf of Categories

5.1 Dg-Categories

This section is centered around the representation theory of C-Arb singularities with cyclicstructures. Of central importance is the notion of a dg-category. Standard references are [3],[14]. Quotients of dg-categories are explained in [7]. The model category structure and theunderlying infinity category is discussed in [34], [8].

Throughtout this paper, we fix an algebraically closed field k of characteristic zero. Allvector spaces will be assumed to be over k.

Definition 5.1.1. A dg-category C is a category enriched over chain complexes. Explicitly,for objects X, Y ∈ C, Hom(X, Y ) has the structure of a chain complex (where the differentialincreases degree). If f ∈ Homk(X, Y ), we say f is a homogeneous element, and write |f | = k.The following axioms are satisfied:

(i) For objects X, Y, Z: The composition morphism:

Hom(Y, Z)× Hom(X, Y )→ Hom(X,Z)

Is bilinear.

(ii) For homogeneous functions f ∈ Hom(X, Y ), g ∈ Hom(Y, Z), |gf | = |g|+ |f |.

(iii) With f and g as in (ii), we have the graded Leibniz rule:

d(fg) = (df)g + (−1)|f |f(dg)

69

Definition 5.1.2. If C and D are dg-categories, a dg-functor is a functor F : C → D suchthat, for any objects X, Y ∈ C, the map:

F : HomC(X, Y )→ HomD(FX,FY )

Is a map of chain complexes.

Definition 5.1.3. The graded homotopy category H•C of a dg-category C is the cate-gory whose objects are the objects of C, and for X, Y ∈ Ob(C), we have HomH•C(X, Y ) =H•(HomC(X, Y )). The homotopy category H0C is the degree-zero portion of the gradedhomotopy category.

H•(HomC(X, Y )) is a graded k−linear category. An isomorphism in such a category isa degree zero invertible map. A morphism in C whose image in H•C is an isomorphism isa homotopy equivalence. Two objects in C are homotopic if they are isomorphic in H•C.

Note that a dg-functor F : C → D induces H•F : H•C → H•D.

Definition 5.1.4. A dg-functor F : C → D is:

• Quasi-fully faithful if H•F is fully faithful (it induces isomorphisms on hom sets).

• Quasi-essentially surjective if H•F is essentially surjective (Any object in H•Dis isomorphic to H•F (X) for some X ∈ Ob(H•C)).

• A quasi-equivalence if H•F is an equivalence of categories, that is, it is fully faithfuland essentially surjective.

Definition 5.1.5. Let C be a dg-category.

• If A ∈ Ob(C), an k-tranlastion of A is an object B such that there exist closed mapsf : A→ B and g : B → A with fg = IdB, gf = IdA, |f | = −k and |g| = k.

• If f : A→ B is a closed degree-zero morphism, a cone of f is an object C, with maps:

C

A B

πA πB

iA

f

iB

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With |πB| = |iB| = 0, |πA| = 1, |iA| = −1, πBiB = IdB, πAiA = IdA, iBπB + iAπA =IdC, πAiB = πBiA = diB = dπA = 0, diA = iBf , dπB = −fπA.

• A zero object Z ∈ Ob(C) is an object with IdZ = 0Z.

• A dg category is strongly pretriangulated if it has a zero object, a k-translation ofany object, and a cone of any closed degree zero morphism. Note that these objects areunique up to unique isomorphism, and are preserved by dg functors.

Definition 5.1.6. If C is strongly pretriangulated, a sequence of closed degree zero maps:

Af→ B

g→ C

is exact if the map g ◦ πB : Cone(f) → C is a homotopy equivalence, equivalently if iB[1] ◦f [1] : A[1]→ Cone(g) is a homotopy equivalence. If F,G,H : C → D are dg-functors and Dis strongly pretriangulated, a sequence of natural transformations:

Fη−→ G

φ−→ H

is called exact for any object X ∈ Ob(C), the sequence:

FXηX−→ GX

φX−→ HX

is exact.

If C is strongly pretriangulated, then the homotopy category H0C is a triangulated cate-gory whose distinguished triangles are the image of exact sequences.

To any dg-category C there exists a strongly pretriangulated dg-category Cpre−tr calledthe pretriangulated hull of C and a canonical fully faithful functor C ↪→ Cpre−tr (see [5]).The pretriangulated hull satisfies a universal property:

Lemma 5.1.1. If D is a strongly pretriangulated category and F : C → D is a functor, thereis a unique (up to unique natural isomorphism) extension of F to a functor Cpre−tr → D.

We turn to the 2-categorical structure underlying dg-categories.

Definition 5.1.7. Let F,G : C → D be dg-functors between dg-categories. A natural trans-formation η : F → G is:

71

(i) An isomorphism if ηX : FX → GX is an isomorphism for all X ∈ Ob(C).

(ii) A quasi-isomorphism if ηX is a homotopy equivalence for all X ∈ Ob(C).

We will often make use of the following proposition in order to ‘lift’ properties of a dg-category C to properties of Cpre−tr.

Lemma 5.1.2. Let C be a dg-category, D be a strongly pre-triangulated dg-category, F,G :Cpre−tr → D be dg-functors. Let FC, GC denote the restrictions of F and G to C. Then ifηC : FC → GC is a a natural transformation, there is a unique extension to η : F → G.Furthermore:

• If ηC is an isomorphism, so is η.

• If ηC is a quasi-isomorphism, so is η.

• If H : Cpre−tr → D is another dg-functor, and:

FCηC−→ GC

φC−→ HC

Is an exact sequence of natural transformations, the unique extensions:

Fη−→ G

φ−→ H

Also give an exact sequence.

5.2 Sheaves of dg-Categories

In this paper we will be dealing with sheaves of dg-categories on C-Arb singularities PT . Bya sheaf, we mean a sheaf in the∞−category sense- for details, see appendix B. The followingdefinition will be important:

Definition 5.2.1. Let C be an ordinary category, and let dg-Cat denote the ∞−category ofdg-categories (over k). A 2-functor Q : C → dg-Cat consists of the following data:

(i) For each object X ∈ Ob(C), a dg-category Q(X).

(ii) For each morphism f : X → Y in C, a dg-functor cf : Q(X)→ Q(Y ).

(iii) For each pair of morphisms f, g such that the composition fg exists, a quasi-isomorphismηf,g : cf ◦ cg → cfg.

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(iv) Such that, for each triple f, g, h of morphisms such that the composition fgh exists,the following diagram commutes:

cf ◦ cg ◦ ch cfg ◦ ch

cf ◦ cgh cfgh

ηf,g � 1ch

1cf � ηg,h ηfg,h

ηf,gh

Where � refers to the horizontal composition of functors.

Remark 5.2.1. By lemma 5.1.2, a 2-functor Q0 : C → dg-Cat can be extended to a 2-functorQ : C → dg-Cat whose image lies in the subcategory of strongly pretriangulated dg-categories,by setting Q(X) = Q0(X)pre−tr, and extending the rest of the data.

Now, let C be a category in which every morphism is monic. For A ∈ Ob(C), recall theconstruction of the poset PA = Pos(C/A) from Chapter 2. Then a 2-functor Q : Cop → dg-Catdetermines a sheaf of dg-categories QA on PA for A ∈ Ob(C), where:

(i) The stalk QA,[f ] of QA at f : B → A is Q(B).

(ii) When [f ] ≥ [g], meaning f = gh, the restriction QA,[g] → QA,[f ] is given by ch.

Definition 5.2.2. Let C be a category, and Q,Q′ 2-functors C → dg-Cat, with associateddata c, η, resp. c′, η′. A left morphism F : Q → Q′ consists of:

(i) For each X ∈ Ob(C), a dg-functor F(X) : Q(X)→ Q′(X).

(ii) For each morphism f : Y → X, a quasi-isomorphism γf : c′f ◦ F(Y )→ F(X) ◦ cf .

(iii) Such that, for morphisms f : Y → X and g : Z → Y , the following diagram commutes:

c′f ◦ c′g ◦ F(X) c′f ◦ F(Y ) ◦ cg F(Z) ◦ cf ◦ cg

c′fg ◦ F(X) F(Z) ◦ cfg

1c′f � γg γf � 1g

η′f,g � 1F(X) 1F(Z) � ηf,g

γfg

73

A right morphism is defined similarly, except γf : F(X) ◦ cf → c′f ◦ F(Y ), and in thecommutative diagram in (iii), the horizontal arrows point in the other direction.

A left or right morphism between 2-functors Q → Q′ determines a morphism QA → Q′Aof sheaves of dg-categories on PA for any A ∈ Ob(C)- see appendix B for a detailed demon-stration of this fact.

There is also a notion of when left/right morphisms are quasi-inverses, and thus inducequasi-equivalences on sheaves. In a later section, when we need this, we will outline explicitlywhat data we need. For more details see appendix B.

5.3 Representations of Directed Trees

We describe the construction of a sheaf of dg-categories on the cyclic C-Arb singularity(PT ,Oµ), where µ is an orientation on a tree T , by defining a 2-functor DirTreeop → dg-Cat.For other perspectives on this construction, see [20] or [28]1.

To begin, we define a dg category associated to the tree (T, µ).

Definition 5.3.1. Let (T, µ) be a directed tree. The category Rep0(T, µ) is a k−linearcategory generated where:

• For each v ∈ V (T ), there is an object Pv.

• If there exists a directed path from v to w in (T, µ), there is a distinguished nonzeromorphism ew,v : Pw → Pv, such that Hom(Pw, Pv) = Span(ew,v).

• ev,w ◦ eu,v = eu,w.

• If there does not exist a directed path from v to w in (T, µ), then Hom(Pw, Pv) = 0.

This can be thought of as a dg-category where all ew,v are in degree zero and all differentialsare zero. The category Rep(T, µ) := (Rep0(T, µ))pre−tr.

We make some remarks regarding the theory of quiver representations- see [6] for proofsand a detailed exposition. First, one has the classical notion of the abelian category Mod(T, µ)of finite-dimensional representations of (T, µ). Objects of this category assign a finite-dimensional k−vector space Xv to each v ∈ V (T ), and a morphism αXe : Xt(e) → Xh(e)

1While both of those papers deal explicitly with rooted trees, there are no essential differences

74

to each edge e ∈ E(T ). Morphisms X → Y of this category are a collection of mapsfv : Xv → Yv satisfying αYe ft(e) = fh(e)α

Xe for all e ∈ E(T ).

For each v ∈ V (T ), there is a corresponding projective object Pv ∈ Mod(T, µ), whichassigns:

(Pv)u =

{k There exists a directed path from v to u0 Else

And all arrows αe are the identity when both endpoints are assigned k, and zero otherwise.

One can see easily that the category Rep0(T, µ) is isomorphic to the subcategory ofMod(T, µ) on the objects {Pv}v∈V (T ). Furthermore ([6]), any object in Mod(T, µ) has afinite projective resolution whose objects are direct sums of the Pv- hence Rep(T, µ) is adg-enhancement of the bounded derived category of Mod(T, µ). This category is sometimescalled the ‘bounded derived dg category’, and can also be presented as the dg-quotient ([7])of the dg-category of finite-dimensional chain complexes of (T, µ)−modules by the subcate-gory of acyclic complexes.

Returning to the goal at hand: We want to define a 2-functor Q : DirTreeop → dg-Cat.

Definition 5.3.2. We define Q0 : DirTreeop → dg-Cat as follows:

(i) For a directed tree (T, µ), we set Q0(T, µ) = Rep0(T, µ).

(ii) For a correspondence p = (Rq� S

i↪→ T ), define the functor c0

p : Rep0(T, µ) →Rep0(R, λ), where λ = p∗µ, by:

• c0p(Pv) =

{Pq(v) v ∈ V (S)0 Else

• c0p(ev,w) =

{eq(v),q(w) v, w ∈ V (S)0 Else

(iii) Note that c0p′ ◦ c0

p = c0p◦p′, hence we can set η0

p,p′ to be the identity transformation.

We now define Q to be the pretriangulated hull of Q0, as in remark 5.2.1.

To verify that c0p is well-defined, first notice that by the definition of λ = p∗µ, if there is

a path from w to v in (T, µ) there will be a path from q(w) to q(v) in (R, λ). It also bearsmentioning that if eu,w = ev,w ◦ eu,v and u,w ∈ V (S), then we also must have v ∈ V (S),since no directed path can leave S and reenter it. (It would be a problem if it were not, sincethen we would have c0

p(eu,v) = c0p(ev,w) = 0 but c0

p(eu,w) = eq(u),q(w) 6= 0).

75

In the following sections we will analyze some properties of this sheaf.

5.4 The Trivalent Vertex

The simplest case to investigate is the trivalent vertex: PA2 . A2 denotes the tree with twovertices, labelled 1 and 2, and a single edge connecting them. We let µ denote the orientationwith the arrow pointing from 1 to 2 (Figure 5.1).

1 2

Figure 5.1: The directed tree (A2, µ)

We use the notation (1), (2), (12) for the three correspondences ?→ PA2 (see, for example,Figure 2.4). We have the following classical fact about Rep(A2, µ):

Lemma 5.4.1. The indecomposable objects in Rep(A2, µ) are (up to shifts and homotopy):

• P1

• P2

• P2[1]e2,1−→ P1

These correspond to the indecomposables k → k, k → 0, and 0 → k, respectively, in

Mod(A2, µ). We are using the notation of twisted complexes to describe P2[1]e2,1−→ P1

∼=Cone(e2,1) ∈ Rep(A2, µ). See definition 5.7.1 later in this chapter.

We can compute the restrictions cpX, for p = (1), (2), (12), this is given in Table 5.1,and illustrated geometrically (with the cyclic order Oµ agreeing with the counterclockwiseorder) in Figure 5.2. The figure should be interpreted as showing the ‘support’ of theindecomposables, i.e. indicating where the restrictions are nonzero.

p P1 P2 P2[1]→ P1

(1) k 0 k(2) 0 k k[1](12) k k 0

Table 5.1: The Restrictions of Indecomposable Elements of Rep(A2, µ).

76

(1)

(2)

(12)

P2[1]→ P1

P2

P1

Figure 5.2: Geometric Picture of the Indecomposable Representations

Note that:

H•(Hom(P2, P1)) ∼= H•(Hom(P1, P2[1]→ P1)) ∼= H•(P2[1]→ P1, P2[1]) ∼= k

While:

H•(Hom(P1, P2)) ∼= H•(Hom(P2, P2[1]→ P1)) ∼= H•(P2[1]→ P1, P1) ∼= 0

As suggested by this computation, the sheaf Qµ encodes the data of the cyclic structure Oµ.That’s the content of the following proposition:

Proposition 5.4.1. Let ϕ ∈ Aut(PA2). Then ϕ∗Qµ ∼= Qµ if and only if ϕ ∈ S(KA2) is aneven permutation (that is, if and only if ϕ preserves the cyclic order Oµ).

Proof. We prove ‘only if’ completely. The proof of ‘if’ is a consequence of the formalism ofthe remainder of the section, but we will sketch the argument here as well.

For ‘Only If’:, assume ϕ is the transposition (e0e2). The other cases are similar (orfollow from ‘if’).

For any object X ∈ Ob(Rep(A2, µ)), we get a sheaf HomQµ(X,X) of chain complexesover k (see appendix B for details). This gives a presheaf of graded algebras E•(X) =H•(HomQµ(X,X)).

Let X = P1 ⊕ P2. If ϕ∗Qµ were quasi-equivalent to Qµ, there would be an objectY ∈ Rep(A2, µ) such that E•(Y ) ∼= ϕ∗E•(X). Looking at the stalks of E•(X) we get:

77

(a 0b c

)0 (

a 00 0

)(0 00 c

) (2)(1)

(We do not need to consider the stalk at (12)). We know Y , if it exists is homotopic toa direct sum of indecomposables. By the stalks E•(Y )(1)

∼= E•(Y )(2)∼= k, using Table 5.1 we

see Y is either a shift of P2[1]→ P1, or a shift of P1 ⊕ P2[n]. The stalk at {0} eliminates allbut a shift of P1 ⊕ P2,i.e. Y is homotopic to X.

But we do not have ϕ∗E•(X) ∼= E•(X): If we take products of ideals in E•(X)0 we getker(c(1))ker(c(2)) = 0 but ker(c(2))ker(c(1)) 6= 0, implying an asymmetry between c(1) and c(2).

For ‘If’ (Sketch): Assume ϕ is the permutation (1) 7→ (2) 7→ (12) 7→ (1). A quasi-equivalence Q → ϕ∗Q can be specified by a morphism of 2-functors, that is, a quasi-equivalence F0 : Rep(A2, µ) → Rep(A2, µ), and for each correspondence p : ? → A2 aquasi-equivalence Fp : Rep(?, ·) → Rep(?, ·), such that the following diagram commutes upto homotopy:

Rep(A2, µ) Rep(A2, µ)

Rep(?, ·) Rep(?, ·)

cp

F0

Fp

cϕ(p)

We define F 00

: Rep0(A2, µ) → Rep(A2, µ) by F 00(P1) = P2[1] −→ P1, and F 0

0(P2) = P1,

and let F0 : Rep(A2, µ)→ Rep(A2, µ) denote the extension guaranteed by lemma 5.1.2. Wealso have: F(12) = F(2) is the identity, while F(1) is the shift operator X 7→ X[1].

The reader can check that these maps do indeed satisfy the required properties.

5.5 Reflection Functors

Now, recall the definition of the category of pointed trees DirTree±, whose objects aretriples (T, µ, v), with v ∈ V±(T, µ).

Our goal of this section will be to ‘categorify’ the reflection functor R± : DirTree± →DirTree∓ described in Chapter 4. The machinery here is effectively an extention of theclassical BGP reflection functors- see [2]- to this sheaf of categories.

78

Because of a small sign issue, we need to slightly modify the category:

Definition 5.5.1. The category DirTree±,s of signed ± pointed directed trees is thecategory whose objects are quadruples (T, µ, v, s), where s is a sign function s : n(v) →{−1,+1}, where n(v) denotes the neighbors of v. When v = 0 is the null vertex, we setn(v) = ∅. When deg(v) = 2, we require that s assigns opposite signs to the two neighbors ofv.

Suppose p = (Rq� S

i↪→ T ) is a correspondence and v ∈ V+(T ), then there is a functorial

injection j : n(w) → n(v). Specifically, if v ∈ V (S) with q(v) = w, and q−1(w) = {v}, thenj is the unique function with q ◦ j = id, and in all other cases n(w) = ∅. So a sign functions : n(v) → {−,+} pulls back to p∗s = s ◦ j : n(w) → {+,−}- this defines morphisms inDirTree+,s as those which pullback sign functions correctly.

Finally, we extend the functor R by declaring that it reverses the sign function, i.e.R(T, µ, v, s) = (T, rvµ, v, s), where s(u) = −s(u).

By a ’categorification’ of R± involves the following data:

(i) For an object (T, µ, v, s) ∈ DirTree±,s, we will construct a functor Γ±v : Rep(T, µ) →Rep(T, rvµ). Note that Γ±v will depend on µ and s, but we omit these from the notationto avoid clutter.

(ii) For a correspondence p : (R, λ, w, s′) → (T, µ, v, s), with v ∈ V+(T, µ), we will con-struct a quasi-isomorphism:

γ+p : crvp ◦ Γ+

v → Γ+w ◦ cp

Satisfying, for p : (R, λ, w, s′) → (T, µ, v, s) and p′ : (R′, λ′, w′, s′′) → (R, λ, w, s′), thecommutativity of:

crwp′ ◦ crvp ◦ Γ+v crwp′ ◦ Γ+

w ◦ cp Γ+w′ ◦ cp′ ◦ cp

crv(p◦p′) ◦ Γ+v Γ+

w′ ◦ cp◦p′

1crvp′ � γ+p γ+

p′ � 1cp

ηrvp,rwp′ � 1Γ+v

1Γ+w′� ηp,p′

γ+p◦p′

79

(iii) When v ∈ V−(T, µ), a quasi-isomorphism:

γ−p : Γ−w ◦ cp → crvp ◦ Γ−v

Satisfying, for p : (R, λ, w, s′) → (T, µ, v, s) and p′ : (R′, λ′, w′, s′′) → (R, λ, w, s′), thecommutativity of:

crwp′ ◦ crvp ◦ Γ−v crwp′ ◦ Γ−w ◦ cp Γ−w′ ◦ cp′ ◦ cp

crv(p◦p′) ◦ Γ−v Γ−w′ ◦ cp◦p′

1crvp′ � γ−p γ−p′ � 1cp

ηrvp,rwp′ � 1Γ−v1Γ−

w′� ηp,p′

γ−p◦p′

(iv) A quasi-isomorphism:ε+v : Γ−v ◦ Γ+

v → Id

Satisfying, for p : (R, λ, w, s′)→ (T, µ, v, s):

Γ−w ◦ crvp ◦ Γ+v cp ◦ Γ−v ◦ Γ+

v

Γ−w ◦ Γ+w ◦ cp cp

γ−p � 1Γ+v

1Γ−w� γ+

p 1cp � ε+v

ε+w � 1cp

(v) A quasi-isomorphism:ε−v : Id→ Γ+

v ◦ Γ−v

Satisfying, for p : (R, λ, w, s′)→ (T, µ, v, s):

cp cp ◦ Γ+v ◦ Γ−v

Γ+w ◦ Γ−w ◦ cp Γ+

w ◦ cp ◦ Γ−v

1cp � ε−v

ε−w ◦ 1cp γp+ � 1Γ−v

1Γ+w� γ−p

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If we temporarily let π± : DirTree±,s → DirTree denote the forgetful functors, the bul-let points (i), (ii) define a left 2-functor from Q◦ π+ to Q◦R+ ◦ π+, and hence a morphismof sheaves Qµ → ω∗vQrvµ on the poset PT , whenever (T, µ, v, s) ∈ Ob(DirTree+,s).

Similarly, bullet point (iii) gives a morphism of sheaves Qµ → ω∗vQrvµ when (T, µ, v, s) ∈Ob(DirTree−,s).

The bullet points (iv) and (v) show that these morphisms are quasi-inverses, hence areeach quasi-equivalences. Hence as a result of this construction we will have:

Proposition 5.5.1. Let (T, µ) be a directed tree, v ∈ V (T ) be a source or sink of degree ≤ 2,and ωv : (PT ,Oµ)

∼→ (PT ,Orvµ) be the directed reflection isomorphism. Then there exists aquasi-equivalence:

Qµ∼→ ω∗vQrvµ

We will delay this construction for a bit so that we can emphasize the most importantresult of this chapter.

5.6 The Cyclic Structure Theorem

Assuming proposition 5.5.1, we can show:

Theorem 5.6.1. (Cyclic Structure Theorem) Let (T, µ) and (T ′, µ′) be directed trees,and suppose ϕ : PT

∼→ PT ′ is an isomorphism. Then ϕ∗Qµ′ is quasi-equivalent to Qµ if andonly if ϕ∗Oµ′ = Oµ.

Proof. By 4.4.1, if ϕ∗Oµ′ = Oµ, then ϕ can be written as a product of reflection isomor-phisms and isomorphisms of the form iσ, where σ is an isomorphism of directed trees. Thenproposition 5.5.1 implies ϕ∗Qµ′ is quasi-equivalent to Qµ.

On the other hand, assume ϕ∗Qµ′ is quasi-equivalent to Qµ. Pick a codimension-one cell[p] ∈ PT , where p : A2 → T is a correspondence, and let [p′] = ϕ([p]). We can arrange it sop∗µ = p′∗µ′ = λ. Then we have a commutative diagram:

PT PT ′

PA2 PA2

ip ip′

ϕ

ϕ0

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By assumption, ϕ∗0Qλ ∼= ϕ0i∗p′Qµ′ ∼= i∗pϕ

∗Qµ′ ∼= i∗pQµ ∼= Qλ. So proposition 5.4.1 impliesϕ∗0Oλ = Oλ. Since the codimension-one cell was arbitrary, ϕ∗Oµ′ = Oµ.

In particuar, if (P ,O) is a cyclic C-Arb singularity, then we can define a sheaf of dg-categories Q on P by picking a directed tree (T, µ), an isomorphism ϕ : P ∼→ PT withϕ∗Oµ = O, and setting Q = ϕ∗Qµ. By the Cyclic Structure Theorem, this sheaf of cate-gories doesn’t depend on the choice of (T, µ) and ϕ- it is determined up to quasi-equivalenceby O.

Remark 5.6.1. While a cyclic structure determines a sheaf of categories up to quasi-equivalence, it does not determine a sheaf up to natural quasi-equivalence. The quasi-equivalence can be sensitive to the order in which the reflection functors are applied.

As a concrete example of this phenomenon, consider the tree (A2, µ), where µ pointsfrom vertex 1 to vertex 2. The isomorphism r = ω1ω2 ∈ Aut(PA2) satisfies r∗Oµ = Oµ, andthe above construction gives a quasi-equivalence F : Qµ

∼→ r∗Qµ, which coincides with thefunctor described in the proof of 5.4.1. Then r3 is the identity on PA2, but F 3 is the shiftfunctor X 7→ X[1]. This grading ambiguity thwarts naive attempts to define a canonicalsheaf associated to a cyclic structure, and is part of the motivation for the definition of agraded structure in Chapter 6.

The remainder of this chapter will be a construction of the reflection functors and relateddata, as explained in section 5.5. It can be skipped without consequence for the rest of thepaper.

5.7 Construction of the Reflection Functors

We must proceed through steps (i)− (v) described in section 5.5. As a preliminary step, weintroduce the notion of twisted complexes as a model for the pretriangulated hull, which wewill use for computational purposes.

