Persistent Homology & Category Theory - Wesleyan Universityskalisnikver.web.wesleyan.edu/ApplicationsofPH.pdf · 2018-06-23 · Category theory book MIT Press has published "Category

Persistent Homology & Category Theory

November 8, 2016

Persistent Homology

There are many ‘introductory’ papers written on the topic of persistenthomology...

Persistent Homology

Here is the Topology & Data by Gunnar Carlsson

Persistent Homology

What is Persistent Homology by Shmuel Weinberger

Persistent Homology

Barcodes: the persistent topology of data by Robert Ghrist

Crash Course in Category Theory

Why category theory?

There are various views on what category theory is about, and what it is goodfor. Here are some.

• As a language, it offers economy of thought and expression

• It reveals common ideas in (ostensibly) unrelated areas of mathematics

• A single result proved in category theory generates many results indifferent areas of mathematics.

To each species of mathematical structure, there corresponds a category, whoseobjects have that structure, and whose morphisms preserve it - Goguen

Less flatteringly – it is often referred to as ‘abstract nonsense.’























David SpivakResearch ScientistDepartment of MathematicsMIT

Office: Building 2, Room 180Email: dspivak--math/mit/edu Curriculum Vitae.

Current research projects

Technical Proposal: "Pixel matrices and other compositionalanalyses of interconnected systems". This is a proposal for another awarded AFOSR grant.

Technical Proposal: "Categorical approach to agent interaction". This is the proposal for the awarded AFOSRgrant FA9550-14-1-0031.

Technical Proposal: Category-theoretic Approaches for the Analysis of Distributed Systems. This is theproposal for the awarded NASA grant NNH13ZEA001N-SSAT.

Other grants. These are several grant proposals, some funded, some in the pipeline, others not funded, thatexplain various facets of my research.

Categorical informatics. What is the underlying mathematical structure of information itself?

FQL. Functorial query language. Build your own categories, functors, and database instances, and push themaround with data migration functors. (Joint with Ryan Wisnesky).

Category theory book

MIT Press has published "Category theory for the sciences". The book can also be purchased on Amazon.Here are reviews by the MAA, by the AMS, and by SIAM.

Free HTML version: not as nice to read, but free to the world, on the MIT Press website. An older version,entitled "Category theory for scientists", can be found here.

Course (Spring 2013). Course webpage.

Past research subjects


Categories

A category C is specified by the following data:

• a class C0 = Obj(C), the objects of the category;

• a class C1 = Arr(C) = Mor(C), the arrows or morphisms of the category.We write C (x , y) or MorC(x , y) or Mor(x , y) to denote the set of arrowsx → y .

• There is a composition operation ◦ : C (x , y)× C (y , z)→ C (x , z).


Categories






Categories






Categories

• Composition is associative: (hg)f = h(gf ) when either side is defined.

• Every object x ∈ C0 has an element 1x ∈ C (x , x), which is an identity inthe sense that f = f 1x and g = 1xg .


Categories

• Composition is associative: (hg)f = h(gf ) when either side is defined.

• Every object x ∈ C0 has an element 1x ∈ C (x , x), which is an identity inthe sense that f = f 1x and g = 1xg .


The Category of Sets

• objects are sets (alice)

• morphisms are functions between sets


Further Examples


The Category of Sets

Any partial order (P,≤)

• Objects are the elements of the partial order

• Morphisms represent the ≤ relation.Composition works because of the transitivity of ≤.


FunctorsLet C ,D be categories. A functor F : C → D is specified by the followingdata:

• Every object x ∈ C0 is assigned an object F (x) ∈ D0.

• Every arrow f ∈ MorC (x , y) is assigned an arrowF (f ) ∈ MorD(F (x),F (y)).

• The functor respects composition: F (f ◦ g) = F (f ) ◦ F (g).

• The functor respects identities: F (1x) = 1F (x).

Examples

• forgetful functor Top→ Set which takes a topological space to itsunderlying set, and which takes each continuous function to itself.

• There is a forgetful functor Vect→ Set which takes a vector space to itsunderlying set, and which takes each linear map to itself.

• Hk : Top→ Group is a functor from the category of topological spaces tothe category of groups.







Examples










Examples










Examples










Examples





The mechanism for comparing two functors or diagrams is the naturaltransformation.

Natural TransformationsLet C ,D be categories and F ,G : C → D functors. A natural transformation νfrom F to G , written

ν : F ⇒ G

is defined as follows.

• To each object x ∈ C we assign an arrow νx : F (x)→ G(x) of D .

• For each arrow f : x → y of C we require that the diagram

F (x)F (f ) //

νx

��

F (y)

νy

��G(x)

G(f ) // G(y)

commutes.




ν : F ⇒ G




F (x)F (f ) //

νx

��

F (y)

νy

��G(x)

G(f ) // G(y)

commutes.




ν : F ⇒ G




F (x)F (f ) //

νx

��

F (y)

νy

��G(x)

G(f ) // G(y)

commutes.

Applications of Category Theory to Persistent Homology

There is one important category that you’ve come across in this class..

Let Vect be a category of vector spaces, and P a partially ordered set.Recall that we regard P as a category with object set P, and with a uniquemorphism from x to y whenever x ≤ y .

The Category of Persistence Vector Spaces

A P-persistence vector space is a functor φ : P → Vect.(In class P = R+. This coincides with our definition: it means a family ofF-vector spaces {Vr}r∈[0,∞), together with linear transformationsLV (r , r ′) : Vr → Vr′ whenever r ≤ r ′, so that LV (r ′, r ′′) · LV (r , r ′) = LV (r , r ′′)for all r ≤ r ′ ≤ r ′′.).


There is one important category that you’ve come across in this class..Let Vect be a category of vector spaces, and P a partially ordered set.