Twisted Complexes

We discuss twisted complexes as a model for Cpre−tr when C is a k−linear category- that is,a dg-category in which all morphisms are in degree zero and all differentials are zero. For amore general notion, see (ref). Bondal-Kapranov.

Definition 5.7.1. Let C be a k−linear category. C⊕ is the graded k−linear category whichcontains formal shifts and finite direct sums of objects in C (including the zero object). A

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twisted complex over C is a pair (C, d), where C ∈ Ob(C⊕) and d : C → C is a degree-onemorphism satisfying d2 = 0. It can also be written as a sequence:

· · · di−2−→ Ci−1di−1−→ Ci

di−→ Ci+1di+1−→ · · ·

Where each Ci ∈ C⊕ is an object concentrated in degree i, each di is a degree-one morphism,di+1di = 0, and only finitely many Ci are nonzero.

The category of twisted complexes is a dg-category in a straightforward way, and it ad-mits a natural embedding of C. We have:

Lemma 5.7.1. The category of twisted complexes is a model for the dg-category Cpre−tr. Inparticular, shifts and cones of twisted complexes can be computed in the usual way.

Suppose C and D are k−linear categories, and we have a functor F : C → Dpre−tr. If wetake a complex in Cpre−tr:

C = · · · fi−2−→ Ci−1fi−1−→ Ci

fi−→ Ci+1fi+1−→ · · ·

The naive extension of F to C is a double complex. Write:

F (Ci) = · · · hi,j−2−→ Di,j−1hi,j−1−→ Di,j

hi,j−→ Di,j+1hi,j+1−→ · · ·

Where the Di,j ∈ D⊕ are concentrated in degree j. Then applying F to the maps fi yieldsa family of degree-one maps vi,j : Di,j → Di+1,j+1. We get an object:

......

· · · Di,j Di,j+1 · · ·

· · · Di+1,j+1 Di+1,j+2 · · ·

......

hi,j−1 hi,j hi,j+1

hi+1,j−1 hi+1,j hi+1,j+1

vi−1,j−1

vi,j

vi+1,j+1

vi−1,j

vi,j+1

vi+1,j+2

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Where we have v2 = h2 = 0 and vh = hv. Such an object is a double complex.

Definition 5.7.2. The totalization of a double complex as in the above diagram is anobject T ∈ Dpre−tr:

T = · · · dj−2−→ Tj−1dj−1−→ Tj

dj−→ Tj+1hj+1−→ · · ·

Where:Tj = ⊕iDi,j

dj =∑i

vi,j − (−1)ihi,j

In this way, we can compute the extension of F to a functor Cpre−tr → Dpre−tr.

The Functors Γ±

We now return to our goal at hand- defining the objects and verifying the relations outlinedin steps (i)− (v) earlier in this chapter.

Definition 5.7.3. Let (T, µ, v, s) ∈ DirTree±,s We define a functor Γ±v : Rep(T, µ) →Rep(T, rvµ) by first defining Γ0,±

v : Rep0(T, µ) → Rep(T, rvµ), and extending it as in lemma5.1.2. Let {Pw}w∈V (T ) denote the objects in Rep0(T, µ), and {Pw}w∈V (T ) the correspondingobjects in Rep0(T, rvµ). Also, let ew,u, ew,u denote the morphisms.

• Set Γ0,±v (Pw) = Pw if w 6= v, and Γ0,±

v (eu,w) = eu,w for u,w 6= v.

• If v = 0, we’ve completely defined Γ0,±v .

• If v is a source, then:

Γ0,+v (Pv) = Pv[1]

d→⊕w∈n(v)

Pw

d =∑w∈n(v)

s(w) · iwev,w

Where iw is the inclusion of the summand Pw, and s is the sign function. (Note that vis a sink with respect to rvµ). If n(v) = ∅ (i.e. T has only one vertex) we let the emptydirect sum equal zero. On morphisms, first Γ0,+

v (ev,v) is the identity on Γ0,+(Pv). Nowsuppose there exists a directed path from v to u in (T, µ), and u 6= v. There is a uniquew ∈ n(v) such that there is a directed path from w to u. Then:

Γ0,+v (eu,v) = iweu,w

84

• If v is a sink:

Γ0,−v (Pv) =

⊕w∈n(v)

Pwd→ Pv[−1]

d =∑w∈n(v)

s(w) · ew,vπw

Where πw is the projection onto the Pw summand. Again, the empty sum is zero.On morphisms, again Γ0,−

v (ev,v) is the identity on Γ0,−v (Pv). Then suppose there is a

directed path from u to v in (T, µ), and u 6= v. There is a unique w ∈ n(v) such thatthere is a directed path from u to w. Then:

Γ0,−(ev,u) = ew,uπw

It is clear that these are well-defined and respect composition.

Definitions of γ±p

We’ve finished part (i) of the program outlined in the beginning of the section. We nowexplain the constructions of γ±p .

First assume v ∈ V+(T ). We will construct, for p : (R, λ, w, s′)→ (T, µ, v, s):

γ0,+p : c0

rvp ◦ Γ0,+v

∼→ Γ0,+w ◦ c0

p

Which is a quasi-isomorphism of functors. Then, by lemma 5.1.2, there is a unique

extension of γ0,+p to the desired quasi-isomorphism γ+

p . Let p = (Rq� S

i↪→ T ). There are

a few easy cases, where the definition of γ0 is clear- i.e., both sides are the same object (upto, perhaps, a re-indexing of a direct sum). These include:

• If, u 6= v, u ∈ V (S) we have c0rvp ◦ Γ0,+

v (Pu) = Γ0,+w ◦ c0

p(Pu) = Pq(u).

• If, on the other hand, u 6= v and u /∈ V (S), c0rvp ◦ Γ0,+

v (Pu) = Γ0,+w ◦ c0

p(Pu) = 0

We remark that at this point all that’s left is to define:

γ0,+p (Pv) : c0

rvp ◦ Γ0,+v (Pv)→ Γ0,+

w ◦ c0p(Pv) (5.1)

Continuing with the easy cases:

• If v nor any of its neighbors are in V (S), both sides of (5.1) are zero.

85

• When q(v) = w and q−1(w) = {v}, both sides of (5.1) are:

Pw[1]d→⊕u∈n(w)

Pu

d = Σu∈n(w)s′(u) · iuew,u

We now examine the remaining three nontrivial cases in detail. We remark that once wedefine γ0,+

p on objects, we have to verify that it is, in fact, a natural transformation.

Case I: v has one neighbor in V (S), and either v /∈ V (S) or v ∈ V (S) has the sameimage as its neighbor under q.

Near v, the correspondence looks as follows:

x v

q(x)

vT

R

q R

First suppose the left-hand-side is illustrating p, so v ∈ V (S). Then:

Γ0,+

0◦ c0

p(Pv) = c0rvp ◦ Γ0,+

v (Pv) = Pq(x)

This is again an easy case. However, if the right-hand-side is illustrating p, so v /∈ V (S):

Γ0,+

0◦ c0

p(Pv) = 0

c0rvp ◦ Γ0,+

v (Pv) = Pq(x)[1]d→ Pq(x)

d = ±eq(x),q(x)

The sign depending on the sign function s. We define γ0,+(Pv) = (Pq(x)[1]d→ Pq(x))

0−→ 0,which is an isomorphism in the homotopy category H•(Rep(R, λ)).

Now, suppose y ∈ V (S), and there exists a directed path from x to y in S (and hencealso from v to y). γ0,+

p being a natural transformation requires the commutativity of thediagram:

c0rvp ◦ Γ0,+

v (Py) c0rvp ◦ Γ0,+

(T,v,µ)(Pv)

Γ0,+

0◦ c0

p(Py) Γ0,+

0◦ c0

p(Pv)Γ0,+

0◦ c0

p(ey,v)

γ0,+p (Py)

c0rvp ◦ Γ0,+

v (ey,v)

γ0,+p (Pv)

86

Which reduces to:

Pq(y) Pq(x)[1]→ Pq(x)

Pq(y) 0

eq(y),q(x)

eq(y),q(y) 0

0

Note that this could not be commutative if we had instead tried to define γ+p : Γ+

w ◦ crvp →cp ◦ Γ+

v - this is why the data Γ+v , γ

+ gives a left 2-functor. When v is a sink this samephenomenon forces Γ−v to be a right 2-functor.

Case II: v has two neighbors in V (S), one of which has the same image as v under q.


T

R

q

x v y

q(x) q(y)

x v y

q(x) q(y)

R

We are assuming v is a source, so the right-hand-side is illustrating p and the left-hand-side rvp. Computing both sides of 5.1, we have:

Γ0,+

0◦ c0

p(Pv) = Pq(y)

c0rvp ◦ Γ0,+

v (Pv) = Pq(x)[1]d→ (Pq(x) ⊕ Pq(y))

d = s(x) · iq(x)eq(x),q(x) + s(y) · iq(y)eq(x),q(y)

In this case we set γ0,+p (Pv) = πq(y) + eq(x),q(y)πq(x), which is a homotopy equivalence. This is

the crucial point requiring the sign function: If we had s(x) = s(y) then γ0,+p (Pv) would not

be a closed morphism (much less a homotopy equivalence).

As in case I, to ensure this is a natural transformation, we need to check commutativityof the diagrams, whenever z ∈ V (S) z 6= v, and there exists a directed path from v to z in(T, µ):

87

c0rvp ◦ Γ0,+

v (Pz) c0rvp ◦ Γ0,+

v (Pv)

Γ00◦ c0

p(Pz) Γ0,+

0◦ c0

p(Pv)Γ0,+

0◦ c0

p(ey,v)

γ0,+p (Pz)

c0rvp ◦ Γ0,+

v (ey,v)

γ0,+p (Pv)

When z is to the left of v, the diagram becomes:

Pq(z) Pq(x)[1]→ (Pq(x) ⊕ Pq(y))

Pq(z) Pq(y)

iq(x)eq(z),q(x)

eq(z),q(z) πq(y) + eq(x),q(y)πq(x)

eq(z),q(y)

And both compositions are eq(z),q(y). Similarly, when z is to the right:

Pq(z) Pq(x)[1]→ (Pq(x) ⊕ Pq(y))

Pq(z) Pq(y)

iq(y)eq(z),q(y)

eq(z),q(z) πq(y) + eq(x),q(y)πq(x)

eq(z),q(y)

Again, both compositions are eq(z),q(y).

Case III: v has two neighbors x1, x2 ∈ V (S), and q(v) = q(x1) = q(x2). For brevity,write q(x1) = q(v) = q(x2) = y:

T

R

q

x1 v x2

y

Here rvp = p. Then:Γ0,+

0◦ cp(Pv) = Py

88

And:cp ◦ Γ0,+

0(Pv) = Py

d→ P (1)y ⊕ P (2)

y

d = s(x1) · i(1)y ey,y + s(x2) · i(2)

y ey,y

Where we use the superscripts to distinguish coordinates. Similarly to case II, we letγ0,+p (Pv) = π

(1)y + π

(2)y - the sign function ensures γ0,+

p (Pv) is a homotopy equivalence. Forcommutativity, if there exists a directed path from z to xα in S, α = 1, 2, the commutativediagram reduces to:

Pq(z) Py[1]→ (P(1)y ⊕ P (2)

y )

Pq(z) Py

i(α)y eq(z),y

eq(z),q(z) π(1)y + π

(2)y

eq(z),y

Regardless of α, both compositions are eq(z),y.

When v is a sink, much is the same. All of the cases which were ‘easy’ when v wasa source are still that way. We do have to consider the two special cases. The analysis ismostly identical, though for completeness we include it (omitting some details). One crucialdifference is the direction of γ−, we are looking to construct:

γ0,−p (Pv) : Γ0,−

w ◦ cp(Pv)→ crvp ◦ Γ0,−v (Pv)

The necessity for this can be seen by analyzing Case I:

Case I: v has one neighbor in V (S), and either v /∈ V (S) or v ∈ V (S) has the sameimage as its neighbor under q.


x v

q(x)

x v

q(x)

T

R

q R

Suppose the left-hand-side is illustrating p, so v ∈ V (S). Then:

Γ0,−0◦ c0

p(Pv) = c0rvp ◦ Γ0,−

v (Pv) = Pq(x)

89

If the right-hand-side is illustrating p, so v /∈ V (S):

Γ0,−0◦ c0

p(Pv) = 0

c0rvp ◦ Γ0

v(Pv) = Pq(x)d→ Pq(x)[−1]

d = ±eq(x),q(x)

Again, the sign depends on the sign function s(x). γ0,−p (Pv) = 0

0−→ (Pq(x)d→ Pq(x)[−1]),

which is a homotopy equivalence. Now, suppose y ∈ V (S), and there exists a directed pathfrom y to x, hence also from y to v. We need commutativity of the diagram:

c0rvp ◦ Γ0,−

v (Pv) c0rvp ◦ Γ0,−

v (Py)

Γ0,−0◦ c0

p(Pv) Γ0,−0◦ c0

p(Py)Γ0,−

0◦ c0

p(ev,y)

γ0,−p (Py)

c0rvp ◦ Γ0,−

v (ev,y)

γ0,−p (Pv)

Which reduces to:

Pq(x) → Pq(x)[−1] Pq(y)

0 Pq(y)

eq(x),q(y)

eq(y),q(y)0

0

Case II: Near v, the correspondence looks as follows:

T

R

q

x v y

q(x) q(y)

x v y

q(x) q(y)

R

Now we assume the left-hand-side illustrates p. We have:

Γ0,−0◦ c0

p(Pv) = Pq(x)

c0rvp ◦ Γ0,−

v (Pv) = (Pq(x) ⊕ Pq(y))d→ Pq(y)[−1]

90

d = s(x) · eq(x),q(y)πq(x) + s(y) · eq(y),q(y)πq(y)

We set:γ0,−p (Pv) = iq(x) + iq(y)eq(x),q(y)

Again the sign function ensures γ0,− is closed. Verifying the associated commutative dia-grams is just as before and is left to the reader.

Case III: Near v the correspondence looks like:

T

R

q

x1 v x2

y

We have:Γ0,−

0◦ c0

p(Pv) = Py

c0p ◦ Γ0,−

v (Pv) = P (1)y ⊕ P (2)

yd→ Py[−1]

d = s(x1) · ey,yπ(1)y + s(x2) · ey,yπ(2)

y

We set:γ−p (Pv) = i(1)

y + i(2)y

And again we leave the commutative diagram to the reader.

Compatibility across Compositions of Restrictions

We now verify the commutative diagrams in (ii) and (iii).First, let p : (R, λ, w, s′) → (T, µ, v, s) and p′ : (R′, λ′, w′, s′′) → (R, λ, w, s′) be corre-

spondences in DirTree+,s. We require:

crwp′ ◦ crvp ◦ Γ+v crwp′ ◦ Γ+

w ◦ cp Γ+w′ ◦ cp′ ◦ cp

crv(p◦p′) ◦ Γ+v Γ+

w′ ◦ cp◦p′

1crvp′ � γ+p γ+

p′ � 1cp

ηrvp,rwp′ � 1Γ+v

1Γ+w′� ηp,p′

γ+p◦p′

If we are in DirTree−,s, the diagram is:

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crwp′ ◦ crvp ◦ Γ−v crwp′ ◦ Γ−w ◦ cp Γ−w′ ◦ cp′ ◦ cp

crv(p◦p′) ◦ Γ−v Γ−w′ ◦ cp◦p′

1crvp′ � γ−p γ−p′ � 1cp

ηrvp,rwp′ � 1Γ−v1Γ−

w′� ηp,p′

γ−p◦p′

It is enough to verify this diagram commutes on each generator {Pu}u∈V (T ). As before,when u 6= v both directions yield the identity or the zero map trivially.

Write p = (Rq� S

i↪→ T ) and p′ = (R′

q′

� S ′i′

↪→ R). Note that if v /∈ V (q−1(S ′)), thenthe bottom-right corner of the + diagram is zero, meaning both compositions are zero. Thesame is true of the top-right corner of the − diagram.

If v ∈ V (q−1(S ′)), and (q′ ◦ q)−1(v) = {v}, then all objects in the diagram are the same(up to perhaps a reindexing of a direct sum) and all the maps are the canonical isomorphismrealizing this.

We now check a few nontrivial cases. We will only check the cases when v is a source,the cases where v is a sink are similar. We assume p is represented on the left of each diagram.

Cases I: v has one neighbor x in V (q−1(S ′)), and q′(q(v)) = q′(q(x)).

Case I.a:

T

R

q

R′q′

R

R

x v

z

x v

z

It is easy to see all objects are Pz and all maps are ez,z in the commutative diagram.

Case I.b:

92

T

R

q

R′q′

R

R

x v

y w

z

x v

y w

z

All objects are Py. Looking at the top-left horizontal arrow of the commutative diagram:

γ+p (Pv) : crvp ◦ Γ+

v (Pv)→ Γ+w ◦ cp(Pv) is the identity map on Pw[1]→ Py, and applying crwp′

yields the identity on Py. All other maps are easily ey,y.

Case I.c:

T

R

q

R′q′

R

R

v

w

v

w

This works identically to case I.b.

Case I.d:

T

R

q

R′q′

R

R

x1 v x2

y1 y2

z

x1 v x2

y1 y2

z

All objects are Pz. Looking at the top-left horizontal arrow: γ+p (Pv) : crvp ◦ Γ+

v (Pv) →Γ+w ◦ cp(Pv) is the map (Py2 → Py2 ⊕ Py1)→ Py1 given by πy1 + ey2,y1πy2 . Applying crwp′ gives

the identity ez,z. All other maps are clearly ez,z.

Cases II: v has two neighbors x1, x2 in V (q−1(S ′)), and q′(q(x1)) = q′(q(v)), which caneither equal q′(q(x2)) or not.

Case II.a:

93

T

R

q

R′q′

R

R

v

z

v

z

The diagram reduces to:

Pz[1]→ P(1)z ⊕ P (2)

z Pz Pz

Pz[1]→ P(1)z ⊕ P (2)

z Pz

π(1)z + π

(2)z

π(1)z + π

(2)z

Where all unlabelled arrows are the identity map, which commutes.

Case II.b:

T

R

q

R′q′

R

R

v v

Case II.c:

T

R

q

R′q′

R

R

v

w

v

w

Verification of Case II.b and II.c is straightforward and left to the reader.

Case II.d: Either q(x1) = q(v) = q(x2), or q(x1) 6= q(v) 6= q(x2) and q′(q(x1)) =q′(q(v)) = q′(q(x2)). In each of these cases, each correspondence is fixed by R.

94

Defining ε±v

We now come to the final parts of our program: (iv) and (v). We will construct two naturalquasi-isomorphisms:

ε+v : Γ−v ◦ Γ+

v → Id

ε−v : Id→ Γ+v ◦ Γ−v

As before, we define ε±,0v on Rep0(T, µ). We see that the only nontrivial part of the definitionis ε±,0v (Pv).

As a shorthand, write:

Pn(v) :=⊕w∈n(v)

Pw

First, let (T, µ, v, s) be an object in DirTree+, with v 6= 0. We have:

Γ−v ◦ Γ+v (Pv) = Γ−v (Pv[1]

d→ Pn(v))

This computation can be carried out using the totalization of a complex (definition 5.7.2):

Γ−v

Pv[1]

Pn(v)

d

= Tot

Pn(v)[1] Pv

Pn(v)

d′

Γ−v (d)

=

Pn(v)[1]

Pv ⊕ Pn(v)

d′′=d′+Γ−v (d)

We have:Γ−v (d) =

∑w∈n(v)

s(w) · iwΓ−v (ev,w) =∑w∈n(v)

s(w) · iwπw

d′ =∑w∈n(v)

s(w) · ivew,vπw

And so:d′′ =

∑w∈n(v)

s(w) · iwπw +∑w∈n(v)

s(w) · ivew,vπw

Define:ε+,0v (Pv) = πv +

∑w∈n(v)

ew,vπw

ε+,0(Pv) is a homotopy equivalence- we remark that ε+,0(Pv) being closed depends on ourconvention that s(w) = −s(w).

We need to show that ε+,0v is, in fact, a natural transformation. So take x 6= v ∈ V (T )

such that there exists a directed path from v to x in T . We need to check the commutativityof:

95

Γ−v ◦ Γ+v (Px) Γ−v ◦ Γ+

v (Pv)

Px Pv

Γ−v ◦ Γ+v (ex,v)

ε+,0(Px) ε+,0(Pv)

ex,v

There is a unique u ∈ n(v) such that there is a path from u to x. The diagram becomes:

Px Pn(v)[1]→ Pv ⊕ Pn(v)

Px Pv

iuex,u

ex,x πv + Σew,vπw

ex,v

And we see both compositions equal ex,v.

Next, let (T, µ, v, s) be an object in DirTree−, with v 6= 0. A similar computation givesus:

Γ+v ◦ Γ−v (Pv) = Pv ⊕ Pn(v)

d→ Pn(v)[−1]

d =∑w∈n(v)

s(w) · iwew,wπw +∑w∈n(v)

s(w)iwev,wπv

Define:ε−v (Pv) = iv +

∑w∈n(v)

iwev,w

For the requisite commutative diagram: Assume there is a directed path from x to v in T .We check the commutativity of:

Pv Px

Γ+v ◦ Γ−v (Pv) Γ+

v ◦ Γ−v (Px)Γ+v ◦ Γ−v (ev,x)

ε−,0v (Px)ε−,0v (Pv)

ev,x

Which becomes:

96

Pv Px

Pv ⊕ Pn(v) → Pn(v)[−1] Px

ev,x

iv +∑iwev,w ex,x

eu,vπu

And we see both compositions equal ev,x.

Hence these definitions give natural quasi-isomorphisms ε±v .

Compatibility of ε±v

The last step is to check the commutativity diagrams in parts (iv) and (v) of our program.

Let p : (R, λ, w, s′) → (T, µ, v, s) be a correspondence. We need to check the commuta-tivity of:

Γ−w ◦ crvp ◦ Γ+v cp ◦ Γ−v ◦ Γ+

v

Γ−w ◦ Γ+w ◦ cp cp

γ−p � 1Γ+v

1Γ−w� γ+

p 1cp � ε+v

ε+w � 1cp

As in all cases so far, we only need to check this when applied to Pv. Writing p = (Rq�

Si↪→ T ), if v /∈ V (S) the bottom-right corner of the diagram is 0, hence we get commuta-

tivity automatically. The other cases will take some care.

Case I: q−1(q(v)) = {v}. Then each object is Γ−w ◦ Γ+w(Pw), with the right and bottom

arrows being ε+w . Furthermore, γ±ρ are canonical isomorphisms on each object, which implies

that the top and left arrows are the identity.

Case II: v has one neighbor x in V (S), and q(v) = q(x) = z.

Then w = 0 and the bottom arrow is just the identity ez,z. Also, the map γ+p (Pv) is ez,z,

hence the left arrow is ez,z. The right arrow is:

cp

[(Px[1]→ Px ⊕ Pv) Pv

ex,vπx+πv]

97

= (Pz[1]→ P(1)z ⊕ P (2)

z ) Pzπ(1)z +π

(2)z

The top arrow is the map γ−p (Pv[1]→ Px). Note that we have:

γ−p (Pv) = 00→ (Pz[1]→ Pz)

Andγ−p (Px) = Pz

ez,z→ Pz

So the top arrow is:

Pzi(1)z→ (Pz[1]→ P (1)

z ⊕ P (2)z )

Putting this together yields:

Pz Pz[1]→ P(1)z ⊕ P (2)

z

Pz Pz

i(1)z

ez,z π(1)z + π

(2)z

ez,z

Which certainly commutes.

Case III: v has two neighbors x, y in V (S), and q(x) = q(v) = z.

Write u = q(y), this demonstration will not depend on whether u = z or not.

We have w = 0, and the bottom arrow is ez,z. Also, the left arrow is effectively γ+p , i.e.

it is the map: (Pu[1]→ Pz ⊕ Pu)→ Pz given by πz + eu,zπu.

The right arrow is:

cp

[(Pn(v)[1]→ Pv ⊕ Pn(v)) Pv

ex,vπx+ey,vπy+πv]

= (Pzu[1]→ P(1)z ⊕ P (2)

z ⊕ Pu) Pzπ(1)z +π

(2)z +eu,zπu

And the top arrow is again obtained using:

γ−p (Pv) = Pu (Pu ⊕ Pz → Pz[−1])eu,u+izeu,z

γ−p (Px ⊕ Py) = Pz ⊕ PuId−→ Pz ⊕ Pu

Hence the top arrow is:

98

Pu[1] Puz[1]

Pz ⊕ Pu P(1)z ⊕ P (2)

z ⊕ Pu

eu,u + izeu,z

i(1)z ez,zπz + iueu,uπu

For each case, we need to verify two things: First, that these are actually natural trans-formations (i.e. that they commute with morphisms), and second that they are compatiblewith the restriction functors.

In the case where v is a source, suppose x ∈ V (T ) with x 6= v and suppose there is adirected path from v to x in µ. Let w ∈ n(v) be the vertex such that there exists a directedpath from w to x. We need to check commutativity of the diagram, but it is enough to checkcommutativity in degree zero, since the bottom-right entry only has degree zero components.We get:

Pz ⊕ Pu P(1)z ⊕ P (2)

z ⊕ Pu

Pz Pz

i(1)z πz + iuπu

πz + eu,zπu

ez,z

π(1)z + π

(2)z + eu,zπu

Which, again, commutes.

99

Chapter 6

Graded Structures and GradedBranes

6.1 Graded Structures

Definition of Graded Structures

Definition 6.1.1. Let P be an arboreal poset. A graded structure G on P consists of thefollowing data:

(i) For each top cell x, a Z−torsor Gx.

(ii) For each triple x, y, z where x, y are top-cells and z is a codimension-one cell withz ≤ x, y, an isomorphism ϕzx,y : Gx

∼→ Gy. These are called the structure maps.

Satisfying the Monodromy Relations. Any automorphism ϕ of Gx is of the form a 7→ a+ n,for some n ∈ Z. Define the monodromy of such an automorphism to be m(ϕ) = n. Then:

(M1) m(ϕzx,yϕzy,x) = 0.

(M2) If x, y, z are the three top-cells containing a codimension-one cell t, m(ϕtz,xϕty,zϕ

tx,y) =

±1 (Figure 6.1).

(M3) Suppose w is a codimension-two cell, and x1, x2, x3, z1, z2, z3 are as in Figure 6.2.Then: m(ϕz1x3,x1ϕ

z3x2,x3

ϕz2x1,x2) = 0

For the remainder of this section we will need some simple facts about the monodromyfunction:

Lemma 6.1.1.

100

tx

y

z

ϕtx,y

ϕty,z

ϕtz,x

Monodromy = ±1

Figure 6.1: Illustration of Mondromy Axiom (M2)

w z1

z2

z3

x1

x2

x3

ϕz2x1,x2

ϕz3x2,x3

ϕz1x3,x1Monodromy = 0

Figure 6.2: Illustration of Monodromy Axiom (M3)

• If G0 is a Z−torsor, the map m : Aut(G0)→ Z is a group isomorphism.

• If G0, G1 are Z−torsors, and a : G0 → G1 and b : G1 → G0 are isomorphisms, thenm(ab) = m(ba).

Proof. Straightforward

We note some simple consequences of the definition. First, (M1) implies that ϕzx,y =(ϕzy,x)

−1. Hence if t, x, y, z are as in (M2), andm(ϕtz,xϕty,zϕ

tx,y) = 1, then we havem(ϕty,xϕ

tz,yϕ

tx,z) =

−1. Also, if we have x1, x2, x3, z1, z2, z3 as in (M3), then ϕz3x2,x3ϕz2x1,x2

= ϕz1x1,x3 .

Definition 6.1.2. Let P be an arboreal poset with graded structure G. The associated pre-cyclic structure OG is defined as follows: Let t be a codimension-one cell, and x, y, z thethree top-cells containing it. Then the cyclic order is induced by x < y < z if and only ifm(ϕtz,xϕ

ty,zϕ

tx,y) = −1.

101

Referring to Figure 6.1: If the cyclic order is the counterclockwise order, the monodromyof the illustrated map is −1.

Proposition 6.1.1. For any graded structure G, OG is a cyclic structure.

Proof. By theorem 3.3.1, it is enough to show that OG satisfies the A3 and S4 coherenceconditions. First we show A3- so let G be a graded structure on PA3 . Recall the setup forthe arrow diagram in Chapter 3 (see Figures 3.4 and 3.5), where we use letters to abbreviatethe two-cells for convenience (Figure 6.3).

∅ {e0}

{e3}

{e1}

{e2}

xy

z t

Figure 6.3: Arrow Diagram Setup With Labelled Cells

We will compute m(ϕ{e0}t,x ϕ

{e2}z,t ϕ

{e1}y,z ϕ

{e3}x,y ) in two ways. First, let a = {e2, e3}. By M3 we

can factor ϕ{e1}y,z = ϕ

{e2}a,z ϕ

{e3}y,a and ϕ

{e0}t,x = ϕ

{e3}a,x ϕ

{e2}t,a . We get:

m(ϕ{e0}t,x ϕ

{e2}z,t ϕ

{e1}y,z ϕ

{e3}x,y ) = m((ϕ{e3}a,x ϕ

{e2}t,a )ϕ

{e2}z,t (ϕ{e2}a,z ϕ

{e3}y,a )ϕ{e3}x,y )

= m(ϕ{e2}t,a ϕ

{e2}z,t ϕ

{e2}a,z ϕ

{e3}y,a ϕ

{e3}x,y ϕ

{e3}a,x )

= m(ϕ{e2}t,a ϕ

{e2}z,t ϕ

{e2}a,z ) +m(ϕ{e3}y,a ϕ

{e3}x,y ϕ

{e3}a,x )

Similarly, let b = {e0, e1}. Factoring ϕ{e3}x,y = ϕ

{e1}b,y ϕ

{e0}x,b and ϕ

{e2}z,t = ϕ

{e0}b,t ϕ

{e1}z,b , we get:

m(ϕ{e0}t,x ϕ

{e2}z,t ϕ

{e1}y,z ϕ

{e3}x,y ) = m(ϕ

{e0}t,x (ϕ

{e0}b,t ϕ

{e1}z,b )ϕ{e1}y,z (ϕ

{e1}b,y ϕ

{e0}x,b ))

= m(ϕ{e0}x,b ϕ

{e0}t,x ϕ

{e0}b,t ϕ

{e1}z,b ϕ

{e1}y,z ϕ

{e1}b,y )

= m(ϕ{e0}x,b ϕ

{e0}t,x ϕ

{e0}b,t ) +m(ϕ

{e1}z,b ϕ

{e1}y,z ϕ

{e1}b,y )

Now we note that these monodromies only depend on the pre-cyclic structure OG, whichcan be encoded using an arrow diagram. For example, m(ϕ

{e2}t,a ϕ

{e2}z,t ϕ

{e2}a,z ) = −1 if and only

102

if the arrow on edge {e2} points from z to t.

Given an arrow diagram, we can define ci to be −1 if the arrow on edge {ei} pointscounterclockwise around the origin, and 1 otherwise. Then the preceeding algebra gives:

m(ϕ{e0}t,x ϕ

{e2}z,t ϕ

{e1}y,z ϕ

{e3}x,y ) = c2 + c3 = c0 + c1

Since each ci = ±1, the only options are if all ci = 1, all ci = −1, or c0, c1 have oppositesigns and c2, c3 have opposite signs.

It is now straightforward to see that the arrow diagram is either (see definition 3.4.2):

• Coherent, if m(ϕ{e0}t,x ϕ

{e2}z,t ϕ

{e1}y,z ϕ

{e3}x,y ) = 0

• Spiral, if m(ϕ{e0}t,x ϕ

{e2}z,t ϕ

{e1}y,z ϕ

{e3}x,y ) = ±2

(See Figure 6.4). Meaning OG is a cyclic structure.

∅ {e0}

{e3}

{e1}

{e2}

xy

z t

∅ {e0}

{e3}

{e1}

{e2}

xy

z t

Figure 6.4: Arrow Diagrams Giving m(ϕ{e0}t,x ϕ

{e2}z,t ϕ

{e1}y,z ϕ

{e3}x,y ) = −2 (left) or 0 (right)

Now let G be a graded structure on PS4 . We know OG is A3−coherent, so it suffices torule out the possibility of a ‘pinwheel orientation’ discussed in Chapter 3. WLOG we canconsider the arrow diagram pictured in Figure 6.5, in which arrows on the x2 = 0 and x3 = 0hyperplanes point counterclockwise about the x1−axis (as viewed from the positive side),and arrows on the x1 = 0 hyperplane point toward x1 > 0.

Focussing on a neighborhood of {v1}, we get Figure 6.6.

Let u denote the top-cell {v2, e2, v3, e3}. By the reasoning in the previous section, wehave:

m(ϕ{v1,v2}t,x ϕ

{v1,e3}z,t ϕ{v1,e2}y,z ϕ{v1,v3}x,y ) = −2

103

{e1}

{v1}{e2}

{v2}

{e3}

{v3}

Figure 6.5: A Pinwheel Orientation

{v1}{v1, v2}

{v1, v3}

{v1, e2}

{v1, e3}

xy

z t

Figure 6.6: A Neighborhood of {v1} in the Pinwheel Orientation

On the other hand, we have the closed subset in a neighborhood of {v3} pictured in Figure6.7 which allows us to apply (M3).

(M3) gives us: ϕ{v1,v3}x,y = ϕ

{v3,e2}u,y ϕ

{v2,v3}x,u . A similar identity holds for ϕ

{v1,e2}y,z , etc. We get:

m(ϕ{v1,v2}t,x ϕ

{v1,e3}z,t ϕ{v1,e2}y,z ϕ{v1,v3}x,y )

= m((ϕ{v2,v3}u,x ϕ{e3,v2}t,u ) ◦ (ϕ

{e3,v2}u,t ϕ{e2,e3}z,u ) ◦ (ϕ{e2,e3}u,z ϕ{v3,e2}y,u ) ◦ (ϕ{v3,e2}u,y ϕ{v2,v3}x,u ))

= m(ϕ{v2,v3}u,x (ϕ{e3,v2}t,u ϕ

{e3,v2}u,t )(ϕ{e2,e3}z,u ϕ{e2,e3}u,z )(ϕ{v3,e2}y,u ϕ{v3,e2}u,y )ϕ{v2,v3}x,u )

= m(ϕ{v2,v3}u,x ϕ{v2,v3}x,u ) = 0

With the last deduction using (M1). This contradicts our previous calculation.

104

{v3}

{v2, v3}{v3, e2}

{v1, v3}

xy

u

Figure 6.7: Cells Near {v3} In PS4

Note on Notation

Due to the nature of graded structures, we will frequently be referencing top-cells in PT .Recall from Chapter 2 (corollary 2.3.1) that these are in bijection with subtrees of T . Forbrevity, in the remainder of this section, we will use the notation [S] to designate the top-cellin PT determined by S. Additionally, if S is an explicit list of vertices, say S = {v1, v2}, wewill write [v1, v2] to denote [S] instead of the lengthier [{v1, v2}].

Additionally, in this section we will be doing many computations on A2 and A3. We willbe using notation from Chapter 2 to express correspondences p : R → A2 and p : R → A3

(The Hasse diagrams are shown in Figures 6.8, 6.9 for convenience).

(1) (2) (12)

(1)(2)


As a final remark: We have a general notation for one-cells: For k ∈ KT we have〈k〉 ∈ C1(PT )- and this introduces a somwhat confusing conflict in the case when T = A2.Namely, when v ∈ Vt(T ), the notation 〈v〉 refers to deleting the vertex v, hence comparingthose notations when T = A2 we have:

[(1)] = 〈2〉[(2)] = 〈1〉

[(12)] = 〈e〉

105

(1)(2) (1)(23) (12)(3) (2)(3)

(1) (12) (2) (123) (23) (3)

(1)(2)(3)


Construction of Gµ

We have seen that any graded structure determines a cyclic structure. In this section wewill see that, locally, graded structures are effectively equivalent to cyclic structures.

First, we will construct a graded structure which inducesOµ when (T, µ) is a directed tree:

Definition 6.1.3. Let (T, µ) be a directed tree. We define the graded structure Gµ on PTas follows: For all top-cells x, we take Gµ,x = Z. Then an isomorphism Gµ,x

∼→ Gµ,y canbe identified with an integer. To emphasize that this notation is additive, if ϕ : Z → Z isdefined by ϕ(x) = x + n, we will write ϕ = +n. (Hence the notation ϕ = +0 denotes theidentity, not the zero map).

Let z be a codimension-one cell represented by p = (A2

q� S ↪→ T ), where the edge

in p∗µ points from 1 to 2, and write z1 = [p ◦ 〈1〉] = [p ◦ (2)], z2 = [p ◦ 〈2〉] = [p ◦ (1)],ze = [p ◦ 〈e〉] = [p ◦ (12)]. Then the map ϕzz1,z2 = +1 = −ϕzz2,z1, and all other maps are +0.

Another way of interpreting this definition: Suppose S, S ′ are subtrees with [S], [S ′] shar-ing a codimension-one cell z in their closure. We have ϕz[S],[S′] = −1 if S and S ′ are disjoint

and the edge that connects them points from [S] to [S ′], +1 if the edge points in the oppositedirection, and +0 in all other cases. This is illustrated in Figure 6.10.

Proposition 6.1.2. Gµ as defined above is a graded structure, and OGµ = Oµ.

Proof. The cyclic order on z1, ze, z3 in O[p]µ is the total order induced by z1 < ze < z2, and we

see this agrees with the cyclic order in O[p]Gµ

. So we need to show that Gµ is actually a gradedstructure. Monodromy axioms (M1) and (M2) are clear from the definition, so we need toshow (M3). The maps involved (M3) are all in the neighborhood of a codimension-two cell

106

S ′ S

z = [p] = [(A2

q� S ∪ S ′ i

↪→ T )]

ϕz[S],[S′] = −1 ϕz[S′],[S∪S′] = +0

ϕz[S∪S′],[S] = +0

Figure 6.10: Illustration of the Graded Structure Gµ

w, meaning that it is enough to verify this axiom when T = A3.

To verify the axiom, we need to select three one-cells z1, z2, z3. This uniquely determinesthe three top-cells x1, x2, x3 referenced in the axiom. Note that applying some permutationto {z1, z2, z3} can only impact the monodromy m(ϕz1x3,x1ϕ

z3x2,x3

ϕz2x1,x2) by changing its sign,so we only need to show the monodromy is zero once for each of the four subsets of threeone-cells of PA3 .

Suppose the three one-cells are z1 = (1)(2), z2 = (1)(23), and z3 = (2)(3). Then x1 = (1),x2 = (23), and x3 = (2).

Then ϕz2x1,x2 = ±1, depending on the orientation of the edge {1, 2}- it is −1 when theedge points from 1 to 2 and +1 otherwise. ϕz3x2,x3 = +0. ϕz1x3,x1 = ±1, also depending on theorientation of the edge {1, 2}, but this time it is +1 when the edge points from 1 to 2 and−1 otherwise. So the total monodromy is zero as desired.

The case when z1 = (2)(3), z2 = (12)(3), z3 = (1)(2), is the same as this by applying anautomorphism of A3 switching vertices 1 and 3.

Suppose we have z1 = (1)(2), z2 = (1)(23), z3 = (12)(3). Then x1 = (1), x2 = (123),

107

x3 = (12).

In this case, ϕz2x1,x2 = ϕz3x2,x3 = ϕz1x3,x1 = 0, so the total monodromy is zero. Applying theautomorphism of A3 switching vertices 1 and 3 gives the last case.

Isomorphisms of Graded Structures

In this section, we show graded structures are locally determined up to isomorphism by cyclicstructures:

Definition 6.1.4. Let P be a combinatorial arboreal space, and G,G′ be graded structureswith structure maps ϕ, ϕ′. An isomorphism τ : G → G′ consists of an isomorphism τx :Gx

∼→ G′x for each top-cell x, such that τyϕzx,y = ϕ′zx,yτx for all top-cells x, y and codimension-

one cells z ≤ x, y.

We say τ intertwines ϕzx,y and ϕ′zx,y when the above relation holds.

Proposition 6.1.3. If G and G′ are graded structures on PT with OG = OG′, then G andG′ are isomorphic. Furthermore, the group of automorphisms of any graded structure G isnaturally isomorphic to Z.

Proof. We prove both statements simultaneously by induction on the number of vertices ofT . When T has a single vertex, a graded structure is the same as a single G−torsor, so bothstatements are true.

For the inductive step, we can assume after applying an automorphism to PT (and usingthe fact that ‘isomorphic’ is a transitive relation) that G = Gµ for some orientation µ ofT . Let G′ be a graded structure with OG′ = OG = Oµ. If v is a terminal vertex of T , letp = (T\{v} ∼← T\{v} ↪→ T ). By induction, there exists an isomorphism τ : i∗pG

′ → i∗pG.Then we are done if we can prove:

Claim: There is a unique extension of τ to an isomorphism G′ → G.

To prove the claim- we first use the commutativity property to show there is only onechoice for τ[S] when [S] ∈ PT\N([p]), i.e. when v ∈ S. We then use (M2) and (M3) to showthis extension is an isomorphism.

108

Let S be a subtree containing v which is not equal to {v}, write S0 = S\{v}. Let z be

the class of the correspondence {A2

q� S

i↪→ T}, with q(v) = 1 and q(S\{v}) = 2. The

following diagram needs to commute:

G′[S] G′[S0]

Z Z

ϕ′z[S],[S0]

τ[S]ϕz[S],[S0]

τ[S0]

Since all other maps in this diagram are defined, this defines τ[S]. If S = {v}, let v denote

the unique vertex connected to v. Write z for the class of (A2

q� {v, v} i

↪→ T ). We’ve definedτ[v,v] in the previous step, hence the diagram:

G′[v] G′[v,v]

Z Z

ϕ′z[v],[v,v]

τ[v]ϕz[v],[v,v]

τ[v,v]

Defines τ[v] uniquely. We are done if we can show τ intertwines all structure maps ϕz[S],[S′]

and ϕ′z[S],[S′]. When v /∈ S, S ′, this is guaranteed by our assumption on τ . Aside from this,

there are a handful of cases, each of which is solved by a suitable application of (M2) or (M3).

Case I: Assume S = {v}, and v /∈ S ′. This case is a little bit tricky: We know v ∈ S ′,where v is the unique vertex connected to v, but S ′\{v} may have multiple connected com-ponents. We induct on the number of connected components of S ′\{v}, with the base casebeing zero connected components, i.e. S ′ = {v}.

For the base case, let p = (A2

q� {v, v} i

↪→ T ), so z = [p]. Then we have a diagram:

G′[v] G′[v,v] G′[v]

Z Z Z

ϕ′z[v],[v,v] ϕ′z[v,v],[v]

τ[v] τ[v,v] τ[v]ϕz[v],[v,v] ϕz[v,v],[v]

The left square commutes by the definition of τ[v], while the right square commutes bythe definition of τ[v,v]. Since OG′ = OG, we have an equality of monodromies:

m(ϕ′z[v],[v]ϕ′z[v,v],[v]ϕ

′z[v],[v,v]) = m(ϕz[v],[v]ϕ

z[v,v],[v]ϕ

z[v],[v,v])

109

Hence τ intertwines ϕ′z[v],[v] and ϕz[v],[v].

For the inductive step, we can take a subtree S0 with S0 ⊆ S ′ and v ∈ S0, such thatS0\{v} has one fewer connected component than S ′\{v}, and S ′\S0 has a single connected

component. Let p = (A3

q� S ′

i↪→ T ), where q(v) = 1, q(S0) = 2, and q(S ′\S0) = 3.

Then z = [p ◦ (1)(23)]. Let z1 = [p ◦ (1)(2)] and z2 = [p ◦ (2)(3)]. Then consider thediagram:

G′[v] G′[S0] G′[S′]

Z Z Z

ϕ′z1[v],[S0] ϕ′z2[S0],[S′]

τ[v] τ[S0] τ[S′]ϕz1[v],[S0] ϕz2[S0],[S′]

The left square commutes by the inductive assumption. The right square commutesbecause v /∈ S0, S

′. By (M3) we have ϕz[v],[S′] = ϕz2[S0],[S′]ϕz1[v],[S0], and likewise for ϕ′, so τ

intertwines ϕz[v],[S′] and ϕ′z[v],[S′].

Case II: Assume v ∈ S, but S 6= {v}, and S ′ = {v}. Letting S0 = S\{v} and p = (A2

q�

Si↪→ T ) with q(v) = 1 and q(S0) = 2, we have z = [p]. Then in the following diagram:

G′[S] G′[S0] G′[v]

Z Z Z

ϕ′z[S],[S0] ϕ′z[S0],[v]

τ[S] τ[S0] τ[v]ϕz[S],[S0] ϕz[S0],[v]

The left square commutes by definition of τ[S], and the right square commutes by CaseI. Then, since OG = OG′ , we have equality of monodromies:

m(ϕ′z[v],[S]ϕ′z[S0],[v]ϕ

′z[S],[S0]) = m(ϕz[v],[S]ϕ

z[S0],[v]ϕ

z[S],[S0])

τ also intertwines ϕz[S],[v] and ϕ′z[S],[v].

Case III: Assume v ∈ S, S 6= {v}, and S, S ′ are disjoint. If there is a codimension-onecell z less than S, S ′, then there is one edge separating S and S ′. Let S0 = S\{v}. If we let

p = (A3

q� S ∪ S ′ i

↪→ T ), where q(v) = 1, q(S0) = 2, q(S ′) = 3, then z = [p ◦ (12)(3)].

We set z1 = [p ◦ (1)(2)], and z2 = [p ◦ (2)(3)]. Consider the diagram:

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G′[S] G′[S0] G′[S′]

Z Z Z

ϕ′z1[S],[S0] ϕ′z2[S0],[S′]

τ[S] τ[S0] τ[S′]ϕz1[S],[S0] ϕz2[S0],[S′]

The left square commutes by the definition of τ[S]. The right square commutes, sinceneither S0 nor S ′ contain v. So the whole rectangle commutes, and by (M3) we haveϕz[S],[S′] = ϕz2[S0],[S′]ϕ

z1[S],[S0], and likewise for ϕ′. So τ intertwines ϕz[S],[S′] and ϕ′z[S],[S′].

Case IV: Assume {v} ∈ S, S 6= {v}, and S ⊆ S ′. Let p = (A3

q� S ′

i↪→ T ), where

q(v) = 1, q(S\{v}) = 2, and q(S ′\S) = 3. Then z = [p ◦ (12)(3)]. Let z1 = [p ◦ (1)(2)], andz2 = [p ◦ (1)(23)]. Consider the diagram:

G′[S] G′[v] G′[S′]

Z Z Z

ϕ′z1[S],[v] ϕ′z2[v],[S′]

τ[S] τ[v] τ[S′]ϕz1[S],[v] ϕz2[v],[S′]

Both squares commute by Case II (the left side using the inverse of Case II), hence thefull rectanges commute, which implies the desired result using (M3).

Case V: In the last case: Assume {v} ∈ S but S 6= {v}, v /∈ S ′, and S ′ ⊆ S. Write

S0 = S\S ′. Use the correspondence p = (A2

q� S

i↪→ T ), with q(S0) = 1, and q(S ′) = 2.

Then z = [p]. Consider the diagram:

G′[S] G′[S0] G′[S′]

Z Z Z

ϕ′z[S],[S0] ϕ′z[S0],[S′]

τ[S] τ[S0] τ[S′]ϕz[S],[S0] ϕz[S0],[S′]

The left square commutes by (the inverse of) Case IV, and the right square commutesby either Case I or Case III, depending on whether S0 = {v} or not. Then, since OG = OG′ ,we have equality of monodromies:

m(ϕ′z[S′],[S]ϕ′z[S0],[S′]ϕ

′z[S],[S0]) = m(ϕz[S′],[S]ϕ

z[S0],[S′]ϕ

z[S],[S0])

τ also intertwines ϕz[S],[S′] and ϕ′z[S],[S′].

The above options (and their inverses) cover all cases.

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6.2 Branes

Pre-Branes

Let P be a combinatorial arboreal space.

Definition 6.2.1. A subset L ⊆ P is a pre-brane if it satisfies the following criteria:

(i) L is equal to the closure of its interior. Equivalently, L is closed and, for x ∈ L, thereexists a y ∈ L with x ≤ y and y maximal in P.

(ii) If x ∈ L is a codimension one cell, so N(x) ∼= PA2, L contains exactly two out of thethree maximal cells greater than x (See Figure 6.11).

(1)

(2)

(12)

Figure 6.11: Pre-Branes on PA2

Note the similarity of Figure 6.11 to Figure 5.2. This is no accident- We will see there isa close relationship between branes and the representation theory of (T, µ).

Let Brpre(P) denote the set of pre-branes in P . Note that a pre-brane is determined bythe set of top-cells it contains, by property 1. If we let C0(P) denote the set of top-cells,then we can think of Brpre(P) as a subset of Fun(C0(P),Z/2Z). In fact, we can see that itis a subspace of this Z/2Z−vector space: If we let:

∂ : Fun(C0(P),Z/2Z)→ Fun(C1(P),Z/2Z)

∂(f)(y) =∑y<x

f(x)

Then condition (ii) implies Brpre(P) = Ker(∂).

112

Whenever a sum of pre-branes is referenced it will be with respect to this structure. Theidentity element is the empty brane ∅ ⊂ P- we will typically refer to it as 0.

We remark that for an arbitrary C-Arb space P , the assignment U 7→ Brpre(U) defines asheaf of vector spaces on P .

For an arboreal singularity PT , we have a nice way to find a basis for Brpre(PT ).

Definition 6.2.2. Let v be a vertex in T . We define Lv ⊂ PT by:

Lv = {[p] | p = (Rq� S

i↪→ T ), v ∈ V (S)}

For a vertex v ∈ V (T ), we have a correspondence pv = (?∼← {v} ↪→ T ). We can define

the linear functional χv : Brpre(PT )→ Z/2Z by:

χv(L) =

{1 [pv] ∈ L0 [pv] /∈ L

Lemma 6.2.1. Lv is a pre-brane. Furthermore, {Lv}v∈V (T ) and {χv}v∈V (T ) are dual bases.

Proof. First we show that Lv satisfies the local A2 condition. Let p = {A2

q� S

i↪→ T} be a

correspondence with [p] ∈ Lv (so, v ∈ V (S)). The three top-cells containing [p] are:

[p ◦ 〈2〉] = (?� q−1(1) ↪→ T ) [p ◦ 〈1〉] = (?� q−1(2) ↪→ T ) [p ◦ 〈e〉] = (?� S ↪→ T )

Then [p ◦ (12)] ∈ Lv, and exactly one of [p ◦ 〈1〉], [p ◦ 〈2〉] are in Lv depending on whetherq(v) = 1 or q(v) = 2.

For the second assertion, we have χv(Lw) = δv,w, so it remains to show that if χv(L) = 0for all v ∈ V (T ), then L = 0.