Recall that we regard P as a category with object set P, and with a uniquemorphism from x to y whenever x ≤ y .




There is one important category that you’ve come across in this class..Let Vect be a category of vector spaces, and P a partially ordered set.Recall that we regard P as a category with object set P, and with a uniquemorphism from x to y whenever x ≤ y .


A P-persistence vector space is a functor φ : P → Vect.

(In class P = R+. This coincides with our definition: it means a family ofF-vector spaces {Vr}r∈[0,∞), together with linear transformationsLV (r , r ′) : Vr → Vr′ whenever r ≤ r ′, so that LV (r ′, r ′′) · LV (r , r ′) = LV (r , r ′′)for all r ≤ r ′ ≤ r ′′.).


There is one important category that you’ve come across in this class..Let Vect be a category of vector spaces, and P a partially ordered set.Recall that we regard P as a category with object set P, and with a uniquemorphism from x to y whenever x ≤ y .




P-persistence vector spaces form a category in their own right, where amorphism F from φ to ψ is a natural transformation.

In more concrete terms, a morphism from one persistence vector space intoanother is a persistence linear map, i.e. is a family of linear transformationsfr : Vr →Wr , so that for all r ≤ r ′, all the diagrams

Vr

LV (r,r′) //

fr

��

Vr′

fr′

��Wr

LW (r,r′)// Wr′

commute in the sense that

fr′ ◦ LV (r , r ′) = LW (r , r ′) ◦ fr .


P-persistence vector spaces form a category in their own right, where amorphism F from φ to ψ is a natural transformation.In more concrete terms, a morphism from one persistence vector space intoanother is a persistence linear map, i.e. is a family of linear transformationsfr : Vr →Wr , so that for all r ≤ r ′, all the diagrams

Vr

LV (r,r′) //

fr

��

Vr′

fr′

��Wr

LW (r,r′)// Wr′

commute in the sense that

fr′ ◦ LV (r , r ′) = LW (r , r ′) ◦ fr .


In class we showed from scratch that we have a decomposition theorem forfinitely presented persistence vector spaces (there is no such theorem forgeneral R-persistence vector spaces). This is not how it is done in most surveypapers on the topic.

The main ingredient is to note that the category of N-persistence vector spacesover F is equivalent to another category, one of non-negatively graded vectorspaces over F[t].(Z/2Z[t] consists of polynomials f (t) with coefficients in Z/2Z in variable t.For example, t + 1 and t7 + t2 are both polynomials with coefficients in Z/2Z.)


In class we showed from scratch that we have a decomposition theorem forfinitely presented persistence vector spaces (there is no such theorem forgeneral R-persistence vector spaces). This is not how it is done in most surveypapers on the topic.

The main ingredient is to note that the category of N-persistence vector spacesover F is equivalent to another category, one of non-negatively graded vectorspaces over F[t].(Z/2Z[t] consists of polynomials f (t) with coefficients in Z/2Z in variable t.For example, t + 1 and t7 + t2 are both polynomials with coefficients in Z/2Z.)


Let {Vn} be any N-persistence vector space over F. We define an associatedgraded module θ({Vn}) over the graded polynomial ring F[t] as follows:

θ({Vn}) =⊕s≥0

Vs ,

where the n-th graded part is the vector space Vn. The action of thepolynomial generator t is given by

t · (v0, v1, v2, . . .) = (0, φ01v0, φ12v1, . . .).

It is readily checked that θ is a functor from the category of N-persistencevector spaces over F to the category of graded F[t]-modules.

It is in fact an equivalence of categories, since an inverse functor can be givenby V∗ → {Vn}, where the morphisms φmn are given by multiplication by tn−m.


There is a classification theorem for finitely generated graded F[t]-modules(Structure theorem for finitely generated modules over a principal idealdomain).

Classification theorem for finitely generated graded F[t]-modules

Let V∗ denote any finitely generated non-negatively graded F[t]-module. Thenthere are integers {i1, . . . , im}, {j1, . . . , jn}, {l1, . . . , ln}, and an isomorphism

V∗ ∼=m⊕s=1

t isF[t]⊕n⊕

k=1

t jk (F[t]/t lkF[t]),

The decomposition is unique up to permutation of factors.

This classification theorem has a natural interpretation. The free portions arein bijective correspondence with those homology generators which come intoexistence at parameter is and which persist for all future parameter values. Thetorsional elements correspond to those homology generators which appear atparameter jk and disappear at parameter jk + lk .Finitely presented persistence F-vector spaces mapped under θ are finitelygenerated non-negatively generated F[t]-modules.


There is a classification theorem for finitely generated graded F[t]-modules(Structure theorem for finitely generated modules over a principal idealdomain).

Classification theorem for finitely generated graded F[t]-modules

Let V∗ denote any finitely generated non-negatively graded F[t]-module. Thenthere are integers {i1, . . . , im}, {j1, . . . , jn}, {l1, . . . , ln}, and an isomorphism

V∗ ∼=m⊕s=1

t isF[t]⊕n⊕

k=1

t jk (F[t]/t lkF[t]),

The decomposition is unique up to permutation of factors.This classification theorem has a natural interpretation. The free portions arein bijective correspondence with those homology generators which come intoexistence at parameter is and which persist for all future parameter values. Thetorsional elements correspond to those homology generators which appear atparameter jk and disappear at parameter jk + lk .Finitely presented persistence F-vector spaces mapped under θ are finitelygenerated non-negatively generated F[t]-modules.

Documents

Persistent Homology & Category Theory - Wesleyan Universityskalisnikver.web.wesleyan.edu/ApplicationsofPH.pdf · 2018-06-23 · Category theory book MIT Press has published "Category