To show this, assume L 6= 0. Pick a cell [S] ∈ L, with S having a minimal numberof vertices. If S has at least two vertices, we can construct a correspondence p = (A2 �S ↪→ T ). Since L is closed, [p] ∈ L, and by the A2− condition we must have exactly one of[q−1(1)] ∈ L or [q−1(2)] in L. But |q−1| and |q−1(2)| are both less than |S|, which contradictsthe minimality assumption. So |S| = 1, meaning S = {v} for some v ∈ V (T ), and soχv(L) 6= 0.

We end this section with a brief discussion about pullbacks:

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Let p = (Rq� S

i↪→ T ) denote a correspondence, which induces the open embedding

ip : PR → PT . Then there is a linear map i∗p : Brpre(PT ) → Brpre(PR) which sends L toi−1p (L).

There is a nice description of this linear map in the basis discussed above:

Lemma 6.2.2.

i∗p(Lv) =

{Lq(v) v ∈ V (S)0 v /∈ V (S)

Proof. For a subtree W of R, write q = (? � W ↪→ R). Then i∗p(Lv) contains the top-cell[q] if and only if Lv contains the top-cell [p ◦ q] = [(? � q−1(W ) ↪→ T )]. This is true if andonly if q−1(W ) contains v, i.e. if and only if v ∈ V (S) and q(v) ∈ W .

The similarity of Lemma 6.2.2 and the restriction functors cp reinforce an analogy betweenbranes and representations: Lv ↔ Pv. This is a special case of a more general correspondence.

Branes

Let (PA3 ,O) denote a cyclic A3 singularity. By proposition 4.4, c = ∆(O) is a cyclic order onKA3 = {e0, e1, e2, e3}. Fix an isomorphism KA3

∼= [4] := {1, 2, 3, 4} such that c is induced bythe total order 1 < 2 < 3 < 4, and identify PA3 with its image {S ⊆ [4] | |S| ≤ 2} ⊂P([4]).

Now, write L0 = {{1, 2}, {2, 3}, {3, 4}, {4, 1}} = PA3\{{1, 3}, {2, 4}} ∈ Brpre(PA3). Thenwe define the set of branes on PA3 . to be Br(PA3 ,O) := Brpre(PA3)\{L0}.

Note that L0 does not depend on our choice of isomorphism KA3∼= [4], since it is fixed

under cyclic-order-preserving automorphisms of [4].

Definition 6.2.3. Let (P ,O) be a cyclic C-Arb space. A brane is a pre-brane L satisfying anA3−coherence condition: If i : PA3 ↪→ P is an open embedding, then i∗(L) ∈ Br(PA3 , i

∗O).Denote the set of branes by Br(P ,O).

Then a brane is a pre-brane satsifying a coherence condition in codimension two. Thereare geometric reasons for imposing this coherence condition- see Chapter 8 for more on this.As we will see in this section, this definition of brane can be justified by representation theoryof Rep(T, µ).

One can understand the above restriction using the arrow diagram from Chapter 3. Re-moving {e0, e1} and {e2, e3} from PA3 , the remaining cells form a pre-brane. The cyclic

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structure on PA3 can be illustrated by drawing arrows crossing the one-cells, forming adiagram which is either coherent or spiral (definition 3.4.2). Then the pre-brane L =PA3\{{e0, e1}, {e2, e3}} is a brane if and only if the diagram is coherent. (Figure 6.12).

∅ {e0}

{e2}

{e1}

{e3}

{e0, e2}{e1, e2}

{e1, e3} {e0, e3}

∅ {e0}

{e2}

{e1}

{e3}

{e0, e2}{e1, e2}

{e1, e3} {e0, e3}

Figure 6.12: An Illustration of L = PA3\{{e0, e1}, {e2, e3}} with two cyclic structures- Onthe left, L is not a Brane.

We now turn to a concrete description of the set of branes on the cyclic C-Arb singularity(PT ,Oµ) in terms of the basis {Lv}v∈V (~T ) for Brpre(PT ).

First, we analyze the case T = A3. We introduce the term ‘non-brane’ for a pre-branewhich is not a brane. For each orientation of A3, we can use Table 3.1 in Chapter 3 toidentify the unique non-brane. The results are listed in Table 6.1.

Tree Non-Brane

1 2 3 L1 + L2 + L3

1 2 3 L1 + L2 + L3

1 2 3 L1 + L3

1 2 3 L1 + L3

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Table 6.1: Non-Branes In (PA3 ,Oµ)

Now, let T be a directed tree, and A a subtree of T isomorphic to An (we call A anAn−subtree). We define V+(A) ⊂ V (A) to be the set of sources in A, and V−(A) to be theset of sinks- we only consider whether they are a source/sink in A (not in T - so, for example,the endpoints of A always qualify), and by convension if A consists of a single vertex we willcall it a source.

Proposition 6.2.1. Given an An−subtree A of T , define:

LA =∑

v∈V+(A)∪V−(A)

Lv

Then the set of nonzero branes Br(PT ,Oµ)\{0} is in bijection with the set of An subtrees viaA↔ LA.

Proof. Given A, we know LA ∈ Brpre(PT ), so we first show that it is a brane. Let p = (A3

q�

Si↪→ T ). Write λ = p∗µ.

Referring to Table 6.1, we want to show i∗pLA is not L1 + L2 + L3 or is not L1 + L3,depending on λ. Assume we have i∗pLA = L1 + aL2 + L3, where a ∈ Z/2Z. Then A mustintersect q−1(1) and q−1(3), which means a is the number of sources or sinks of A containedin q−1(2), counted mod 2. If both arrows in λ are pointing the same direction (i.e. both tothe left or both to the right), there are an even number of sources or sinks in A containedin q−1(2), meaning a = 0. Similarly, if both arrows in λ are pointing in opposite directions,a = 1. So by Table 6.1, i∗LA ∈ Br(PA3 ,Oλ).

Now, let L ∈ Br(PT ,Oµ), and write:

L =∑v∈W

Lv

Where W ⊂ V (T ). We will use Table 6.1 along with certain correspondences p : A3 → T toshow that W = V+(A) ∪ V−(A) for some An−subtree A.

Step 1: If v ∈ V (T ) is a vertex of degree > 2, we show W cannot contain a vertex inmore than two connected components of T\{v}.

Suppose for the sake of contradiction that it did. Then we could find v1, v2, v3 ∈ W suchthat no vertices between vi and v are in W for each i. Let e1, e2, e3 be the edges adjacent tov in the connected components of v1, v2, v3 respectively. Then at least two of the three, say

116

e1 and e2, must either both be pointing toward v or both away from v.

Let p = (A3

q� S

i↪→ T ), where q−1(1) is the path connecting v1 to e1 and q−1(3) is the

path connecting v2 to e2. If v /∈ W , let q−1(2) = {v}, and if v ∈ W , let q−1(2) be the pathconnecting v to v3. Then i∗pL = L1 +L3, and by assumption on the orientations of e1 and e2

and table 6.1, this is not a brane.

Step 2: By step 1, we know all the vertices in W lie in some An−subtree. Let A be thesmallest An−subtree containing all the vertices in W (so, in particular, the endpoints of Aare in W ). Then for a vertex v ∈ V (A) which is not an endpoint, we show v ∈ W iff v is asource or sink in A.

To see why, since A is an An−tree and both endpoints of A are in W , there exist twovertices v1, v2 ∈ W such that no vertices between vi and v are in W for each i. Lete1, e2 be the edges adjacent to v in the connected components of v1, v2 respectively. Let

p = (A3

q� S

i↪→ T ), where q−1(1) is the path connecting v1 to e1, q−1(2) = {v}, and q−1(3)

is the path connecting v2 to e2.

So we have i∗pL = L1 +L3 + aL2, where a ∈ Z/2Z depends on whether v ∈ W or not. ByTable 6.1, if v is a source or sink, since L is a brane we must have i∗pL = P1 + P2 + P3, i.e.v ∈ W . Similarly, if v is neither a source or sink, i∗pL = P1 + P3, so v /∈ W .

Graded Branes

Let (P , G) be a graded C-Arb space.

Definition 6.2.4. A graded brane is a pair (L, g), where L ∈ Brpre(P), and g is a choicegx ∈ Gx for all top-cells x ∈ L, such that whenever z is a codimension-one cell in L andx, y ∈ L are top-cells containing z in their closure, ϕzx,y(gx) = gy. Denote the set of gradedbranes by Brgr(P , G).

The use of the term graded brane (rather than graded pre-brane) is explained in thefollowing lemma:

Lemma 6.2.3. If (L, g) ∈ Brgr(P , G), then L ∈ Br(P ,OG).

Proof. It is enough to prove this when P = PA3 . In our discussion following definition 6.2.3,we see that if L ∈ Brpre(PA3) is a non-brane (with respect to the cyclic structure OG), wecan represent the cyclic structure as an arrow diagram on L, with the arrows representing a

117

cyclic orientation. But by our argument in the proof of proposition 6.1.1, for such an arrowdiagram, the monodromy going counterclockwise around the four top-cells of L equals ±2,hence there is no possible grading g on L.

In fact, the above is an if-and-only-if locally:

Proposition 6.2.2. If T is a tree, the map Brgr(PT , G)→ Br(PT ,OG) of (L, g) 7→ L realizesthe former as a Z−bundle over the latter (where the Z−action is the obvious shifting).

Proof. Step 1: First, let g1, g2 be two gradings of a brane L. We want to show theydiffer by the same constant on any top-cell in L- that is, if S, S ′ are subtrees of T in L,g1

[S] − g2[S] = g1

[S′] − g2[S′].

If there is a codimension-one cell z with z ≤ [S], [S ′], then this conclusion is clear. Moregenerally, we can construct a graph HL whose vertices are labelled by top-cells [S] ∈ L, andwhose edges connect pairs of subtrees sharing a codimension-one cell in their closure. It isenough to show that HL is connected. In fact, HL is connected more generally for pre-branes,as we show.

We use strong induction on the number of vertices of T , with the base case T = ? beingobvious. For the inductive step, let L ∈ Brpre(PT ), and [S], [S ′] ∈ V (HL). If S, S ′ are dis-joint and there is a single edge between them in T , there is a codimension-one cell z withz ≤ [S], [S ′], so there is an edge between [S] and [S ′] in HL. If S, S ′ are disjoint and there

are multiple edges of T between them, there is a correspondence p = (A3

q� S

i↪→ T ) such

that q−1(1) = S and q−1(3) = S ′. Letting q = p ◦ (1)(2), the A2 condition at N([q]) impliesL contains either [q−1(2)] or [q−1({1, 2})], each of which are connected to [S] and [S ′] in HL.

Now suppose S ∩ S ′ is nonempty. WLOG S\S ′ 6= ∅, let S1 be a connected component of

S\S ′. There is a correspondence p = (A2

q� S

i↪→ T ) with q(S1) = 1 and q(S\S1) = 2. The

A2 condition at N([p]) implies L contains either [S1] or [S\S1]. In the former case, [S1] isconnected to [S] and [S ′] in HL. In the latter case, [S] is connected to [S\S1], so it remainsso show [S] and [S\S1] lie in the same connected component of HL.

Letting S = (S ∪ S ′)\S1, and setting q = (Sq← S

i↪→ T ), we have i∗qL ∈ Brpre(S), and

there is a map of graphs Hiq∗L → HL which sends [R] to [R] when R ⊆ S. By induction,Hiq∗L is connected, so [S\S1] and [S ′] are in the same connected component in Hiq∗L , and soalso in HL.

Step 2: Now we show that any brane has a grading.

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By proposition 6.1.3, we can assume WLOG that G = Gµ, so OG = Oµ. By proposition6.2.1, we can write L = LA for some An−subtree A.

If A = {v} has a single vertex, then LA = Lv. For any subtree S with v ∈ S, defineg[S] = 0. Then for any pair of top-cells [S], [S ′] ∈ L, if there exists a codimension-one cell zin the closure of both, we must have S ⊆ S ′ or S ′ ⊆ S. In either case, the definition of Gµ

says ϕz[S],[S′] = +0, so this satisfies the condition of being a grading.

If A has multiple vertices, let V+(A) denote the sources in A and V−(A) the sinks. Forany two elements in V+(A) there is an element of V−(A) between them, and likewise with thesigns reversed. So for any subtree S, the quantity χA(S) = #(S ∩V+(A))−#(S ∩V−(A)) iseither 1, 0, or −1. [S] ∈ LA if and only if S has an odd number of elements of A, meaningχA(S)± 1. Define g[S] = 0 if χA(S) = 1, g[S] = −1 if χA(S) = −1.

Now, let [S], [S ′] ∈ LA, and assume there is a codimension-one cell z in the closure ofboth. If [S] and [S ′] are disjoint, suppose the edge e in between is pointing from [S] to [S ′](so ϕz[S],[S]′ = −1). Then the closest element in S ∩ (V+(A) ∪ V−(A)) to e is a source, which

means #(S ∩ V+(A)) ≥ #(S ∩ V−(A)), and so χA(S) = 1 ⇒ g[S] = 0. For the same reasonχA(S ′) = −1, so g[S′] = −1, we get ϕz[S],[S′](g[S]) = 0− 1 = −1 = g[S′] as desired.

Lastly, if S ⊆ S ′, then A∩S ⊂ A∩S ′ are two An trees in T . Also, one of the endpoints ofA∩S must coincide with one of the endpoints of A∩S ′. If the closest vertex in V+(A)∩V−(A)to that endpoint is a source in A, χA(S) = χA(S ′) = 1, and both are −1 if that closest vertexis a sink, so g[S] = g[S′]. We also have ϕz[S],[S′] = +0, so ϕz[S],[S′](g[S]) = g[S′].

6.3 The Representation Theory of Graded Branes

We now turn to a representation-theoretic interpretation of graded branes. To begin, fix adirected tree (T, µ), and an object X ∈ Rep(T, µ) = Qµ(PT ). We define the support of Xto be the set supp(X) = {[p] ∈ PT | cp(X) � 0}, which is a closed subset of PT .

Definition 6.3.1. Let X ∈ Rep(T, µ). For any maximal element [S] ∈ PT , we have c[S](X) ∈Rep(?, ·) ∼= C(k), the dg-category of finite-dimensional chain complexes over k. X is rank-one if for all maximal elements [S] ∈ supp(X), we have H•(c[S](X)) is one-dimensional.As a consequence, c[S](X) is homotopy equivalent to a shift of k. If X is rank one and[S] ∈ supp(S), define gX[S] ∈ Z by c[S](X) ∼ k[−gX[S]].

Proposition 6.3.1. For any rank-one object, (supp(X), gX) ∈ Brgr(Pµ, Gµ).

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Proof. By definition, is it clear that if p : R→ T is a correspondence and X ∈ Rep(T, µ) isrank one, then cp(X) is also rank-one.

So assume X is rank-one, we show that (supp(X), gX) is a graded brane. Note that thecondition of being a graded brane is determined by restriction to open neighborhoods ofcodimension-one cells, so by the above comment it is enough to prove this for T = A2.

WLOG we can assume in µ the single edge points from vertex 1 to vertex 2. X is a directsum of indecomposable objects. First assume X is indecomposable. The indecomposableobjects in Rep(A2, µ) are, up to shifts and homotopy, P1, P2, and P2[1] → P1 (see lemma5.4.1). Since homotopy does not change the support, and shifts only change g up to aconstant, neither will affect (supp(X), gX) being a graded brane. Thus we only need tocheck these three objects, which is routine. We can also see from Table 5.1 that if X is adirect sum of more than one indecomposable object then X cannot be rank-one.

We remark briefly that the object P2[1]→ P1 has gX[1] = 0 and gX[2] = −1, which fixes ourconvention on Gµ that the monodromy of structure maps in the cyclic order is −1.

We also have the following:

Proposition 6.3.2. The map X 7→ (supp(X), gX) gives a surjection:{Rank-One Objects

in Rep(T, µ)

}/Homotpy Equivalence→

{Graded Branes

in (Pµ, Gµ)

}Proof. For any graded brane (L, g) ∈ (Pµ, Gµ), we construct a rank one object X with(supp(X), gX) = (L, g). We have L = LA for some An−subtree A, and by proposition 6.2.2up to shifts we can assume g is the grading given in that proof. We repeat the explanationof g here: If [S] ∈ LA, then χA(S) := #(S ∩ V+(A)) − #(S ∩ V−(A)) = 1 or −1, we getg[S] = 0 in the first case and g[S] = −1 in the second case.

The object we claim works is given by the twisted complex (see Chapter 4):

XA =⊕

v∈V−(A)

Pv[1]d→

⊕w∈V+(A)

Pw

Where d = Σev,w, and the sum is over all v ∈ V−(A) and w ∈ V+(A) such that there is adirected path from w to v in (T, µ).

We check that (supp(XA), gXA) = (LA, g), as claimed. Let P? denote the generator ofRep(?), then for a subtree S we have:

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c[S](XA) =⊕

v∈S∩V−(A)

P?[1]d→

⊕w∈S∩V+(A)

P?

∼= kS∩V−(A)[1]d→ kS∩V+(A)

We can write d in matrix notation as follows: First suppose [S] ∈ LA. If g[S] = 0,then moving along S ∩ A we have an alternating sequence w1, v1, w2, . . . , wn, vn, wn+1 ofsources/sinks, starting and ending with a source. Then the matrix entries are (d)vi,wi =(d)vi,wi+1

= 1, and all everything else zero- from this we easily see c[S](XA) ∼= k, hence

[S] ∈ supp(A) and gXA[S] = 0 = g[S]. Similarly, when g[S] = −1, S∩A alternates between sinks

and sources starting and ending with a source, and c[S](XA) ∼= k[1], meaning gXA[S] = −1 = g[S].

Lastly, suppose [S] /∈ LA. If S does not intersect V−(A)∪ V+(A), c[S](XA) = 0. If it doesintersect, then since [S] /∈ LA, #(S ∩ V−(A)) = #(S ∩ V+(A)) and the map d in c[S](XA) isan isomorphism, so c[S](XA) ∼= 0. Either way, [S] /∈ supp(XA).

We also see from this calculation that X is rank-one.

In fact, the above correspondence is actually a bijection. In order to show this, we provea representation-theoretic lemma:

Lemma 6.3.1. Let (T, µ) be a directed tree with more than one vertex, and let v be aterminal vertex of T . Write pv = (?

∼← {v} ↪→ T ), and p◦v = (T\{v} ∼← T\{v} ↪→ T ). Ifwe let Rep\v(T, µ) denote the subcategory of Rep(T, µ) consisting of objects X whose supportdoes not contain [pv], the functor cp◦v : Rep\v(T, µ)→ Rep(T\{v}, µ) is a quasi-equivalence.

Proof. Define the functors j0v : Rep0(?, ·)→ Rep0(T, µ) and j\v : Rep0(T\{v}, µ)→ Rep0(T, µ)

by:j0v(P?) = Pv

j0\v(Pw) = Pw

Furthermore, if v is a source, we have natural transformations fitting into an exact sequence:

j0vc

0pv → Id→ j0

\vc0p◦v

Which, when applied to Pv yields:Pv

∼→ Pv → 0

When applied to Pw for w 6= v yields:

0→ Pw∼→ Pw

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And, when applied to ew,v for w 6= v, yields the map of complexes:

0 Pw Pw

Pv Pv 0

0

0

ew,w

ew,v 0

ev,v 0

By lemma 5.1.2 this data can be extended into an exact sequence of natural transformationsin Rep(T, µ). Since the image of j\v lies in Rep\v(T, µ), and cp◦vj\v is the identity, j\v isquasi-fully faithful. Additionally, if X ∈ Ob(Rep\v(T, µ)), the exact sequence gives a homo-topy equivalence X ∼ j\vcp◦vX, so j\v is quasi-essentially surjective. It follows that j\v is aquasi-equivalence, and so cp◦v is as well.

When v is a sink, the direction of the triangle is reversed but the argument is unchanged.

Proposition 6.3.3. The correspondence in proposition 6.3.2 is a bijection. In particular,any two rank-one objects in Rep(T, µ) with the same support are homotopy-equivalent up toa shift.

Proof. Let L ∈ Br(PT ,Oµ), and suppose supp(X) = supp(Y ) = L, with X, Y rank-oneobjects. By the above lemma, if there is a terminal vertex v such that [v] /∈ L, we can reduceto the tree T\{v}. The classification result says there is always such a terminal vertex unlessT = An.

In the case of An, for n ≥ 2, by applying reflection isomorphisms we can assume thatall edges point from vertex i to vertex i + 1. If L contains both terminal vertices, then wemust have L = L1 +Ln. Recall (proposition 4.4.1) there is an automorphism ω of (PAn ,Oµ)which acts as a cyclic rotation on KAn : Writing KAn = {e0, e1, . . . , en} we have ω(ei) = ei+1.We claim that [1] /∈ ω(L).

Indeed, considered as a subset of KAn , we have [1] = {e0, e1}c, and [An] = {e0, en}c, andnote that [An] /∈ L, and ω([An]) = [1], so [1] /∈ ω(L). Hence we can apply this rotation, andthen reduce to the tree An\{1}.

This reduces us, inductively, to the base case T = A1 where the claim is obvious.

The above proposition suggest we should think of graded branes as objects in our sheafof categories. This idea is discussed briefly at the end of this chapter.

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6.4 Branes on An Singularities

In this section we give a brief discussion of Br(PAn ,O), where O is a cyclic structure.

Proposition 6.4.1. Let µ be the orientation on An in which all vertices point from vertexi to i+ 1. Then:

Br(PAn ,Oµ) = {∅} ∪ {Lv | v ∈ V (An)} ∪ {Lv + Lw | v 6= w ∈ V (An)}

Proof. This is an easy corollary of proposition 6.2.1.

Our first goal will be to describe branes in a more invariant way. Writing PAn ∼= {X ⊆KAn | |X| ≤ n− 1}, with KAn = {e0, e1, . . . , en}, one can show:

• For 1 ≤ i ≤ n, X ∈ Li iff X does not contain the cyclic interval [e0, ei−1], nor the cyclicinterval [ei, ej−1].

• For 1 ≤ i < j ≤ n, X ∈ Li +Lj iff X does not contain the cyclic interval [ei, ej−1], northe cyclic interval [ej, ei−1].

See appendix A for the notion of cyclic intervals. We see that we each nonzero brane corre-sponds with a way of dividing KAn into two disjoint nonempty cyclic intervals.

Notice that such a division can be seen as drawing a line connecting two ‘gaps’. We canalso think of a gap as an adjacent pair gi = {ei−1, ei} of elements in KAn . Then branes arelines connecting two such pairs. We diagram this identification explicitly for A3 in Table 6.2:

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Brane ‘Division’ Diagram ‘Gap’ DiagramSet-TheoreticDescription

L1

e0

e1 e2

e3

g0

g1

g2

g3

{X ⊆ KA3 |[e0, e0] * X[e1, e3] * X}

L2

e0

e1 e2

e3

g0

g1

g2

g3

{X ⊆ KA3 |[e0, e1] * X[e2, e3] * X}

L3

e0

e1 e2

e3

g0

g1

g2

g3

{X ⊆ KA3 |[e0, e2] * X[e3, e0] * X}

L1 + L2

e0

e1 e2

e3g0

g1

g2

g3

{X ⊆ KA3 |[e1, e1] * X[e2, e0] * X}

L1 + L3

e0

e1 e2

e3

g0

g1

g2

g3

{X ⊆ KA3 |[e1, e2] * X[e3, e0] * X}

L2 + L3

e0

e1 e2

e3g0

g1

g2

g3

{X ⊆ KA3 |[e2, e2] * X[e3, e1] * X}

Table 6.2: Branes in PA3 , Four Ways

In particular, there is a full diagrammatic calculus for branes:

Definition 6.4.1. Let ∆cycsurj denote the category of finite sets with a cyclic order whose

morphisms are order-preserving injections. We define a functor Br : ∆cycsurj → Set by:

• For a set C with a cyclic order, Br(C) consists of the zero brane 0, and the line segmentsL(x, y) joining pairs of elements in C.

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• For a surjection f : C → D, the map Bf : Br(C) → Br(D) is defined by Bf (0) = 0,Bf (L(x, y)) = L(f(x), f(y)) if f(x) 6= f(y), and Bf (L(x, y)) = 0 if f(x) = f(y).

The main idea of this section is:

Proposition 6.4.2. Let ArbcycA denote the category whose objects are pairs (PAn ,O), whereO is a cyclic structure on PAn (see proposition 4.4). Then there is a contravariant functor∆◦ : ArbcycA → ∆cyc

surj and a natural identification Br(PAn ,O) ∼= Br(∆◦(PAn ,O)), such thatfor an open embedding i : PAm ↪→ PAn we have the commutative diagram:

Br(PAn ,O) Br(PAm , i∗O)

Br(∆◦(PAn ,O)) Br(∆◦(PAm , i∗O))

i∗

∼B∆◦i

∼

Spelling it out:

• ∆◦(PAn ,O) consists of sets of pairs {x, s(x)} ∈ KAn which are adjacent in the cyclicorder determined by O. Denote this set K◦An.

• An embedding i : PAm ↪→ PAn determines an embedding ∆(i) : KAm ↪→ KAn (seedefinition 3.6.2). Then ∆◦(i) : K◦An � K◦Am sends the pair {x, y} to the unique adjacentpair {a, b} with [x, y] ⊆ [∆(i)(a),∆(i)(b)].

• The identification Br(PAn ,O) ∼= Br(K◦An) is as described at the beginning of this sectionand illustrated for PA3 in Table 6.2.

The proof is straightforward, and we omit it. This can be extended to describe gradedLagrangian branes as well:

Definition 6.4.2. A graded cyclic set is a finite set C equipped with a cyclic order, alongwith:

(i) For each x ∈ C, a Z−torsor Ix.

(ii) For each x ∈ C, an isomorphism ϕx : Ix∼→ Is(x), where s(x) denotes the successor

function.

Note that if y = sk(x) we get an isomorphism ϕkx : Ix∼→ Iy. Then we require:

(M1) If |S| = n, m(ϕnx) = 2− n.

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The set of graded branes Brgr(C) consists of the empty brane 0 as well as pairs (L(x, y), (ιx, ιy)),where x 6= y ∈ S, and if y = sk(x) we have ιy = ϕk(ιx) + k − 1. Note that the monodromycondition implies that this condition is symmetric in x and y.

Then, just as before, we can functorially assign a graded cyclic set to a pair (PAn , G),where G is a graded structure on PAn , such that the graded Lagrangian branes are identified. Explicitly, for a pair {x, y} ∈ K◦An , we define the Z−torsor I{x,y} to equal G{x,y}c , and the

map ϕ{x,y} to equal ϕ{x,y,z}c{x,y}c,{y,z}c . We leave further details to the reader.

6.5 In Search of a ‘Fukaya Category’

The discussion in this section is largely informal, and refers to a ‘Fukaya category’ and‘A∞ categories’ without defining them. For references on Fukaya categories see [1], and forA∞−categories see [16].

To any cyclic C-Arb singularity (PT ,O), we explained in Chapter 5 how to construct asheaf of dg-categories on PT up to quasi-equivalence. This sheaf of categories is meant tomodel the Fukaya category of an arboreal singularity embedded in a symplectic manifold,or (following [25]) a category of microlocal sheaves. We discuss this perspective further inChapter 8.

The objects of the Fukaya category Fuk(M,ω) of a symplectic manifold are (graded)Lagrangian submanifolds of M . As shown in propositions 6.3.2 and 6.3.3, graded branes inPT with respect to the graded structure gµ have a natural interpretation in terms of objectsin the sheaf Qµ.

This raises a hope that, given a graded C-Arb singularity (PT , G) one could define anA∞−category whose objects are the graded Lagrangian branes, which only depends on G(and not on, say, any isomorphism with a directed tree graded structure).

Such a goal has been partially realized in [20]. In that paper, Nadler shows how to (func-torially) construct a (Z/2Z−graded) A∞− category CS associated to any cyclic set S, withOb(CS) = S. The triangulated hull of this category is quasi-equivalent to Rep(An, µ), wheren+ 1 = |S|. This construction can easily be modified using the notion of ‘graded cyclic set’to construct a Z−graded category.

A potential future goal, then, would be to extend Nadler’s construction to arbitrary trees.

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Chapter 7

Arboreal Moves

7.1 The H-to-I Move

In the theory of trivalent ribbon graphs there is a ‘move’ or ‘mutation’ that preserves theassociated Riemann surface (Figure 7.1).


This move can be viewed as a transformation of cyclic C-Arb spaces. Viewing this trans-formation as a continuous deformation over time, the total space of this move is the A3−singularity (Figure 7.2).

In this section, our goal will be to generalize this observation to describe a wide classof moves of C-Arb spaces whose total space is a C-Arb singularity. Before introducing theformal definition, we make a few comments about combinatorial aspects of the H-to-I move.

Combinatorially, we can describe the H-to-I move as follows: Begin with the cyclic C-Arbsingularity (PA3 ,Oµ), with all arrows in µ pointing to the right. We identify elements of PA3

with their image in P(KA3), and recall (section) that the cyclic structure Oµ corresponds

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Figure 7.2: Total Space of the H-to-I move

to the cyclic order e0 < e1 < e2 < e3.

In the H-to-I move, the ‘before’ picture can be thought of as the poset N({e0})∪N({e2}),and the ‘after’ as N({e1})∪N({e3}). We redraw the move in Figure 7.3, this time labellingcells.

{e0} {e2}

{e3}

{e1}

{e0, e3}

{e0, e1}

{e0, e2}

{e2, e3}

{e2, e1}

{e0, e3}

{e0, e1}

{e1, e3}

{e2, e3}

{e2, e1}


Note that in each case the induced cyclic structure is faithfully represented by the fig-ure. For example, the cyclic order on the top-cells containing {e0} is induced by the totalorder {e0, e1} < {e0, e2} < {e0, e3}, and in the figure those cells appear in that order goingcounterclockwise about {e0}.

It is also worth examining some non-examples. If we were to set ‘before’ to N({e0}) ∪N({e1}) ∪N({e2}) and ‘after’ to N({e3}), we would have the situation pictured in 7.4.

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{e1} {e2}

{e0}{e3}

{e0, e3}

{e0, e3} {e2, e3}

{e0, e3}

{e1, e3} {e2, e3}

Figure 7.4: An Invalid Move

More subtlely, a move N({e0})∪N({e1}) to N({e2})∪N({e3}) would look like 7.5. Thecrossing paths in the figure is necessary to preserve the boundary and have the correct cyclicorders at each vertex.

{e0} {e1} {e2} {e3}

{e0, e2}

{e0, e3}

{e0, e1}

{e1, e3}

{e1, e2}

{e0, e2}

{e0, e3}

{e2, e3}

{e1, e3}

{e1, e2}

Figure 7.5: A Second Invalid ‘H-to-I-esque’ Move

We see that there are topological restrictions to valid moves- the move in Figure 7.4 isnot a homotopy equivalence- and also restrictions imposed by the cyclic structure- the movein Figure 7.5 introduces unwanted ‘crossings’ if we want to maintain the cyclic orders abouteach vertex.

To try to capture these qualities in our combinatorial definition, we will use the sheaf ofdg categories constructed in Chapter 5.

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7.2 Definition of Arboreal Moves

Definition 7.2.1. Given a tree T , a combinatorial arboreal move of type T betweencyclic C-Arb spaces (P1,O1) and (P2,O2) consists of the following:

(i) An orientation µ on T

(ii) Open embeddings iα : Pα ↪→ PT , with i∗αOµ = Oα, for α = 1, 2. (PT is called the totalspace of the move).

(iii) Let Uα denote the image of iα, and 0 the minimal element of PT . Then U1 ∪ U2 =PT\{0}.

(iv) U1 is the smallest open set containing U1\U2, and likewise with the indices switched.

(v) Preservation of Categories: The restrictions Γ(PT ;Qµ) → Γ(Uα;Qµ) are quasi-equivalences of dg-categories.

We will frequently abbreviate “combinatorial arboreal move of type T” to simply “moveof type T”.

A brief remark on condition (iv): Note that i2i−11 gives an isomorphism i−1

1 (U2)∼→

i−12 (U1). Letting Z1 = (i−1

1 (U2))c, and Z2 = (i−12 (U1))c, we can think of the move as mutat-

ing Z1 into Z2. For example, In the case of the H-to-I move, Z1, Z2 are the internal closed linesegments. Condition (iv) then says the mutation is ‘local’, in the sense that Pα = N(Zα),for α = 1, 2.

Definition 7.2.2. Two moves of type T , given by the data i1α : (Pα,Oα) ↪→ (PT ,Oµ) andi2α : (Pα,Oα) ↪→ (PT ,Oµ′) are equivalent if there is an isomorphism of total spaces ϕ :(PT ,Oµ)

∼→ (PT ,Oµ′) such that i2α = ϕ ◦ i1α for α = 1, 2.

Then a move is determined up to equivalence by the open sets Uα.

We introduce the following notation: For a subset K ⊂ KT we write:

UK :=⋃k∈K

N(〈k〉)

Lemma 7.2.1. If we have an move of type T then we can find a partition KT = K1 tK2

such that Uα = UKα.

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Proof. Define Kα = {k ∈ KT | 〈k〉 ∈ Uα}. We will show that K1, K2 give a partition of KT ,and Uα = UKα .

Since U1∪U2 = P~T\{0}, we must have 〈k〉 ∈ U1∪U2 for all k ∈ KT . Hence KT = K1∪K2.

If k ∈ K1 ∩K2, the set {〈k〉, 0} is closed and violates property (iii). So KT = K1 tK2.

Since Uα is open, we have UKα ⊆ Uα. To show equality, WLOG set α = 1 and let x ∈ U1.By property (iv), there exists y ∈ U1\U2 with y ≤ x. Then there is k ∈ K with 〈k〉 ≤ y.Since y /∈ U2 we must have k /∈ K2, so k ∈ K1, hence x ∈ UK1 .

Definition 7.2.3. A subset K ⊆ KT is good with respect to an orientation µ on T if therestriction Γ(PT ;Qµ)→ Γ(UK ;Qµ) is a quasi-equivalence.

We can also say K ⊆ KT is good with respect to a cyclic structure O on PT if for anyorientation µ on T and ϕ ∈ Aut(PT ) with ϕ∗Oµ = O, ϕ(K) is good with respect to µ. TheCyclic Structure Theorem (theorem 5.6.1) implies this definition makes sense.

Hence, up to equivalence, moves of type T are given by orientations µ on T and partitionsKT = K1 tK2 into sets which are good with respect to µ.

The following sections will be dedicated to a classification of good subsets of T withrespect to µ.

7.3 The Topology of Branes

In this section we show a negative result: That K = KT is not a good subset of KT . Toaccomplish this we need some topological information about branes of the form Lv, forv ∈ V (T ).

A regular cell complex is a cell complex such that the closure of any open cell is aclosed cell. To a regular cell complex we can associate a poset, called the face poset, whoseelements are cells and whose relation is given by x ≤ y iff x ⊂ y. From Nadler ([20]), (alter-natively, see Chapter 8), we have:

Proposition 7.3.1. Let Lv denote the closed set of all [p], where p = (Rq� S

i↪→ T ) is a

correspondence with v ∈ V (S) (see definiton 6.2.2). Let n = |T |−1 denote the top dimensionof a cell in PT . Then:

131

(i) Lv \ {0} is the face poset of a n− 1−sphere (with the convention that S−1 = ∅).

(ii) For v 6= w, (Lv ∩ Lw) \ {0} is the face poset of a closed n − 1−ball, whose interior is

given by correspondences q = (Rq� S

i↪→ T ) with q(v) = q(w).

Proof. Nadler’s construction explicitly realizes Lv \ {0} as a face poset of the standard unitn−1−sphere Sn−1 ⊂ Rn. Similarly, (Lv ∩Lw)\{0} is realized as the face poset of the closedsubset of Sn−1 ⊂ Rn cut out by a finite (nonzero) number of equations of the form xi ≥ 0,where xi is a coordinate on Rn. This space is homeomorphic to a closed n− 1−ball, and thestatement about the interior is also implied by the construction.

For any Pv, Pw ∈ Rep0(T, µ), the assignment:

U 7→ HomQµ(Pv |U , Pw |U)

Is a sheaf on PT valued in Ch(k) (the underlying ∞−category of bounded chain complexesover k) which we call HomQµ(Pv, Pw).

Lemma 7.3.1. We can describe the sheaf HomQµ(Pv, Pw) explicitly: Let p = (Rq� R

i↪→

T ). Then the stalk:

HomQµ(Pv, Pw)[p] =

kv, w ∈ V (S) and all edges between v and wpointing from v to w are contracted by q

0 Else.

And the maps between stalks are the identity when both stalks are k, and 0 otherwise.

Proof. This follows immediately from the definitions of Rep0(R, λ) for a tree R and therestriction functors cp : Rep0(T, µ)→ Rep0(R, λ).

Proposition 7.3.2. Let (T, µ) be a directed tree, and let K = KT . For v, w ∈ V (T ), writeFv,w = HomQµ(Pv, Pw). Then:

(i) If v ∈ V (T ), the restriction Γ(PT ;Fv,v)→ Γ(UK ;Fv,v) is not a quasi-isomorphism.

(ii) If v 6= w ∈ V (T ) and there exists a directed path from v to w, then the restrictionΓ(PT ;Fv,w)→ Γ(UK ;Fv,w) is not a quasi-isomorphism.

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Proof. It will suffice to show that the cohomology Γ•(PT ;Fv,w) is not isomorphic to Γ•(UK ;Fv,w).Let n = |T | − 1 be the dimension of top-cells.

In case (i), Γ•(PT ;Fv,v) ∼= k. By lemma 7.3.1, we have Fv,v is the constant sheaf on theclosed subset Lv\{0}. Hence by proposition 7.3.1, when n > 0, Γ•(UK ;Fv,v) ∼= H•(Sn−1) ∼=k ⊕ k[1− n]. When n = 0, Γ•(UK ;Fv,v) ∼= H•(S−1) = 0. In either case, Γ•(UK ;Fv,v) is notisomorphic to k.

In case (ii), Γ•(PT ;Fv,w) = 0. Note n ≥ 1, since v 6= w. Fv,w is zero outside the closedsubset (of UK), Zv,w = (Lv ∩Lw)\{0}. Furthermore, since there is a directed path from v tow, the stalk (Fv,w)[p] is nonzero for [p] ∈ Zv,w only when q(v) = q(w). By proposition 7.3.1and lemma 7.3.1, we have:

Γ•(UK ;Fv,w) ∼= H•(Bn−1, ∂Bn−1) ∼= k[1− n]

As a corollary, the restriction Γ(PT ;Qµ) → Γ(UK ;Qµ) is not quasi-fully faithful whenK = KT . This particular result does not help us classify moves (As the decompositionKT = KT t ∅ obviously does not give a move), however the above proposition will serve asa crucial ‘base case’ in the proof of our main result later in this chapter.

7.4 Classification of Good Subsets

The Result

Let us state the main result first, then we will give a detailed proof.

Definition 7.4.1. Let (T, µ) be a directed tree. A generalized cyclic interval is a subsetof KT of the form I(v, w,D), where:

• v, w ∈ V (T )

• D is a subset of the terminal vertices Vt(T ) of T .

• I(v, w,D) ⊂ KT consists of all elements of D, all edges between a point in D and theshortest path joining v and w, and all edges on the shortest path joining v and w whichare pointing from v to w.

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v′

v w

Figure 7.6: A Generalized Cyclic Interval I(v, w,D).

An example is given in Figure 7.6. Elements of I(v, w,D) are thick blue edges and blueterminal vertices, while elements not in I(v, w,D) are thin gray lines and gray terminal ver-tices. D is the set of blue terminal vertices. We remark briefly that the generalized cyclicinterval does not necessarily uniquely determine the data v, w,D: In the figure, we haveI(v, w,D) = I(v′, w,D).

The reason for the term ‘generalized cyclic interval’ will be explained in the next section.The main result of this section is:

Theorem 7.4.1.

(i) UK ↪→ KT is a weak equivalence with respect to HomQµ(Pv, Pw) if and only if K isnot a generalized cyclic interval of the form I(v, w,D).

(ii) K ⊂ KT is good if and only if it is not a generalized cyclic interval.

A proof of this result will involve some careful combinatorial arguments. The reader mayskip directly to the examples for An and Sn trees with no crucial gaps.

Combinatorial Retracts

We introduce two simple combinatorial lemmas which will do most of the heavy lifting withrespect to quasi-equivalences of sheaves of dg-categories. These lemmas can be stated moregenerally using the language of sheaves valued in an arbitrary∞−category with small limits.We leave the proofs and definitions to appendix B. It is enough to understand the statementsof these facts in order to follow the arguments in the following section.

Let P be a poset. We let Sh(P ; C) denote the category of sheaves on P valued in an∞−category C with small limits.

Definition 7.4.2. Let Q ∈ Sh(P ; C). The inclusion of a subset i : A ↪→ P is a weakequivalence with respect to Q if the map of global sections Γ(Q;P) → Γ(i∗Q;A) is anequivalence in C.

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Definition 7.4.3. Let P be a poset, and A any subset. A combinatorial retract from Pto A is a map r : P → A satisfying:

(i) r(x) = x for x ∈ A.

(ii) If x ≤ y, r(x) ≤ r(y).

(iii) Either (a) r(x) ≤ x for all x ∈ P, or (b): r(x) ≥ x for all x ∈ P

If (iii)(a) holds, r is a downward combinatorial retract, and if (b) holds it is an upwardcombinatorial retract.

Lemma 7.4.1. Let A be a subset of a poset P, and Q ∈ Sh(P ; C).

(i) If there exists a downward combinatorial retract r : P → A, i : A ↪→ P is a weakequivalence with respect to Q.

(ii) If there exists an upward combinatorial retract r : P → A and a morphism F : r∗i∗Q →Q such that i∗F : i∗Q → i∗Q is the identity morphism, then i : A ↪→ P is a weakequivalence with respect to Q.

Proof. See appendix B (lemma B.4.1).

In practice, our sheaves will be defined using 2-functors (Q, c, η) : P → dg-Cat, seedefinition 5.2.1. When this is the case, when we have an upward combinatorial retract r,the required morphism in (ii) can be specified by a left or right morphism of 2-functorsi∗r∗Q → Q (definition B.3.1) given by the data F, γ, such that, when x ∈ A, Fx = IdQx ,and when x ≤ y ∈ A, γxy = Idcxy .

We will also deal with the case C = Ch(k), the underlying∞−category of bounded chaincomplexes over k with the standard (Quillen) model structure. In this case, sheaves of chaincomplexes will all specified by functors P → Ch(k), and morphisms of these specified bynatural transformations.

We also have the following lemma:

Lemma 7.4.2. Let Q ∈ Sh(P ; C). Suppose C has a zero object 0 (i.e. an object whichis initial and terminal). If j : U ↪→ P is the inclusion of an open subset of P such thatj∗Q ∼= 0, then i : P\U ↪→ P is a weak equivalence with respect to Q.

Proof. See appendix B (lemma B.4.2)

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Edge Expansions, Cuts, and Contractions

Let (T, µ) be a directed tree. We describe three combinatorial retracts which will be instru-mental in all our arguments in this section.

If e ∈ E(T ) is an edge, write Ae ⊂ PT for the subset consisting of all [p], where

p = (Rq� S

i↪→ T ), such that either (i) e /∈ E(S) or (ii) e ∈ E(S) and is not contracted by

q.

Definition 7.4.4. If e ∈ E(T ) is an edge, we have the edge expansion map expe : PT → Ae

defined as follows: Let p = (Rq� S

i↪→ T ) be a correspondence. Letting e = {v, w}, if

v, w ∈ S and q(v) = q(w), we define expe([p]) = [p′], where p′ = p except the edge e is notcontracted by q′. For all other p, expe([p]) = [p]. (See Figure 7.7)

e e

expe

e e e

Figure 7.7: Illustrating The Edge Expansion expe

Then expe is a downward combinatorial retract. Now, any edge e divides T into two

disjoint subtrees Σ1,Σ2. For a subtree Σ of T , let pΣ := (Σ∼← Σ

i↪→ T ).

Definition 7.4.5. For e,Σ1,Σ2 as above, define the edge cut:

cute : Ae\N([pΣ2 ])→ N([pΣ1 ])

As follows: If p = (Rq� S

i↪→ T ) and [p] ∈ Ae\N([p]), then we know S intersects Σ1, and

q(Σ1) and q(Σ2) are disjoint in R (by virtue of e not being contracted). We define:

cute([p]) = [(R ∩ q(Σ1)q� S ∩ Σ1

i↪→ T )]

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(See Figure 7.8).

cute

Σ2Σ1

e e

q(Σ1)

e q(Σ2)

Σ2Σ1

e

S ∩ Σ1 q(Σ1)

Figure 7.8: Illustrating the Edge Cut cute.

Then cute is an upward combinatorial retract. Finally, we define edge contraction- whichis just as it sounds. If e = {v, w}, let Ze ⊂ PT be the closed subset Lv ∩ Lw- that is, classes

of correspondences [(Rq� S

i↪→ T )] with v, w ∈ V (S) (so, e ∈ E(S)).

Definition 7.4.6. Define the edge contraction map cone : Ze → Ze ∩ N(〈e〉), wherecone([p]) is the same as [p] except the edge e is contracted.

Then cone is an upward combinatorial retract.

Proof of Theorem 7.4.1

We will need some preliminary results:

Definition 7.4.7. If Σ is a subtree of T , we define the subsheaf QΣ,µ of Qµ as follows: Let

Rep0Σ(T, µ) denote the full subcategory of Rep0(T, µ) on the objects {Pv}v∈V (Σ). If p = (R

q�

Si↪→ T ) is a correspondence, write p∗Σ = {q(w), w ∈ V (Σ ∩ S)}, and λ = p∗µ. Then the

restriction functor c0p maps Rep0

Σ(T, µ) to Rep0p∗Σ(R, λ).

Then, this data assembles to define the subsheaf QΣ,µ.

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Before we state our first result, let us introduce the following notation: Let Σ be a subtreeof T , such that each terminal vertex of Σ is either a terminal vertex of T or is incident toa unique edge in E(T )\E(Σ). Then we have an inclusion κ : KΣ ↪→ KT , which sends eachedge in E(Σ) to its image in E(T ), and every terminal vertex v ∈ Vt(Σ) to either its imagein Vt(T ) or the unique edge incident to V in E(T )\E(Σ).

Our first main result is the following:

Proposition 7.4.1. Let K ⊂ KT , and suppose e ∈ KT\K is an edge in T . Removinge divides T into two subtrees Σ1,Σ2, where we assume e points from Σ1 to Σ2. Writeκα : KΣα ↪→ KT for the inclusions. Then:

(i) The inclusion N([pΣα ]) ∩ UK ↪→ UK is a weak equivalence with respect to QΣα,µ forα = 1, 2.

(ii) the inclusion UK ↪→ PT is a weak equivalence with respect to QΣα,µ if and only if[pΣα ] ∈ UK or κ−1

i (Σi) is good with respect to (Σα, µ).

(iii) There is a functorial exact sequence of functors on Γ(UK ;Qµ):

F1 → Id→ F2

Where the image of F1 lies in Γ(UK ;QΣ1,µ), and the image of F2 lies in the image ofΓ(UK ;QΣ2,µ).

Proof. (i) WLOG α = 1 (the direction of the edge e will not impact this step).

Write Ae for the set of correspondences in which e is either avoided or contracted, andlet expe : PT → Ae denote the edge expansion, which is a downward combinatorialretract. Since e /∈ K, UK is invariant under expe. Abbreviate Ae,K = Ae ∩ UK- lemma7.4.1 then implies that the inclusion Ae,K ↪→ UK is a weak equivalence with respect toany sheaf on UK .

Note that QΣ1,µ is zero when restricted to N([pΣ2 ]), hence (ref) the inclusion:

Ae,K \N([pΣ2 ]) ↪→ Ae,K

is a weak equivalence with respect to QΣ1,µ. Then:

cute : Ae,K \N([pΣ2 ])→ Ae,K ∩N([pΣ1 ])

Is an upward combinatorial retract. Writing r = cute, we need to construct a map:

i∗r∗QΣ1 → QΣ1

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Notice that, for p = (Rq� S

i↪→ T ) with [p] ∈ Ae,K \N([pΣ2 ]), we have:

QΣ1,[p] = Repq(Σ1)(R, λ) QΣ1,cute([p]) = Repq(Σ1)(R ∩ q(Σ1), λ)

There is a natural equivalence between these categories, which establishes that theinclusion:

Ae,K ∩N([pΣ1 ]) ↪→ Ae,K \N([pΣ2 ])

Is a weak equivalence with respect to QΣ1 . Then since N([pΣ1 ]) ⊂ Ae:

Ae,K ∩N([pΣ1 ]) = UK ∩N([pΣ1 ])

And we are done.

(ii) Consider the diagram of subsets of PT :

N([pΣ1 ]) ∩ UK N(pΣ1)

UK PT

The left arrow is a weak equivalence with respect to QΣ1 , as was established in part(i), and the right arrow is clearly a weak equivalence. Therefore, the bottom arrow isa weak equivalence with respect to QΣ1 if and only if the top arrow is.

If [pΣ1 ] ∈ UK , the top arrow is the identity map. If not, then all elements of K areeither edges in E(Σ1) or terminal vertices of T which lie in V (Σ1). So for q : R→ Σ1,[p ◦ q] ∈ N(pΣ1) ∩ UK if and only if [q] ∈ Uκ−1

1 (K):

N([pΣ1 ]) ∩ UK Uκ−11 (K)

N([pΣ1 ]) PΣ1

∼

∼

So in this case, N(pΣ1) ∩ UK ↪→ N([pΣ1 ]) is a weak equivalence with respect to QΣ1 ifand only if κ−1

1 (K) is good with respect to (Σ1, µ).

The proof for α = 2 is identical.

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(iii) Since Ae,K is a downward combinatorial retract of UK , we can instead construct atriangle on Γ(Ae,K ; i∗Qµ), where i : Ae,K ↪→ UK . We will do this by constructingfunctors of sheaves : i∗Qµ → i∗Qµ, and then taking global sections.

The definition is straightforward. Let p = (Rq� S

i↪→ T ) be a correspondence with

[p] ∈ Ae,K , and let Pv ∈ Rep0(R, λ). Since e is not contracted, the subsets q(Σ1), q(Σ2)partition R (though one of these may be empty). Then if v ∈ q(Σ1), the exact sequenceis:

Pv → Pv → 0

And if v ∈ q(Σ2), it is:0→ Pw → Pw

We also need to know the behavior on morphisms: ev,w should be taken to a map ofsequences. For ev,w, if v, w ∈ q(Σ1) or v, w ∈ q(Σ2), the definition is clear. If not,recall that the edge e points from Σ1 to Σ2, so if ev,w exists we must have v ∈ q(Σ2)

and w ∈ q(Σ1), and to this the functor assigns:

0 Pv Pv

Pw Pw 0

0

0

ev,v

ev,w 0

ew,w 0That this is

compatible with restrictions is evident from the definition.

Recall that for v, w ∈ V (T ), we have the sheaf HomQµ(Pv, Pw). This is a sheaf valued inthe ∞−category Ch(k), and is therefore amenable to the methods in the previous section.Namely:

Lemma 7.4.3. Let K ⊆ K(T ), and v, w ∈ V (T ) with v 6= w. Then if either:

(i) There exists an edge e ∈ K between v and w pointing from w to v, or:

(ii) There exists an edge e /∈ K between v and w pointing from v to w

Then the inclusion i : UK ↪→ PT is a weak equivalence with respect to HomQµ(Pv, Pw).

Proof. Abbreviate Fv,w = HomQµ(Pv, Pw).

(i) Let Ze be the closed subset of PT of correspondences containing the edge e, and cone :

Ze → Ze ∩N(〈e〉) the edge contraction map. Any correspondence p = (Rq� S

i↪→ T )

with v, w ∈ V (S) must be in Ze, so Fv,w ∼= 0 on UK \Ze, meaning Ze ↪→ UK is a weak

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equivalence with respect to Fv,w.

Since e points from w to v, contracting e cannot change whether there is a directedpath from w to v. This means, for [p] ∈ Ze,

(Fv,w)[p] → (Fv,w)r[p]

Is an isomorphism of chain complexes. This implies by lemma 7.4.1 that the inclusion:

Ze ∩N(〈e〉) ↪→ Ze

Is a weak equivalence with respect to Fv,w. Now, consider the diagram:

Ze ∩N(〈e〉) UK

N(〈e〉) PT

We’ve shown the top arrow is a weak equivalence with respect to Fv,w. The left arrowis one since both subsets admit a downward combinatorial retract to 〈e〉. The bottomone is a weak equivalence since the map of stalks:

(Fv,w)0 → (Fv,w)〈e〉

Is an isomorphism. Hence the right arrow is as well.

(ii) Since there is an edge between v and w pointing from v to w, Γ(PT ;Fv,w) ∼= 0.

Let i : UK ↪→ PT denote the inclusion. Writing Ae,K = Ae ∩ UK as before, weknow the inclusion j : Ae,K ↪→ UK is a weak equivalence with respect to any sheafof ∞−categories on UK . Since e points from v to w, Fv,w is zero on Ae,K , meaningΓ(Ae,K ; j∗i∗Fv,w) ∼= Γ(UK ; i∗Fv,w) ∼= 0.

We are now ready for a proof of theorem 7.4.1:

(i) UK ↪→ KT is a weak equivalence with respect to HomQµ(Pv, Pw) if and onlyif K is not a generalized cyclic interval of the form I(v, w,D).

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(ii) K ⊂ KT is good if and only if it is not a generalized cyclic interval.

Proof. (i) Let v, w ∈ V (T ), K ⊆ KT . If K = I(v, w,D) for some D, then D mustequal the set of terminal vertices of T which are in K. Unless explicitly stated other-wise, for the rest of this proof D will refer to this set. To abbreviate notation, we letFv,w = HomQµ(Pv, Pw). We divide the analysis into cases.

Case 1: There exists an edge in K between v and w pointing from w to v, or an edgenot in K between v and w pointing from v to w.

By lemma 7.4.1 the inclusion UK ↪→ PT is a weak equivalence with respect to Fv,w. Ifeither of these hold, K 6= I(v, w,D).

Case 2: There is a vertex u ∈ D, and an edge e /∈ K which is between u and theshortest path joining v and w.

Then K 6= I(v, w,D). Additionally, e separates T into two disjoint subtrees, Σ1 andΣ2, WLOG v, w ∈ V (Σ1). Since 〈u〉 ≤ [pΣ1 ] we have [pΣ1 ] ∈ UK , and so proposition7.4.1 implies the inclusion UK ↪→ PT is a weak equivalence with respect to QΣ1,µ. Pvand Pw are both objects of the subcategory QΣ1,µ(PT ), and so UK ↪→ PT is a weakequivalence with respect to Fv,w.

Case 3: Cases 1,2 do not hold. This gives I(v, w,D) ⊆ K. Suppose there is an edgee not between v and w which is not in K.

Then e separates T into two subtrees Σ1,Σ2, WLOG v, w ∈ Σ1- by our assumption,then Σ2 cannot contain any elements of D. If [pΣ1 ] ∈ UK , then by the same reasoningas the previous paragraph, UK ↪→ PT is a weak equivalence with respect to Fv,w, andK contains some edge in Σ2, hence is not equal to I(v, w,D). If [pΣ1 ] /∈ UK , we candraw the same diagram of subsets as in the proof of 7.4.1:

N(pΣ1) ∩ UK N(pΣ1)

UK PT

By the same reasoning as in that proof, the bottom arrow of this diagram is a weakequivalence with respect to Fv,w if and only if the top arrow is. Furthermore, K =

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I(v, w,D) ⊂ KT if and only if κ−11 (K) = I(v, w,D) ⊂ KΣ1 (recall D ⊂ Σ1). So we

can inductively assume K contains all edges except those which are between v and wpointing from w to v.

Case 4: K contains all edges except those which are between v and w pointing fromw to v.

If this is the case, K is a generalized cyclic interval if and only if D contains everyterminal vertex in Vt(T ), except for v or w (if those happen to be terminal vertices).To see why, if there were a terminal vertex u /∈ D which is not equal to v or w, thenwe have u /∈ D, but the unique edge incident to u is in K, which cannot be the case ifK = I(v, w,D).

Case 4a: K is not a generalized cyclic interval.

That is, there is a terminal vertex of u not in D which is not equal to v or w. Letωu denote the reflection isomorphism at u. UK ↪→ PT is a weak equivalence withrespect to Fv,w if and only if Uωu(K) ↪→ PT is a weak equivalence with respect to

HomQruµ(Pv, Pw). Since ωu acts as a transposition of u and the unique edge e inci-dent to u, and e ∈ K, this puts us in Case 2.

Case 4b: K is a generalized cyclic interval.

Finally, we assume K contains all edges except for the ones between v and w pointingfrom v to w, and all terminal vertices not equal to v or w, so K is a generalized

cyclic interval. Let Zv,w denote the set of correspondences [p] = [(Rq� S

i↪→ T )] with

v, w ∈ V (S). Let Av,w be the open subset of Zv,w consisting of correspondences whereall edges pointing from w to v (those which are not in K) are contracted. Lastly, let

pv,w denote the correspondence (Rq� T

∼→ T ), where q contracts all the edges not inK. Note that Av,w = Zv,w ∩N([pv,w]). We have the following diagram of subsets:

Av,w ∩ UK Zv,w ∩ UK UK

N([p]) \ {[pv,w]} N([pv,w]) PT

Now, 7.3.2 implies the bottom-left horizontal arrow is not a weak equivalence withrespect to Fv,w. It will follow that the right vertical arrow also is not, as long as the

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other four arrows are.

The top-right horizontal arrow is a weak equivalence with respect to Fv,w, since Fv,wis zero on the complement of Zv,w. The left vertical arrow is a weak equivalence forthe same reason: K contains all edges not contracted by pv,w, and all terminal verticesnot equal to v or w (and no correspondence in Av,w is in N(〈v〉) or N(〈w〉) if either isa terminal vertex) we get:

Av,w ∩ UK = (N([pv,w])\{[pv,w]} ∩ Zv,w

Now consider the top-left horizontal arrow. The map r : Zv,w → Av,w given by applyingall edge contractions cone for e between v and w pointing from w to v is an upward com-binatorial retract. Also, contracting edges pointing from w to v will not affect whetherthere is a directed path from w to v, hence the restriction Fv,w(N(x))→ Fv,w(N(r(x)))is an isomorphism for any x. This implies Av,w ↪→ Zv,w is a weak equivalence withrespect to Fv,w. Furthermore, since none of the contracted edges are in K, r mapsZv,w ∩ UK to Av,w ∩ UK .

Lastly, the map of stalks:(Fv,w)0 → (Fv,w)[pv,w]

Is an isomorphism, which implies the bottom-right horizontal arrow is also a weakequivalence with respect to Fv,w. This completes the proof of Case 4b, and thus of (i).

(ii) By part (i), we know that the functor Γ(PT ;Qµ) → Γ(UK ;Qµ) is quasi fully faithfulif and only if K is not a generalized cyclic interval, so it suffices to show that, if K isnot a generalized cyclic interval, the restriction is quasi essentially surjective.

If K = KT , then K is a generalized cyclic interval. If K contains all edges but omitsat least one terminal vertex, we can apply a reflection isomorphism at that terminalvertex. So we can assume there is an edge e which is not in K. Then e divides T intotwo subtrees Σ1 and Σ2.

Since the restriction functor is quasi fully faithful, its essential image is a stronglypretriangulated subcategory of Γ(UK ;Qµ). By proposition 7.4.1, if its essential imagecontains Γ(UK ;QΣ1,µ) and Γ(UK ;QΣ2,µ), it is quasi essentially surjective. Thus it suf-fices to show the inclusion UK ↪→ PT is a weak equivalence with respect to QΣ1,µ andQΣ2,µ.

By proposition 7.4.1, if [pΣ1 ], [pΣ2 ] ∈ UK , this requirement is satisfied. If not, say[pΣ1 ] /∈ UK , all edges and terminal vertices in K are internal to Σ1. Since K is

144

nonempty, we get [pΣ2 ] ∈ UK . If Σ1 = {u} were a single (necessarily terminal) vertex,this would force K = ∅ or K = {u}, each of which are generalized cyclic intervals, sothis cannot occur. So assume Σ1 has more than one vertex. Using the notation from thebeginning of this section: If κ−1

1 (K) were a generalized cyclic interval I(v, w,D) ⊂ KΣ1 ,we would have K = I(v, w,D) is also a generalized cyclic interval. Hence we can as-sume κ−1(K) is not a generalized cyclic interval, and therefore by induction is goodwith respect to (Σ1, µ). Then by proposition 7.4.1, UK ↪→ PT is a weak equivalencewith respect to QΣ1,µ.

7.5 Examples for An and Sn Trees

In this section, we discuss moves with total space An and Sn.

Moves of type An

Recall (proposition 4.4) that cyclic structures on PAn are in bijection with cyclic orders onKAn . The key result is that, in this case, generalized cyclic intervals are cyclic inter-vals. (See appendix A for the notion of a cyclic interval). This explains the terminology.This is the content of the following proposition:

Proposition 7.5.1. A subset K ⊆ KAn is good with respect to a cyclic structure if and onlyif K is not a cyclic interval in the associated cyclic order on KAn. (See appendix A for thedefinition of cyclic interval).

Proof. It suffices to consider the case where the cyclic structure is Oµ, and µ is the orienta-tion where all edges are pointing to the right. If we let ei denote the edge {i, i+1}, e0 denotethe vertex 1 and en denote the vertex n, proposition 3.6.2 implies that the cyclic order onKAn is induced by the total order e0 < e1 < · · · < en.

Consider the generalized cyclic interval I(i, j,D). We consider possible cases for i, j,D.

If i < j and D = ∅, then I(i, j,D) is the closed interval [ei, ej−1]. If D = {1}, thenI(i, j,D) is [e0, ej−1]. If D = {2}, I(i, j,D) = [ei, en]. If D = {1, 2}, then I(i, j,D) = KAn .Each of these are cyclic intervals.

If i ≥ j and D = ∅, then I(i, j,D) = ∅. If D = {1}, I(i, j,D) = [e0, ej−1]. If D = {2},I(i, j,D) = [ei, en]. Finally, if D = {1, 2}, I(i, j,D) = [ei, ej−1] (where this interval wraps

145

around). Each of these are cyclic intervals.

Then we see any generalized cyclic interval is a cyclic interval, and every cyclic intervalis a generalized cyclic interval.

We can illustrate a partition KAn = K1 tK1 by representing elements of KAn as pointson a circle (respecting the cyclic order), and filling in the elements of K1 while keeping theelements of K2 empty circles. Let us examine some small cases.

For KA3 there are four possible partitions up to equivalence and switching 1 and 2. SeeFigure 7.9. Only the fourth gives a move-the ‘H-to-I’ move. The second and third are thenon-moves illustrated at the beginning of this chapter (Figures 7.4 and 7.5). The first withK2 = ∅ clearly does not give a move.

Figure 7.9: Possible Partitions of KA3- Only the Fourth Gives a Move

When T = A4, up to equivalence and switching 1 and 2 there is only one possible movewith total space PA4 , pictured in Figure 7.10.

Figure 7.10: The Partition Givng The Unique A4 Move Up To Equivalence/Reversal

We can give a topological picture of this move, as a continuous move of cell complexes(Figure 7.11). It looks a bit like a tetrahedral H-to-I move.

The above picture does not really capture the symplectic geometry of the move. We’llinvestigate that more in the final section.

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Figure 7.11: A Topological Picture of The A4 Move

Moves of type Sn

By proposition 3.7.1, we know cyclic structures on PSn are in bijection with polarizations onKSn .

Proposition 7.5.2. A subset K ⊆ KSn is a good with respect to a cyclic structure if andonly if either:

(i) There are at least three equivalence classes of KSn such that K contains exactly oneelement of that equivalence class.

(ii) There are two equivalence classes of KSn such that K contains exactly one element ofthat equivalence class, and those two elements have the same sign with respect to thepolarization.

Proof. As in the previous part, we can prove this by looking at all possible generalized cyclicintervals. Denote the elements of KSn by ei, vi, with 1 ≤ i ≤ n. Let us use the term ‘prohib-ited set’ to denote the subsets of KSn which are not of the form described in the proposition.Then our goal is to show that generalized cyclic intervals are exactly the prohibited sets.

WLOG we can assume the cyclic structure is Oµ, where all edges are pointing outward.The ρ(ei) = − and ρ(vi) = + (see the proof of proposition 3.7.1).

Letting 0 denote the central vertex, the generalized cyclic interval I(0, 0, D) either con-tains each equivalence class {ei, vi} or is disjoint from it, depending on whether vi ∈ D, henceis a prohibited set. Furthermore, each prohibited set which either contains or is disjoint fromeach equivalence class is of this kind.

The generalized cyclic intervals I(0, vi, D) and I(vi, 0, D) are disjoint from each equiva-lence class {ej, vj} with j 6= i and vj /∈ D, and contains each equivalence class {ej, vj} with

147

j 6= i and vj ∈ D, so are prohibited sets. Furthremore, each prohibited set which eithercontains or is disjoint from all but one equivalence class if of this kind.

I(vi, vi, D) contains each equivalence class {ej, vj} with j 6= i and vj ∈ D, and is disjointfrom each equivalence class {ej, vj} with j 6= i and vj /∈ D, so is a prohibited set.

I(vi, vj, D) contains each equivalence class {ek, vk} with k 6= i, j and vk ∈ D, and is dis-joint from each equivalence class {ek, vk} with k 6= i, j and vk /∈ D. Furthermore, I(vi, vj, D)contains ej, but not ei, by the direction of the arrows. Therefore if I(vi, vj, D) containsexactly one element of {ei, vj} and of {ej, vj}, it contains ej and vi, which have oppositesigns. (This occurs when vi ∈ D but vj /∈ D). Hence I(vi, vj, D) is a prohibited set, and anyprohibited set which either contains or is disjoint from all but two equivalence classes is ofthis kind.

As a corollary, we can enumerate moves with total space PS4 , up to equivalence. We candiagram KS4 as three pairs of objects, each pair consisting of a + object and a − object withrespect to the polarization. As before, we fill in the elements of K1 while keeping elementsof K2 empty. Then the four possible moves, up to equivalence and switching 1 and 2, arepictured in Figure 7.12.

+

+

+

−−

−

+

+

+

−−

−

+

+

+

−−

−

+

+

+

−−

−

Figure 7.12: The Four Distinct Partitions of KS4 Giving Moves

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Chapter 8

The Geometry of Arboreal Spaces

This chapter is an overview of the geometric theory of arboreal singularities. We make somecomments relating the geometric situation to the combinatorial situation, and present someconjectures at various levels of precision.

8.1 Stratified Spaces

Before we introduce arboreal singularities, we make some comments on stratified spaces.The definition below (and much of the following discussion) is inspired by [17].

Definition 8.1.1. A stratified space is a (Hausdorff, locally compact, second countable)topological space X which is can be written as a disjoint union: X = tα∈AXα, where the Xα

are called strata. In addition we have:

(i) Each Xα is equipped with the structure of a smooth manifold of some dimension, whichis compatible with the subspace topology on Xα.section

(ii) For α 6= β ∈ A, if Xβ intersects Xα, then Xβ ⊂ Xα and dim(Xβ) < dim(Xα).

A stratified space is called n-dimensional if any strata Xα is contained in the closureof some Xβ with dim(Xβ) = n.

Definition 8.1.2. Given a stratified space X = tα∈AXα, the strata poset S (X) of X isthe poset whose elements are A, and whose relation is given by β ≤ α iff Xβ ⊂ Xα.

We have some basic constructions on stratified spaces: Any open subset U of a stratifiedspace X is also a stratified space, whose strata Uα are the nonempty intersections Xα ∩ U .

149

Additionally, for any stratified spaces X and Y , the cross product X × Y has the structureof a stratified space whose strata are cross products of strata of X and Y .

Definition 8.1.3. Let X and Y be stratified spaces. An open embedding is a map i :X ↪→ Y which is topologically an open embedding, such that for any x ∈ Xα, there exists anopen subset U of Xα containing x and a strata Yβ of Y such that ϕ(U) ⊂ Yβ and ϕ |U is adiffeomorphism.

We close with two lemmas which will be useful in the next section:

Lemma 8.1.1. Let i : X ↪→ Y be an open embedding of stratified spaces. If the strata of Xare connected, then i determines an map si : S (X)→ S (Y ) of posets, such that wheneverx ∈ Xα, i(x) ∈ Ysi(α). Furthermore si is order-preserving and has open image.

Proof. Let Xα and Yβ be strata of X and Y respectively. The definition of open embeddingimplies that i−1(Yβ) ∩Xα is an open subset of Xα. Since Xα is connected, the image i(Xα)is contained in a unique strata of Y , this defines si.

To show si is order-preserving: we need to show Xα ⊂ Xβ implies Ysi(α) ⊂ Y si(β). If wetake x ∈ Xα, then any open set containing X intersects Xβ, hence any open set containingi(x) intersects i(Xβ), which establishes this fact.

To show si has open image: we need to show Ysi(α) ⊂ Y β, implies β is in the image ofsi. This follows from i being an open embedding, hence the image of X is open and so mustintersect Yβ.

Due to our definition of the category Pos (definition 2.1.2), we would like to identifytopological conditions which imply si is an open embedding of posets- the only piece theabove lemma doesn’t give us is the injectivity of si.

Definition 8.1.4. We say a stratified space X is a local model if the poset S (X) has aunique minimal element and the strata of X are connected. X is locally connected if, forevery strata Xα and every x ∈ Xα, there is an open set U containing x such that Xα ∩ U isconnected.

Lemma 8.1.2. Let i : X ↪→ Y be an open embedding of stratified spaces. If X is a localmodel and Y is locally connected, the map si is injective (and therefore an open embeddingof posets).

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Proof. Let α0 denote the minimal element of S (X), and pick x ∈ Xα0 . Fix a strata Yβ withβ in the image of si- it follows from the continuity of si that i(x) ∈ Y β, so we can find anopen set U containing i(x) such that Yβ ∩ U is connected.

Now suppose α ∈ S (X) with si(α) = β, so i(Xα) ⊆ Yβ. From the continuity of si, we

have i(x) ∈ i(Xα), so i(Xα)∩U is nonempty. Furthermore, the definition of open embeddingimplies i(Xα) is open in Yβ. This implies:

Yβ ∩ U =∐

si(α)=β

i(Xα) ∩ U

Is a covering of Yβ∩U by disjoint nonempty open sets, so the connectedness of U∩Yβ impliessi is injective.

8.2 Arboreal Singularities and Arboreal Spaces

We now explain the geometric construction of arboreal singularities, as presented in [20]:

Definition 8.2.1. Given a tree T , for v ∈ V (T ), we define the Euclidean space LT (v) =RV (T )\{v}, whose coordinates are written xw(v), for w ∈ V (T )\{v}. The arboreal singu-larity LT is the quotient space:

LT =

∐v∈V (T )

LT (v)

/ ∼

Where ∼ is the equivalence relation generated by the edge relations: If e ∈ E(T ), withe = {v, w}, we set (xu(v))u∈V (T )\{v} ∼e (xu(w))u∈V (T )\{w} if:

xw(v) = xv(w) ≥ 0

xu(v) = xu(w) for all u ∈ V (T )\{v, w}

It is easy to see that the quotient map is a closed embedding restricted to LT (v), hencewe can consider LT (v) to be a subset of LT . For a fixed x ∈ LT . We have a subset Sx ofV (T ) given by:

Sx = {v ∈ V (T ) | x ∈ LT (v)}In [20] it is proven that Sx is a subtree. Now, for e = {v, w} ∈ E(Sx), by assumption wehave xw(v) = xv(w) ≥ 0. Define the subset Ex ⊂ E(Sx) to be those edges e = {v, w} suchthat xw(v) = xv(w) > 0. Then we get:

q : Sx � Rx

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The quotient map obtained by contracting the edges in Ex. This gives a correspondence:

px = (Rx

q� Sx

i↪→ T )

Definition 8.2.2. Let T be a tree, and p : R→ T a correspondence. Define:

LT (p) = {x ∈ LT | [px] = [p]}

The following statement is proven in [20]:

Proposition 8.2.1.

(i) LT (p) is an open cell of dimension |V (T )| − |V (R)|. In particular, for v ∈ V (S),LT (p) ⊂ LT (v) is cut out by equations of the form xw(v) = 0, xw(v) > 0, xw(v) < 0.

(ii) For correspondences p : R→ T , q : R′ → T , LT (q) ⊂ LT (p) if and only if q ≤ p.

Corollary 8.2.1. LT is a stratified space whose strata poset is naturally identified with PT .(The smooth structure on LT (p) is given by its description as an open subset of Euclideanspace).

From this definition we can establish the following lemma, which implies lemma 7.3.2:

Lemma 8.2.1.

(i) LT (p) ⊆ LT (v) if and only if p = (Rq� S

i↪→ T ) satisfies v ∈ V (S)- i.e. if and only if

[p] ∈ Lv (definition 6.2.2). Hence Lv \ {0} is the face poset of Sn−1 ⊆ LT (v).

(ii) For v 6= 0, LT (v)∩LT (w) is the closed subset of LT (v) given by xu(v) ≥ 0 for u ∈ V (T )either equal to w or between v and w. LT (p) is in the interior of LT (v)∩LT (w) iff alledges between v and w are contracted by q. It follows that (Lv ∩ Lw) \ {0} is the faceposet of Sn−1 ∩ {xu(v) ≥ 0}, with the interior given by [p] for which all edges betweenv and w are contracted.

Proof. Both are straightforward from the definition of px above.

Note that LT has a unique zero-dimensional strata- call it {0}.

Definition 8.2.3. A arboreal space is an n−dimensional stratified space L such that, forany x ∈ L there is a tree T and an open embedding i : LT × Rk ↪→ L such that i(0,0) = x.

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It is worth noting that LT itself is an arboreal space according to the above definition.Actually, we can say more: Note that LT is a local model and locally connected, hence so isLT × Rk for any k. The following is shown in [20]:

Lemma 8.2.2. Let p : R → T be a correspondence. Then there exists an open embeddingi : LR ×Rk ↪→ LT such that the image of {0} × Rk is the strata LT (p), and furthermore theopen embedding si : S (LR × Rk)→ S (LT ) is naturally identified with ip : PR ↪→ PT .

Definition 8.2.4. If L is an arboreal space, an arboreal chart is a triple (U, T, ϕ), whereU is an open subset of L, T is a tree, and ϕ : LT × Rk → L is an open embedding whoseimage is U .

For arboreal charts (U, T, ϕ) and (V,R, ϕ′), with V ⊆ U , ϕ−1 ◦ ϕ′ is an open embeddingLR × Rm → LT × Rk, which determines an open embedding of posets PR ↪→ PT .

Our discussion in this section will focus on the relationship between the combinatorialunderstanding of C-Arb singularities and a geometric understanding of arboreal singulari-ties. There are not many concrete results here, but there will be a few conjectures (statedat varying levels of precision) as well as a handful of specific constructions and pictures inlow-dimensional cases.

8.3 Embeddings

Let L be an n−dimensional arboreal space.

For a symplectic vector space (V, ω), if W is a subspace, write W⊥ = {v ∈ V | ω(v, w) =0∀w ∈ W}. Recall that W is isotropic if W ⊂ W⊥, coisotropic if W⊥ ⊂ W , and La-grangian if it is both isotropic and coisotropic. A submanifold X of a symplectic manifold(M,ω) is isotropic/coisotropic/Lagrangian iff TxX ⊆ TxM is the appropriate type of sub-space, for all x ∈ X.

Definition 8.3.1. A Lagrangian embedding of L consists of a symplectic manifold M ofdimension 2n and a topological embedding i : L ↪→M such that:

(i) For each strata Lα ⊂ L, i : Lα ↪→M is a smooth embedding whose image is an isotropicsubmanifold of M . (By an abuse of notation we will sometimes identify Lα and i(Lα)when it will not cause confusion).

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(ii) Whitney’s Condition B: ([17]) Suppose Lα ⊂ Lβ are two strata, and {xi}i∈N and{yi}i∈N are subsequences of Lα, Lβ respectively, converging to x ∈ Lα. Then if thetangent planes TyiLβ converge to V ⊂ TxM and the secant lines xiyi converge to a line` ⊂ TxM

1, we have ` ⊂ V .

(iii) Suppose Lα is a strata, and {xi}i∈N and {yi}i∈N are subsequences of L converging tox ∈ Lα. If the secant line xiyi converges to a line ` ⊂ TxM , then ` ⊂ (TxLα)⊥.

Definition 8.3.2. A pre-cyclic structure on L is the data of, for every arboreal chart(U, T, ϕ), a pre-cyclic structure OU,ϕ on PT , such that if (V,R, ϕ′) is an arboreal chart withV ⊆ U and i : PR ↪→ PT denotes the associated open embeding, i∗OU,ϕ = OV,ϕ′. A cyclicstructure is a pre-cyclic structure in which OU,ϕ is a cyclic structure for every arborealchart.

Note that by our definition of pre-cyclic structures on posets, a pre-cyclic structure on Lis determined by its behavior on arboreal charts of the form (U,A2, ϕ). Additionally, we seepre-cyclic structures on LT are in bijection with pre-cyclic structures on PT .

The following proposition connects symplectic geometry and combinatorics:

Proposition 8.3.1. Lagrangian embeddings of L induce pre-cyclic structures on L.

Proof. It is enough to show that a Lagrangian embedding i : LA2 × Rn−1 → (M,ω), whereM is a 2n−dimensional symplectic manifold, determines a canonical cyclic order on PA2 .

Denote the strata of LA2 by {0}, a, b, c. Write X = i(0 × Rn−1), Xα = i(α × Rn−1) forα = 1, 2, 3, and L = i(LA3 × Rn−1).

The symplectic normal bundle E, given by Ex = (TxX)⊥/(TxX), is a 2-dimensional sym-plectic vector bundle over X. The isotropic embedding theorem (ref) states that we can findan open neighborhood U of X which is symplectomorphic to an open neighborhood of thezero section in T ∗X ⊕ E. Let πX : U → X denote the associated projection.

Fixing a Riemannian metric on the vector bundle T ∗X⊕E determines a smooth distance-squared function ρ : U → R such that X = ρ−1(0). Whitney’s condition B implies (reference:Mather) that we can pick U small enough such that (πX , ρ) : U → X × R is a submersionwhen restricted to any of the strata Xα, α = a, b, c.

1With respect to a coordinate chart around x

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Fix x ∈ X, then π−1(x) is a submanifold diffeomorphic to an open neighborhod of theorigin in T ∗X ⊕ Ex, and Xα ∩ π−1(X) is a one-dimensional submanifold. Letting p : T ∗xX ⊕Ex → Ex, we claim that p is injective on L ∩ π−1(x) in some neighborhood of x. If not,we could find xi, yi ∈ L ∩ π−1(X) be sequences in L ∩ π−1(x) converging to x such thatxi− yi ∈ T ∗xX. By the compactness of projective space, we can take some subsequence suchthat Span(xi − yi) converges to ` ⊂ T ∗xX, contradicting requirement (iii).

So, for small enough ε, q(L ∩ π−1(x) ∩ ρ−1(ε)) ⊂ Ex is three distinct points on anoriented circle, with each point contained in one Xα giving a cyclic order on the strata{Xa, Xb, Xc}. Furthermore, these vary continuously as x, ε change. Lastly, the isotropicembedding theorem implies that different choices for the symplectomorphism T ∗X ⊕E ∼= Uare related by a continuous family, hence this gives a canonical well-defined cyclic order onPA2 .

As a consequence, we have a map:

{Lagrangian embeddings of LT} −→ {Pre-cyclic structures on PT}

Our main conjecture is a bijection of the following form:{‘Regular’ Lagrangianembeddings of LT

}/equivalence←→ {Cyclic Structures on PT} (8.1)

We discuss various aspects of the main conjecture in the subsequenct sections, includingthe notions of ‘equivalence’ and ‘regularity’ which have not been mentioned yet.

Equivalence

The first part of the main conjecture we discuss is the idea of equivalence of Lagrangianembeddings. We begin with the following ‘too strict’ definition:

Definition 8.3.3. Let L be an arboreal space. Two Lagrangian embeddings i1 : L ↪→ (M1, ω1)and i2 : L → (M2, ω2) are germ equivalent if there exist open neighborhoods Uα of iα(L)in Mα, for α = 1, 2, and a symplectomorphism ϕ : U1 → U2 such that ϕ ◦ i1 = i2.

The main issue with germ equivalence as a notion of equivalence is that symplecto-morphisms act linearly on tangent spaces and are consequently far too rigid. For instance,germ equivalence fails to identify the three embeddings of LA2 into R2 pictured in Figure 8.1.

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Figure 8.1: Embeddings Which Are Not Germ Equivalent

To state our proposed definition, we first extend the notion of germ equivalence:

Definition 8.3.4. If iα : L ↪→ (Mα, ωα), α = 1, 2 are Lagrangain embeddings, and πα :Mα → P are submersions onto some smooth manifold P , i1, i2 are germ equivalent rela-tive to P if there exists U1, U2, ϕ as in definition 8.3.3 with π2 ◦ ϕ = π1.

Definition 8.3.5. An equivalence between two Lagrangian embeddings i1 : L → (M1, ω1)and i2 : L → (M2, ω2) consists of a Lagrangian embedding H : L × R → (M, ω) and asubmersion t : M → R satisfying:

(i) Writing π : L× R→ R→ R for the projection onto R, π = t ◦H.

(ii) For α = 1, 2 and ε > 0, write Bα(ε) = (α−ε, α+ε) ⊂ R. Write jα : Bα(ε) ↪→ T ∗Bα(ε)for the Lagrangian embedding as the zero section. Then there exists ε such that theembeddings

iα × jα : L×Bα(ε) ↪→Mα × T ∗Bα(ε)

and:H : L×Bα(ε) ↪→ t−1(Bα(ε))

Are germ equivalent relative to Bα(ε). (Both right hand sides have a natural projectionto Bα(ε)).

Then we have:

Lemma 8.3.1. If there exists an equivalence between to Lagrangian embeddings, they inducethe same pre-cyclic structure on L.

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Proof. Let (U,A2, ϕ) be an arboreal chart on L. Then (U × R, A2, ϕ) is an arboreal charton L×R. The Lagrangian embedding H then induces a cyclic order on PA2 . For ε > 0, thesubchart (U×Bα(ε), A2, ϕ) inherits the same cyclic order from H, which by germ equivalenceis the same cyclic order as induced by iα × j0. This is the same as the cyclic order inducedby iα.

Thus a rightward-pointing map in the main conjecture (8.1) is well-defined. Showing theinjectivity of this map is a central goal of this program going forward- unfortunately, wewere unable to make much progress on this.

Realizing Cyclic Structures using Cooriented Hypersurfaces

We now turn to the question of the surjectivity of the rightward-pointing map in the mainconjecture (8.1).

Definition 8.3.6. A realization of a cyclic structure O on PT is a Lagrangian embeddingi : LT ↪→ (M,ω) inducing O.

Let us not worry about the adjective ‘regular’ for now. Then the surjectivity of the mapis the same as the assertion that every cyclic structure has a realization. We will begin thisdiscussion by analyzing certain types of Lagrangian embeddings of LA2 × Rk.

LA2 is the union of two 1-dimensional Euclidean spaces: LA2(1) with coordinate x2(1),and LA2(2) with coordinate x1(2). It has four strata: We denote the correspondences by {0},p(1), p(2) and p(12) as in chapter 2. (The notation has been slightly modified to emphasizethat p(1), etc. are correspondences). Then:

LA2({0}) = {0} = {0 ∈ LA2(1)} = {0 ∈ LA2}

LA2(p(1)) = {x ∈ LA2(1) | x2(1) < 0}

LA2(p(2)) = {x ∈ LA2(2) | x1(2) < 0}

LA2(p(12)) = {x ∈ LA2(1) | x2(1) > 0} = {x ∈ LA2(2) | x1(2) > 0}

Let’s define a pair of Lagrangian embeddings i± of LA2 as a conical Lagrangian in T ∗R.Write coordinates on T ∗R as (x, ηx). Then the embeddings are given by:

i±(x) =

{(x2(1), 0) x ∈ LA2(1)(0,±x1(2)) x ∈ LA2((2))

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LA2((1)) LA2((12))

LA2(2)

LA2((1)) LA2((12))

LA2(2)

Figure 8.2: The embeddings i+ (left) and i− (right) of LA2 .

They are drawn in Figure 8.2.

From the figure and our conventions in Chapter 2, we see that i+ induces the cyclicstructure Oµ where µ points from vertex 2 to vertex 1.

More generally, let H be a smooth hypersurface inside an n−dimensional manifold M .Let T ∗HM ⊂ T ∗M denote the conormal bundle to H. We denote by S∗M the sphericalprojectivization of T ∗M , whose points consist of pairs (x, [v]), where x ∈M , v 6= 0 ∈ T ∗xM ,and [v] = [w] if w = λv for some λ > 0. S∗HM ⊂ S∗M is the spherical projectivization ofT ∗HM .

Definition 8.3.7. A co-orientation of a hypersurface H is a section σ of S∗HM .

If H is a co-oriented hypersurface, we have a conical2 Lagrangian submanifold LH ⊂T ∗M = {(x, v) | x ∈ H, [v] = σ(x)}. Furthermore, the singular conical LagrangianLH = LH tM × {0} is an embedded arboreal space.

Locally nearH we can find a coordinate neighborhood U ∼= Rn with coordinates {x1, . . . , xn}such that H ∩ U = {x1 = 0} and [dx1] = σ. Then we have an arboreal chart:

(T ∗U ∩ LH , LA2 × Rn−1, ϕ)

Where, writing T ∗(U) = T ∗R×T ∗Rn−1 ϕ = i+× j, j being the inclusion of the zero section.

As a result, if we let ξ1 denote the dual coordinate to x1, locally near U we see the cyclicstructure is induced by the total order:

{x1 < 0} < {x1 > 0} < {x1 = 0, ξ1 > 0}2A subset of T ∗M is conical if it is invariant under scaling by a positive scalar.

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Where each of these denote subsets of T ∗U ∩ LH- the first two subsets of the zero sectionand the last a subset of LH .

Now, let’s consider two hypersurfaces: Let H1 = {y = 0} and H2 = {x = 0} be curves inR2, cooriented toward y > 0 and x > 0 respectively. Then LH1 ∪LH2 is a conical Lagrangian

in T ∗R2 which is isomorphic to LA3 (Figure 8.3).

H1

H2

Figure 8.3: A Lagrangian Embedding of LA3

Our discussion allows us to determine the induced pre-cyclic structure. One can checkthat it is, in fact, a cyclic structure. Explicitly, labelling the one-cells 1, 2, 3, 4 going in acounterclockwise order starting from {x > 0, y = 0}, we can determine the cyclic order. Inthe neighborhood of 1, the top-cells containing it are {1, 2} (the top-right quadrant), {1, 3}(the conormal direction LH1), and {1, 4} (the bottom right quadrant). Our discussion showsthat the induced cyclic order is induced by the total order:

{1, 4} < {1, 2} < {1, 3}

Using similar considerations, we can deduce that the cyclic structure on LA3 corresponds tothe cyclic order on the one-cells:

1

4

2

3

Another important figure for us is constructed as follows: Let f : R → R be a smoothfunction such that f(x) = 0 for x ≤ 0 and f(x) > 0, f ′(x) > 0 for x > 0. Let H1 = {x, f(x)}

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and H2 = {y = 0} be hypersurfaces in R2 both cooriented toward y > 0. Then L = LH1∪LH2

is a conical Lagrangian in T ∗R2 which is isomorphic to LA3 . This is pictured in Figure 8.4,with the one-cells labelled.

H2

H1

1

2

3

4

Figure 8.4: Another Lagrangian Embedding of LA3

The one-cell labelled 4 is, writing (x, y, ξx, ξy) for coordinates on T ∗R2, the subset {(0, 0, 0, t), t >0}. Looking at a cross-section {ξy = t0} for some fixed t0 > 0, we get the 3-d picture inFigure 8.5.

x

yξx

y = f(x)

ξx = −t0f ′(x)

3 1

2

Figure 8.5: A Cross-Section of the Embedding at ξy = t0

We see by projecting onto the (x, ξx)−plane that the cyclic order on the top-cells isinduced by the total order:

{4, 1} < {4, 3} < {4, 2}These constructions are generalized in [20]3. Explicitly, let T be a tree, and v ∈ V (T ) a

vertex. Nadler constructs an embedding i : LT ↪→ T ∗RV (T )\{v} with the following properties:

3The Constructions in [20] refer to Legendrian embeddings- but the theory is essentially the same

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• For w 6= v, there is a smooth function hw : RV (T )\{v} → R with nonvanishing differen-tial.

• hw yields a coordinted hypersurface Hw = {hw = 0} cooriented toward {hw > 0}.

• The image of LT is: ⋃w 6=v

LHw

• More specifically, the image of LT (v) is the zero section, and the image of LT (w) isLHw and the subset {hw > 0} of the zero section.

See that paper for the details of the construction. The vertex v ∈ V (T ) is called a rootand the pair (T, v) a rooted tree. There is an associated orientation µv of the rooted tree inwhich all edges are pointing toward the root. We can prove:

Proposition 8.3.2. The cyclic structure given by the embedding is Oµv .

Proof. To simplify notation, we identify LT with its image in T ∗RV (T )\{v}. Let p = (A2

q�

Si↪→ T ) be a correspondence. First assume v ∈ V (S), WLOG v ∈ q−1(1). Pick any vertex

w ∈ q−1(2). Then [p] lies in the hyperplane Hw, and the three top-cells containing LT (p)are:

LT (p ◦ p(1)) ⊂ LT (v) \ LT (w)

LT (p ◦ p(2)) ⊂ LT (w)\LT (v)

LT (p ◦ p(12)) ⊂ LT (w) ∩ LT (v)

The situation is pictured in Figure 8.6.Therefore the cyclic order is given by:

[p ◦ (1)] < [p ◦ (12)] < [p ◦ (2)]

So the arrow between the trees is pointing toward v- which is consistent with µ.

Now suppose v /∈ V (S). Let w1 be the closest vertex in S to v, WLOG w1 ∈ q−1(1). Pickw2 ∈ q−1(2). Then LT (w2) ∩ LT (v) ⊂ LT (w1) ∩ LT (v), and we are in the situation picturedin Figure 8.7.

Then LT (p) lies in the fiber of the cotangent bundle over the origin in the image (thoughnote that the origin represents a codimension-two subspace), LT (p ◦ (1)) ⊂ LT (w1)\LT (w2)lies in the fiber of the cotangent bundle over {x > 0}, and LT (p ◦ (12)) ⊂ LT (w1) ∩ LT (w2)

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hw < 0

LT (v) \ LT (w)

hw > 0

LT (v) ∩ LT (w)

LT (w) \ LT (v)

Hw = {hw = 0}

Figure 8.6: The planes LT (v) and LT (w)

Hw1

Hw2

hw1 , hw2 > 0

hw1 , hw2 < 0

Figure 8.7: The planes Hw1 ,Hw2 in Lv(T )

lies in the fiber of the cotangent bundle over {x < 0}. From our previous computation thisis enough to determine the cyclic order:

[p ◦ (2)] < [p ◦ (1)] < [p ◦ (12)]

This is again consistent with Oµ.

Conjecture 8.3.1. Any cyclic structure has a realization.

It is enough to show that any cyclic structure of the form Oµ has a realization. We expectthat there is a similar strategy involving cooriented hypersurfaces, perhaps of the followingform: Fix a source v ∈ V (T ). Then:

• For w 6= v, there is a smooth function hw : RV (T )\{v} → R, and a cooriented hypersur-face Hw as before.

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• The image of LT (w) is the zero section, and the union of LHw and the subset {hw > 0}of the zero section is the image of the strata contained in LA, the brane associated tothe subtree connecting v to w (see proposition 6.2.1).

Regularity and Bad Embeddings

In this section we give two ‘bad’ examples exhibiting behavior we would like to prohibit. Thefirst is already prohibited by requirement (iii) of a Lagrangian embedding, and this examplesjustifies that condition. The second is not prohibited by the given definition, which leads usto believe that the definition given is lacking- this explains our use of the mysterious word‘regular’ in the main conjecture (8.1).

First: Let H1 = {x, f(x)} and H2 = {y = 0} be curves in R2 as constructed in the pre-vious section, and define L = {(x, v) | x ∈ H1, v ∈ T ∗H1,x

R2} ∪ {(x, v) | x ∈ H2, v ∈ T ∗H2,xR2}-

so we incude both directions in the conormal bundle and exclude the zero section (Figure 8.8).

H2

H1

Figure 8.8: A Disallowed Embedding of LA2 × R.

This gives a seemingly well-behaved embedding of LA2 × R- but it violates requirement(iii): The secant line connecting (x, 0, 0, 0) and (x, f(x), 0, 0) approaches the line (0, t, 0, 0)(with coordinates (x, y, ξx, ξy)), but the tangent space to the strata at the origin is (0, 0, 0, s),and these are not orthogonal. Indeed, this embedding is disallowed for a reason: The cyclicorder on the three Lagrangian half-planes surrounding the central strata flips as we cross thezero section!

For another example which is not disallowed by our definition, Figure 8.9 shows a La-grangian embedding of LA3 using cooriented half-hypersurfaces.

One can check using the methods of the previous section that the pre-cyclic structureinduced by the above embedding is not a cyclic structure. This suggests that the notion of‘Lagrangian Embedding’ is missing some extra condition- this is the meaning of the myste-

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Figure 8.9: An Embedding of LA3 Realizing a Non-Cyclic Structure

rious term ‘regular’ in our main conjecture.

The ‘regularity’ condition would be something which prohibits embeddings of the aboveform. At this point, I have no concrete definition for regularity. One (somewhat unsatisfying)idea is:

Proposed Definition: A Lagrangian embedding is regular if the induced pre-cyclic structure is a cyclic structure.

However, the hope is that there’s a natural geometric condition which yields the above asa theorem. By the coherence theorem, it is enough to ensure that the cyclic structures in-duced by the embeddings are A3−coherent and S4−coherent- meaning the desired geometriccondition is a requirement in codimensions 2 and 3.

As a final comment in this section: In Chapter 3 we construct pre-cyclic structures onPS4 which are A3−coherent but not S4−coherent. It is an interesting question whether thereare ‘bad’ embeddings of PS4 realizing these pre-cyclic structures, analogous to Figure 8.9.

A Sheaf of Categories

We conclude our discussion of the ‘main conjecture’ by mentioning the representation the-ory of Lagrangian embeddings. The canonical references for the sheaf of categories discussedhere is [13].

Let M be a smooth manifold. We let Sh(M) denote the dg category of cohomologi-cally constructible complexes of sheaves on M over a fixed field k. To an object F of Sh(M),one can define the singular support SS(F), which is a conical Lagrangian subspace of Sh(M).

To an open set U of T ∗M , one can form the dg category Sh(M,U), which is the dg quo-tient of Sh(M) by the subcategory of sheaves F with SS(F) ∩ U = ∅. This data assembles

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to define a presheaf of dg-categories on T ∗M . We let MSh(M) denote the sheafification ofthis presheaf.

If Λ ⊂ T ∗M is a conical Lagrangian subspace, one can construct a category ShΛ(M)of constructible sheaves with singular support contained in Λ. Similarly, ShΛ(M,U) is thedg-quotient of ShΛ(M) by the subcategory of sheaves whose singular support is disjoint fromU . We let MShΛ(M) denote the associated sheaf of dg-categories on T ∗M .

In [20], Nadler essentially shows that, using the embedding LT ↪→ Rn coming from achoice of root, ShLT (Rn) ∼= Rep(T, µ). where µ is the orientation induced by the root.Furthermore, for a correspondence p : R→ T one has an open neighborhood U of LT (p) inT ∗M , and the restriction:

ShLT (Rn)→ ShLT (p)(U)

Coincides with the restriction functor cp : Rep(T, µ) → Rep(R, λ). It follows that thesheaf MShLT (Rn) is quasi-equivalent to the sheaf Qµ- more precisely, one can construct a‘geometric realization’ functor:

χT : Sh(PT ; dgCat)→ Sh(LT ; dgCat)

And we have χT (Qµ) is quasi-equivalent to the pull-back of MShLT (Rn) to LT .

Definition 8.3.8. Two embeddings iα : L ↪→ T ∗Mα, α = 1, 2, of L as a conical Lagrangianare weakly equivalent if the sheaves MShiα(L)(Mα) are quasi-equivalent sheaves of dg-categories on L.

Conjecture 8.3.2. If i : L ↪→ T ∗M induces the cyclic structure Oµ on PT , the sheafMShL(M) produces (under the functor ref) a sheaf quasi-equivalent to Qµ.

As a corollary of the conjecture and the Cyclic Structure Theorem (theorem 5.6.1), onewould be able to conclude that two embeddings are weakly equivalent if and only if theyinduce the same cyclic structure. This can be seen as a weak form of the main conjecture.

8.4 The Geometry of Branes

We make a brief comment on the geometry of branes. In Chapter 6, we introduce the notionof a pre-brane (definition 6.2.1), and specify a subcollection of such objects as branes (defini-tion 6.2.3). While the definition of brane is partially justified algebraically by the connectionto representation theory- namely, the bijection between graded branes and rank-one objects

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in Rep(T, µ) (propositions 6.3.2 and 6.3.3)- we also hope that the definition will turn outto have geometric content. More specifically, we would hope that, given an embedding, thebranes can be ‘smoothed’ into Lagrangian submanifolds, while non-branes cannot.

We present here an illustration of the non-brane in PA3 , relative to the embeddings inFigures 8.3 and 8.4. This is pictured in Figure 8.10.

H1

H2

H2

H1

Figure 8.10: The non-brane in (PA3 ,O) relative to two Lagrangian embeddings

We remark that in each case there does not exist a constructible sheaf on R2 whosesingular support is the non-brane.

8.5 The Geometry of Moves

We move on from a discussion of embeddings to a discussion of arboreal moves:

Definition 8.5.1. Let LT be an arboreal singularity. An abstract elementary arborealmove with total space LT is a function π : LT → R satisfying the following:

(i) π(0) = 0.

(ii) Away from 0, π is a locally trivial fibration. Explicitly: If t 6= 0 there exists ε > 0 suchthat π−1(t− ε, t+ ε) is isomorphic, as a stratified space, to L× (t− ε, t+ ε), where Lsome arboreal space and the isomorphism preserves projection onto R.

(iii) Away from 0, π is a locally trivial fibration. Explicitly: For every open neighborhoodU of 0, there exists an open set V and ε > 0 such that π−1(−ε, ε) ⊆ U ∪ V , and π |Vis a locally trivial fibration at 0.

This notion will be less important for us than the following:

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Definition 8.5.2. Let LT be an arboreal singularity with cyclic structure O. An embeddedelementary arboreal move with total space (LT ,O) is a Lagrangian embedding i : LT ↪→T ∗(M × R), where M is a smooth manifold, inducing O, such that:

(i) LT is a conical Lagrangian subspace of T ∗(M × R).

(ii) If π : T ∗(M ×R)→ R is the projection, π : LT → R is an abstract elementary arborealmove.

(iii) Preservation of Categories: The restriction ShLT (M) → ShLT (π−1(−∞, 0)) is aquasi-equivalence, as is ShLT (M)→ ShLT (π−1(0,∞)).

An ideal definition would share the quality of the ‘abstract’ notion that doesn’t rely onan explicit embedding, while somehow encoding the preservation of categories requirementof the embedded moves. Such a definition escapes us at the time of writing.

Definition 8.5.3. If (PT ,O) is a cyclic C-Arb singularity, and KT = K1tK2 is a partitionof KT into good sets with respect to O, a realization of the combinatorial move UE1 → UE2

is an embedded elementary arboreal move with total space (LT ,O) such that LT (p) intersectsπ−1(−∞, 0) iff [p] ∈ UE1, and likewise with π−1(0,∞) and UE2.

We have very little to say when it comes to theorems or precise conjectures about arbo-real moves. We have what might be called “dreams”:

Dream 1: The combinatorial situation faithfully represents the geometric situ-ation, in that the combinatorial classification of moves gives a geometric classifi-cation of moves up to equivalence.

Dream 2: Any two arboreal Lagrangian skeleta of a symplectic manifold can berelated by a sequence of arboreal moves.

We will not make any claims regarding dream 2. In the following sections, we will exploresome embedded elementary arboreal moves where the dimension of M is one or two, meaningLT has dimension two or three. These examples will shed light on dream 1 (and add somecomplications!)

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Realization of Combinatorial Moves

Just as in a previous section, where we constructed Lagrangian embeddings using coorientedhypersurfaces, here we will construct Lagrangian embeddings into T ∗(M ×R) using movingcooriented hypersurfaces. We will call these constructions movies.

As a first example, consider the H-to-I move. We can represent it as two coorientedpoints moving past each other in one dimension, pictured in 8.11.

Figure 8.11: A Movie Realizing the ‘H-to-I’ Move

Let us look one dimension higher. Movies of cooriented curves in R2 can yield elemen-tary arboreal moves whose total space is three-dimensional, so T = A4 or S4. By proposition7.5.1, we can see there is effectively one combinatorial move with total space PA4 . It can berealized as the movie in Figure 8.12.

Figure 8.12: Realizing the PA4 move with a movie

Preservation of categories holds for such a movie. Recall that points in the movie areone-cells of LA4 , i.e. elements of KA4- ignoring the t = 0 slice, where the origin is the mini-mal strata 0. We can label all the cells appearing in the zero-section of the t < 0 state, thisis shown in Figure 8.13.

The labelling of the one-cells is arbitrary. We can restrict to neighborhoods of codimension-one strata to deduce pieces of the cyclic order:

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{1, 4}

{3, 5}

{2, 5}

{3, 4}

{2, 4}

1

2

3

{1, 2, 4}

{1, 2}

{1, 3}

{2, 3}{2, 3, 5}{1, 2, 3}

{2, 4, 5}

{3, 4, 5}

{1, 3, 4}

Figure 8.13: Labelled Cells In the PA4 Movie

{1, 3} :

5

2 4

{3, 4} :

2

5 1

{1, 2} :

5

4 3

The pieces above are enough to deduce the full cyclic order, illustrated in Figure 8.14,with (as in Chapter 7) filled in dots correspond to K1, or t < 0.

There are four combinatorial moves with total space PS4 . A first example is given inFigure 8.15, with the polarization diagram (we leave the detailed calculation to the reader).

Here is another, different-looking example, in Figure 8.16.

This second move raises some interesting questions. In particular, we notice that if wejust shifted the horizontal line slightly vertically, the second move would look like a sequenceof two moves: The first S4 move, and another simpler one we haven’t diagrammed yet shown

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3

5

1 2

4

Figure 8.14: The Partition Corresponding to the PA3 Move

+

+

+

−

−

−

Figure 8.15: One Movie Realizing a PS4 Move

in Figure 8.17.

Toward a Classification of Moves

Pursuing ‘Dream ‘1’, it is sensible to wonder what role the movie in Figure 8.17 plays. Onecan show the move in Figure 8.17 has total space LA3 ×R, so it is not, strictly speaking, anelementary arboreal move as defined in the previous section. However, it seems to deservethe status of being an elementary move, which suggests we extend the definition of elemen-tary move to allow total spaces of the form LT × Rk:

170

+

+

+

−

−

−

Figure 8.16: A Second PS4 Movie

Figure 8.17: A movie realizing a new move?

Definition 8.5.4. A generalized elementary move is an elementary move with totalspace LT × Rk for some nonnegative integer k

To justify our assertion that this move has total space LA3 × R, we can interpret it as amove of a move. To explain, look at the ending position in the figure. If we make horizontal

171

slices and read the image from the top-down, it looks like the H-to-I move (two points movingpast each other in one dimension, Figure 8.11), then the H-to-I move run in the oppositedirection. For an illustration of these cross-sections, see Figure 8.18.

H-to-I

H-to-I Backwards

Figure 8.18: Horizontal Cross-Sections of the LA3 × R Movie

We can generalize this phenomenon: Any move with total space LT × Rk can be trans-formed into a move with total space LT × Rk+1 where t < 0 corresponds to ‘running themove from the end to the start to the end’, while t > 0 corresponds to ‘stay at the end’, orthe opposite. Formally:

Definition 8.5.5. Let π : LT × Rk → R be a generalized elementary move. The minus-suspension is the generalized elementary move π− : LT × Rk+1 → R defined by π−(x, s) =π(x)− s2. The plus-suspension is defined by π+(x, s) = π(x) + s2.

Then the move illustrated in Figure 8.17 is the plus-suspension of the H-to-I move. Wecan iterate suspensions as well. Applying two minus-suspensions of the H-to-I move resultsin a move which looks like two intersecting cooriented spheres being pulled apart, whileapplying a minus-suspension and a plus suspension gives two ‘saddles’ moving past eachother.

A more sophisticated conjecture might read:

Conjecture 8.5.1. Any generalized elementary move is, up to equivalence, the iteratedsuspension of an elementary move.

Going beyond this, there is another idea raised by Figure 8.16- namely, that some ofthe combinatorial arboreal moves described in Chapter 7 have realizations which are not‘generic’, in the sense that a small perturbation reveals them to be compositions of other(possibly generalized) elementary moves. My conjecture is that any combinatorial move

172

KT = E+ t E− in which any boundary chain is entirely contained within either E+ or E−is of this form. Formulation of a precise conjecture would require a well defined notion ofgeneric move and of composition of moves, both of which are beyond the scope of this dis-cussion.

To close this section, I want to illustrate two movies which also appear in the context ofWeinstein isotopies of Lagrangian submanifolds- see Figure 8.19.

Figure 8.19: Two New Moves?

These moves don’t appear in any of the pictures I’ve drawn so far and, in fact, are notelementary moves according to the definition. One can check that the total spaces of thesemoves are arboreal- the first is LA4 and the second is LS4 . However, both of them ‘changethe topology at infinity’ (in fact, they look like an H-to-I move at infinity)- we might callthem non-compact moves, while what we have been discussing are compact moves.

Specifically, it is property (iii) in the definition which ensures this idea of compactness,and which both of these moves violate. Looking at the t = 0 slice for both moves, we see thatthere is a one-dimensional cell which is entirely contained within that slice (the conormalvector at the origin).

173

If we want to include such moves, this suggests we should define non-compact elementarymoves by eliminating property (iii)- though the observation that each of these non-compactmoves is an H-to-I move at infinity perhaps suggests something more subtle. We proposethe following notion:

Definition 8.5.6. A Type I Noncompact Elementary Move with total space LT is afunction π : LT → R satisfying all the properties in definition 8.5.1 except property (iii), andadditionally satisfying:

(i) There is a unique non-minimal strata LT (p) which is entirely contained within π−1(0).(It follows that LT (p) is one-dimensional)

(ii) Letting U ∼= LR × R denote an open neighborhood of LT (p), the composition LR ×{0} ↪→ U

π→ R is an elementary move with total space LR.

We can similarly modify the combinatorial definition:

Definition 8.5.7. A Type I noncompact combinatorial move with total space PTequipped with a cyclic structure O is a decomposition KT = K1 t {e0} t E2, such thatK1, K2 are good, and the triple of data (UE1 ∩N(〈e0〉), UE2 ∩N(〈e0〉), N(〈e0〉)) determines acombinatorial arboreal move with total space N(〈e0〉).

With this notion, we can view the Type I noncompact moves in Figure 8.19 as realizationsof noncompact combinatorial moves diagrammed in Figures 8.20 and 8.21, with the doublenested circle denoting {e0}, and the one-cells labelled:

174

2 3

154

1

3

4 2

5

Figure 8.20: Diagramming the Type I Noncompact Move of Type A4

B− C−

A+ A−

B+C+

A+

B+

C+

A−

B−

C−

Figure 8.21: Diagramming the Type I Noncompact Move of type S4

175

Appendix A

Cyclic Orders on Sets

There are multiple notions of a cyclic order, which we discuss here. Many proofs are straight-forward and have been omitted. A reference is [12].

Definition A.0.1. Let S be a finite set. A cyclic order on S is a ternary relation c on Ssatsifying:

(i) Cyclicity: If c(x, y, z), then c(y, z, x).

(ii) Antisymmetry: If c(x, y, z), then not c(x, z, y).

(iii) Transitivity: If c(x, y, z) and c(x, z, t), then c(x, y, t).

(iv) Totality: If x, y, z are distinct, then either c(x, y, z) or c(x, z, y).

Note that, by antisymmetry, if c(x, y, z), then x, y, z are all distinct. Cyclic orders havea standard geometric interpretation:

Lemma A.0.1. Giving a cyclic order on a set is equivalent to choosing an embedding i : S ↪→S1 up to isotopy (Where S1 is the standard oriented unit circle). An embedding i inducesa cyclic order c on S, where c(x, y, z) if and only if x, y, z are distinct, and in travellingcounterclockwise from i(x) to i(z), we pass i(y).

Furthermore, we can understand cyclic orders in terms of successor functions:

Definition A.0.2. A successor function s : S → S is a function such that, for anyx ∈ S, S = {x, s(x), . . . , sn−1(x)}.

176

Lemma A.0.2. Giving a cyclic order on a set S is equivalent to fixing a successor functions : S → S. The successor function determined by a cyclic order c is uniqely determined by:c(x, s(x), y) whenever y 6= x, s(x).

A typical way of constructing a cyclic order is by giving a successor function. Another isusing a total order:

Definition A.0.3. Given a total order < on S, the induced cyclic order is the one wherec(x, y, z) if and only if x < y < z, or y < z < x, or z < x < y.

In particular, given the set S = {0, 1, . . . , n − 1}, the following cyclic orders are allequivalent:

• The cyclic order induced by the total order 0 < 1 < . . . < n− 1.

• The cyclic order described by the successor function s(k) = k+ 1, with addition takenmod n.

• The cyclic order described by the embedding S ↪→ S1 given by k 7→ e2kπin .

Lemma A.0.3. If S has n elements, there are (n− 1)! cyclic orders on S.

Proof. Fix x ∈ S. Then successor functions on S are in bijection with permutations of S\{x}via s↔ (s(x), s2(x), . . . , sn−1(x)).

In Chapters 6 and 7, the notion of a cyclic interval is referenced.

Definition A.0.4. If S is a set with cyclic order c, and x, y ∈ S, then the open cyclicinterval (x, y) is defined to be {z ∈ S | c(x, z, y)}. The closed cyclic interval [x, y] isdefined to be (x, y) ∪ {x, y}. A cyclic interval is an open or closed cyclic interval in S.

177

Appendix B

Infinity-Categories and Sheaves

B.1 Quasi-Categories

We use quasi-categories as a model for ∞−categories (or, (∞, 1)−categories). A canonicalreference is [16]. Information on the∞−category underlying model categories and the modelcategory of dg-categories can be found in [8], [3], [34], see also the review in [32].

Definition B.1.1. An ∞−category is a simplicial set C in which all inner horns can befilled- meaning, every map Λn

k → C with 0 < k < n can be extended to a map ∆n → C.

Recall that a simplicial set is a functor C : ∆op → Set, where ∆ is the simplex categorywhose objects are the totally ordered sets [n] = {0, 1, . . . , n}, and whose morphisms areorder-preserving functions. 1 The set C([n]) is called the set of n−simplices of C. ∆n is thesimplicial set Hom∆(·, [n]), and the horn Λn

k for 0 ≤ k ≤ n is the sub-simplicial set consistingof homs whose image avoids k ∈ [n].

In a simplicial set X, for n ≥ 0 and 0 ≤ k ≤ n we have the face maps dnk : C([n]) →C([n − 1]) given by the inclusions [n − 1] ↪→ [n] whose image avoids k, and the degeneracymaps snk : C([n]) → C([n + 1]) given by the surjections [n + 1] → [n] whose preimage of kconsists of two elements.

By objects of C we will mean the set of zero-simplices, and by morphisms A→ B we willmean one-simplices f such that d1

1(f) = A and d10(f) = B.

When n = 2 and k = 1, the condition that an inner horn Λ21 → C can be filled can be

interpreted as saying that for any pair of morphisms f : A → B and g : B → C, thereis a third morphism (the ‘composition’) h : A → C and a 2-simplex α ∈ C([2]) such that

1In practice, we will usually have class of objects and not a set- this technicality is easily incorporatedinto the theory (for example by replacing ‘set’ with ‘discrete category’) and we will not worry about it here.

178

d20(α) = g, d2

1(α) = h, d22(α) = f :

C

A B

fg

h

α

h and α do not have to be unique, but one can show that there is a contractible space ofchoices for h.

A common source of ∞-categories is as the nerve of small categories:

Definition B.1.2. If C is a small category, the nerve NC is the ∞−category where:

• An element of the set NC([n]) is an n−tuple of composable morphisms

X0 → X1 → · · · → Xn

• The face maps dnk : NC([n])→ NC([n− 1]) are given by removing the kth object Xk,and composing the two maps meeting at Xk if 0 < k < n.

• The degeneracy maps snk : NC([k]) → NC([k + 1]) are given by duplicating Xk andinserting an identity morphism.

If P is a poset, we can form the ∞−category NP by viewing P as a category in whichthere is a unique morphism x→ y when x ≤ y.

We introduce some basic definitions:

Definition B.1.3.

• Given two ∞−categories C,D : ∆op → Set, a functor F : C → D is a naturaltransformation.

• Let J be the category with two objects, 0 and 1, and two non-identity morphisms x :0→ 1 and y : 1→ 0 which are inverses. A morphism f : A→ B in an ∞−category Cis an equivalence if there is a functor F : NJ → C such that F (x) = f .

• If C and D are ∞−categories, the product C ×D is defined by (C ×D)([k]) = C([k])×D([k]), with maps defined component-wise.

• If F,G : C → D are functors, a homotopy F → G is a functor η : C × ∆1 → Dmaking the following diagram commute:

179

C

C ×∆1 D

C

F

G

η

i0

i1

Where, for α = 0, 1, iα : C[k] → (C × ∆1)[k] = C[k] × Hom([k], [1]) is the mapS 7→ (S, πα), πα being the constant function j 7→ α.

When C and D are ∞−categories, Hom(C,D) also has the structure of an ∞−category,where the objects are functors and the morphisms are homotopies- higher simplices can besimilarly defined, see [16].

Another source of ∞−categories is the localization of simplicial model categories- thatis, model categories endowed with a compatible simplicial enrichment. The ∞−categoryconstructed- which we call the ‘underlying ∞−category’, has objects which are the fibrantand cofibrant objects of the model category, and morphisms are the usual morphisms ofthese objects. Model categories are equipped with a subcollection of morphisms called ‘weakequivalences’. The key property of the underlying ∞−category is that the equivalences (inthe sense of definition B.1.3) are exactly the weak equivalences.

For us, the two key examples of simplicial model categories are:

• dg-Cat, the category of dg-categories, equipped with the Dwyer-Kan model structure.The weak equivalences are quasi-equivalences of dg-categories (see [8]).

• Ch, the category of bounded chain complexes 2, equipped with the standard (Quillen)model structure. The weak equivalences are quasi-isomorphisms.

We restate an important definition here:

Definition B.1.4. Let C be an ordinary category, and let dg-Cat denote the ∞−category ofdg-categories (over k). A 2-functor Q : C → dg-Cat consists of the following data:

2In this paper, the differential in chain complexes always increases the degree. These are sometimescalled ‘cochain complexes’

180

(i) For each object X ∈ Ob(C), a dg-category QX .

(ii) For each morphism f : X → Y in C, a dg-functor cf : QX → QY .

(iii) For each pair of morphisms f, g such that the composition fg exists, a quasi-isomorphismηg,f : cf ◦ cg → cfg

(iv) Such that, for each triple f, g, h of morphisms such that the composition fgh exists,the following diagram commutes:

cf ◦ cg ◦ ch cfg ◦ ch

cf ◦ cgh cfgh

ηg,f � Idch

Idcf � ηh,g ηh,fg

ηgh,f

Where � refers to the horizontal composition of functors.

The above notion will allow for the construction of ∞−functors:

Lemma B.1.1. If In denotes the poset with n + 1 objects 0 < 1 < . . . < n (which can alsobe viewed as a category), then:

(i) A 2-functor In → dg-Cat describes an n-simplex in dg-Cat.

(ii) For any ordinary category C, a 2-functor C → dg-Cat determines an∞−functor NC →dg-Cat.

B.2 Sheaves

Now, let X be a topological space. The collection of open sets τ is a poset, where we letU ≤ V if and only if V ⊆ U .

Definition B.2.1. If X is a topological space and C is an ∞−category, a presheaf on Xvalued in C is a functor F : N τ → C.

There is a notion of ∞−categorical limits, which are well-defined up to a contractiblespace of choices. Then we have:

181

Definition B.2.2. If X is a topological space and C is an ∞−category with small limits, asheaf on X valued in C is a presheaf F satisfying the following: Let U be any open set, andU any open cover of U for which, if V1 ∈ U and V2 ⊆ V1 then V2 ∈ U . Then the map:

F(U)→ lim←−V ∈U

F(V ) (B.1)

Is an isomorphism.

We let Sh(X; C) denote the category of sheaves on X. Morphisms in Sh(X; C) are homo-topies of functors. A sheaf on X is determined by its values on a base for the topology ofX. Explicitly:

Proposition B.2.1. Let B be a base for a topology τ on X. Define ShB(X; C) to be thecategory of functors NB → C satisfying condition B.1 restricted to B. Then the inclusionNB ↪→ N τ gives an equivalence of categories ShB(X; C) ∼= Sh(X; C).

As a result, we can define:

Definition B.2.3. If P is a poset and C is an ∞−category with small limits, a sheaf on Pvalued in C is an ∞−functor F : NP → C. For x ∈ P, Fx ∈ C is called the stalk of F atx. The morphisms Fx → Fy when x ≤ y are the generization maps.

This is equivalent to the previous definition. To see why, let U be an open subset of P ,considered as a topological space. If F is a sheaf on P , we define:

Γ(F ;U) = lim←−x∈UFx

Then we have Γ(F ;N(x)) ∼= Fx, and the N(x) form a base of the topology for P . Further-more, any open cover of N(x) must include N(x), meaning there are no extra compatibilityconditions among the stalks.

We can then show:

Lemma B.2.1. Let C be a category in which every morphism is monic, and Q : Cop → dg-Catbe a 2-functor. Then for each object A ∈ Ob(C), Q defines a sheaf QA on the poset PA.

Proof. Let (Q, c, η) denote the triple of data associated to a 2-functor. For A ∈ Ob(C) we set:

182

• For f : B → A, the stalk of QA at [f ] ∈ PA is:

QA,[f ] = Q(B)

• For h : C → B, let g = fh. Then [g] ≥ [f ] and the generization map:

c[f ][g] : QA,[f ] → QA,[g] := ch : Q(B)→ Q(C)

• For j : D → C, let k = gj. Then:

η[f ][g],[g][k] := ηh,j

This defines a 2-functor, and hence a sheaf.

B.3 Morphisms Between Sheaves

We have the following notion:

Definition B.3.1. Let C be a category, and Q,Q′ 2-functors C → dg-Cat, with associateddata c, η, resp. c′, η′. A left morphism F : Q → Q′ consists of:

(i) For each X ∈ Ob(C), a dg-functor F(X) : Q(X)→ Q′(X).

(ii) For each morphism f : Y → X, a quasi-isomorphism γf : c′f ◦ F(Y )→ F(X) ◦ cf .

(iii) Such that, for morphisms f : Y → X and g : Z → Y , the following diagram commutes:

c′f ◦ c′g ◦ F(X) c′f ◦ F(Y ) ◦ cg F(Z) ◦ cf ◦ cg

c′fg ◦ F(X) F(Z) ◦ cfg

1c′f � γg γf � 1g

η′f,g � 1F(X) 1F(Z) � ηf,g

γfg

A right morphism is defined similarly, except γf : F(X) ◦ cf → c′f ◦ F(Y ), and in thecommutative diagram in (iii), the horizontal arrows point in the other direction.

Importantly for us, left/right morphisms define morphisms of sheaves:

183

Lemma B.3.1. Let C be a category in which every morphism is monic, and Q,Q′ : Cop →dg-Cat are 2-functors. If F : Q → Q′ is a left or right morphism, F defines a morphism ofsheaves.

Proof. In light of definition B.1.3, we need to construct an∞−functor NPA×∆1 → dg-Catfitting into the appropriate commutative diagram. Note that NP×∆1 ∼= N (PA×I1), whereI1 is the poset {0 < 1}. We construct a 2-functor (also called F), F : PA × I1 → dg-Cat,with C,N the compatibility data:

• Let f : B → A. We let [f ]0, [f ]1 denote elements of PA × I1. The 2-functor sends:

(i) F[f ]0 = QA,[f ] = Q(B)

(ii) F[f ]1 = Q′A,[f ] = Q(B)

• Now suppose h : C → B, and write g = fh. The generization maps:

C[f ]0[g]0 : F[f ]0 → F[g]0

C[f ]1[g]1 : F[f ]1 → F[g]1

are given by ch and c′h, respectively. We set:

C[f ]0[g]1 = F(C) ◦ ch : Q(B)→ Q′(C)

• Let j : D → C, and k = gj. Then we have:

C[g]0[k]1 ◦ C[f ]0)[g]0 = F(D) ◦ cj ◦ ch

C[f ]0[k]1 = F(D) ◦ cjhAnd we can set N[g]0[k]1,[f ]0[g]0 = ηj,h. We also have:

C[g]1[k]1 ◦ C[f ]0[g]1 = c′j ◦ F(C) ◦ ch

And we can set N[g]1[k]1,[f ]0[g]0 to be the composition:

c′j ◦ F(C) ◦ chγj�1ch−→ F(D) ◦ cj ◦ ch

ηj,h→ F(D) ◦ cjh

We also have N[g]0[k]0,[f ]0[g]0 = ηj,h, and N[g]1[k]1,[f ]1[g]1 = η′j,h.

Then the commutative diagram in the definition of left morphism ensures that all the proper-ties of being a 2-functor are satisfied. The construction is analogous for right morphisms.

Given 2-functors F1,F2 : PA × I1 → dg-Cat such that F1 ◦ i1 = F2 ◦ i0 : PA → dg-Cat,there is a natural way to compose them to get F2 ◦F1 : PA×I1 → dg-Cat, and furthermore,these morphisms fit into a 2-simplex in the ∞−category Hom(NP , dg-Cat):

184

•

• •

F2F1

F2◦F1

We conclude this section with a discussion on our construction in Chapter 5: We constructquasi-isomorphisms:

ε+v : Γ−v ◦ Γ+

v → Id

ε−v : Id→ Γ+v ◦ Γ−v

Satisfying certain commutativity properties. The commutativity properties are exactly whatis needed to create diagrams of 2-simplices in Hom(NP , dg-Cat):

• •

• •

Id

Id

Γ+

Id

ε+

Γ−

• •

• •

Γ+

Γ−

Id

Γ+◦Γ−

ε−

Id

The construction of the required 2-simplex and verification of that claim are just as inB.3.1. The existence of these diagrams imply the data (Γ+

v , γ+) and (Γ−v , γ

−) give quasi-inverse morphisms of sheaves of dg-categories.

185

B.4 Pullback Functors and Homotopy

In this section we discuss the proof of the lemmas in Chapter 7. Fix an ∞−category C withsmall limits.

If f : P1 → P2 is a morphism of posets, it induces a functor of∞−categories f : NP1 →NP2

3.

Definition B.4.1. If Q : NP2 → C is an object in Sh(P2; C), we define the pullbackf ∗Q = Q ◦ f ∈ Sh(P1; C).

Now, let Q : NP → C be a sheaf. There is a poset C(P), the cone over P , which isobtained by adjoining a minimal element 0 to P . The limit:

lim←−x∈PQx

Is given by the universal (in the ∞−categorical sense) functor Q making the following dia-gram commute:

NP

NC(P) C

jPQ

Q

Now, let P be a poset, and i : Z ↪→ P be the inclusion of a subset. This gives a commu-tative diagram:

NZ NP

NC(Z) NC(P)

i

jZ jP

i

Which leads to:

NZ

NC(Z) C

jZ i∗Q

i∗Q

3It should not cause confusion to also call this f

186

By universality, this gives a morphism Γ(P ;Q)→ Γ(Z; i∗Q). We say i is a weak equiv-alence with respect to Q when this morphism is an equivalence. It is enough to constructa morphism H : NC(P)→ dg-Cat making the following diagram commute:

NC(Z)

NC(P) P

C

i∗Q

Q

H

i

jP

Figure B.1: A Commutative Diagram Giving Γ(Z; i∗Q)→ Γ(P ;Q)

Where i∗Q is the universal morphism extending i∗Q : NZ → dg-Cat.

We can now prove the two lemmas from Chapter 7:

Lemma B.4.1. (Lemma 7.4.1 in the text) Let A be a subset of a poset P, and Q ∈ Sh(P ; C).

(i) If there exists a downward combinatorial retract r : P → A, i : A ↪→ P is a weakequivalence with respect to Q.

(ii) If there exists an upward combinatorial retract r : P → A and a morphism F : r∗i∗Q →Q such that i∗F : i∗Q → i∗Q is the identity morphism, then i : A ↪→ P is a weakequivalence with respect to Q.

Proof. We construct a diagram as in Figure B.1. H is determined on all objects and on allsimplices within P . So let

∆ = p < x1 < · · · < xn

, we construct the value H(∆).

First, consider the case where n = 1, i.e. we have a simplex:

p < x

We have a one-simplex f = i∗Q(p < r(x)). If r is a downward combinatorial retract, we haveanother one-simplex g = Q(r(x) < x). Furthermore we have ∂0f = i∗Q(r(x)) = Q(r(x)) =

187

∂1g, hence this data gives a map Λ21 → C, which by assumption is fillable. Other simplices

can be defined inductively in a similar way.

If r is an upward combinatorial retract, we are given the data of a morphism F : r∗i∗Q →Q, i.e. (see definition B.1.3)

NP

N (P × I1) NP

C

r∗i∗Q

Q

Fi0

i1

Then to the one-simplex p < x, let f = i∗Q(p < r(x)), and g = F((x, 0) < (x, 1)).We have ∂1g = F((x, 0)) = r∗i∗Q(x) = Q(r(x)), so again we get a map Λ2

1 → C which byassumption is fillable. The other simplices can similarly be defined.

We also provide a proof of:

Lemma B.4.2. (Lemma 7.4.2 in the text) Let U be open in P, and Z = P\U . Let i : Z ↪→ Pand j : U ↪→ P denote the inclusions. If Q : P → C is a sheaf, and j∗Q = 0, then i is aweak equivalence with respect to Q.

Proof. Since U is open, we can construct H as in Figure B.1 by extending i∗Q to be zero onsimplices which aren’t contained in the image of NC(P).

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A Combinatorial Model of Lagrangian Skeleta · A Combinatorial Model of Lagrangian Skeleta by Alex Zorn Doctor of Philosophy in Mathematics University of California, Berkeley Professor