Categorical Semantics and Topos Theory Homotopy type theory Seminar University … · 2011-12-15 · Categorical Semantics and Topos Theory Homotopy type theory Seminar University

Categorical Semantics and Topos TheoryHomotopy type theory Seminar

University of Oxford, Michaelis 2011

November 16, 2011

Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011

References

I Johnstone, P.T.: Sketches of an Elephant. A Topos-TheoryCompendium. Oxford University Press, 2002. (Main source)

I Caramello, O.: Lectures on Topos Theory. Cambridge, 2010.(Additional source)

I Bell, J.L.: Notes on Toposes and Local Set Theories. 2009.(Additional source)

I Moerdijk, I. and Mac Lane, S.: Sheaves in Geometry andLogic: A First Introduction to Topos Theory. Springer, 1992.

I Abramsky, S. and Tzevelekos, N.: Introduction to Categoriesand Categorical Logic. Oxford.


Introduction

Let’s start with a simple example. The definition of a group canbe realized in many different categories:

Sets G any groupFinSets G finite groupTop G topological groupSh(X), X topological space G sheaf of groups.

We call the purely axiomatic definition of a group the syntax andthe different realizations in categories semantics. Properties ofgroups can be proved on the syntactic level, and then also hold onthe semantic level. Categorical semantics makes thisrelationship, which we often use, precise.Note: All the categories Sets, FinSets, Top, Sh(X) are examplesof toposes.Aim: To give an idea the relation between syntax and semantics,and to advertise toposes as mathematical universes.


Logic and Lattices

Definition: A preorder is a category with at most one morphismfor each pair of objects. A partially ordered set (poset) is asmall preorder where the only isomorphisms are identity morphism.A lattice is a poset L where each pair x , y ∈ L has a supremum(join) x ∨ y and an infimum (meet) x ∧ y .Remark: Note that ∧ and ∨ are product, respectively coproductsin L seen as a category. We say a lattice is complete if is iscomplete (and hence cocomplete) as a category, i.e. all small limits(and colimits) exist.Here, this just means that each subset of L has an infimum andsupremum. In particular,∨∅ =

∧L =: 0, bottom (initial),

∧∅ =

∨L =: 1, top (terminal)

exist. Such a lattice is called bounded.


Logic and Lattices

Definition: A Heyting algebra is a bounded lattice H s.t. for allx , y ∈ H, there exists an element x ⇒ y ∈ H s.t.

z ≤ (x ⇒ y) ⇔ z ∧ x ≤ y , ∀z ∈ H.

This is the largest element z s.t. z ∧ x ≤ y .We write ¬x := x ⇒ 0, the pseudocomplement of x .A Boolean algebra B is a complete latte s.t.

¬(¬x) = x , ∀x ∈ B,

⇔ ¬x ∨ x = 1, ∀x ∈ B.

Proposition: H is a complete Heyting algebra if and only if H is acomplete lattice and

x ∧∨i∈I

yi =∨i∈I

(x ∧ yi ), ∀x , yi ∈ H,

for any set I .Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011

Logic and Lattices

As the notations suggest, these structures are closely related tologic, in particular propositional logic.

Classical propositional logic ↔ Boolean algebrasIntuitionistic propositional logic ↔ Heyting algebrasFree intuitionistic logic (with ∀, ∃) ↔ Complete Heyting algebras.

Classical logic allows the Law of Excluded Middle, whileintuitionistic logic does not. By these correspondences, I meancertain completeness, more about this later.


Signatures and Theories

Brief introduction to how Logic formalizes mathematics. Back toour example, groupsSignature: One type: G ; one constant (function) symbol (0-ary):1; two function symbols inv : G → G (1-ary), and · : G × G → G(2-ary).Axioms:

> `x (x · 1 = x ∧ 1 · x = 1)

> `x (x · x−1 = 1 ∧ x−1 · x = 1)

> `x ,y ,z (x · y) · z = x · (y · z).



More generally:Definition: A signature Σ consists of

I Types (or sorts).

I Function symbols, each such f has a configurationf : A1 × . . .An → B, where A1, . . . ,An,B are types (f isn-ary). 0-ary function symbols are called constants.

I Relation symbols, each such R has a configurationR A1 × . . .× An, where A1, . . . ,An,B are types (R isn-ary). 0-ary relation symbols are called propositions.



Given a signature Σ, we can recursively define terms over Σ

(I) x : A means x is a variable of type A.

(II) f (t1, . . . , tn) : B if f : A1 × . . .× An → B is a functionsymbol, and ti : Ai are terms.



From terms, we can build up formulae recursively by applying

I Relations: R(t1, . . . , tn) is a formula if R A1 × . . .× An isa relation symbol, and ti : Ai are types. We assume that thereis always the relation symbol equality = A× A.

I Truth and falsity: Truth > and falsity ⊥ are formulaewithout variables.

I Binary constructs: conjunction φ∧ ψ, disjunction φ∨ ψ, andimplication φ⇒ ψ are formulae provided φ and ψ are.

I Quantification: (∃x)φ and (∀x)φ are formulae for x avariable and φ a formulae.

I Sometimes infinitary disjunction and conjunction are included.

First order logic does not allow infinitary conjunction anddisjunction. If these occur, we are in higher order logic.



Definition: A sequent over Σ is an expression

φ `x1,...,xn ψ,

where φ, ψ are formulae, and x1, . . . , xn are variables containing atleast all free variables in φ and ψ, i.e. those not being bound by aquantifier ∃ or ∀ (such a context is suitable for φ and ψ).The interpretation of such a sequent is that for all x1, . . . , xn,φ(x1, . . . , xn) implies ψ(x1, . . . , xn).We can now define what a theory T over Σ is: it is just acollection of sequents, which we call the axioms of T.Examples: See above for groups. The theory of categories can beformulated using two sorts O and M, 2-categories with three sortsO, M1, and M2, etc.



In logic, we are interested in proof rather than validity of certainstatements. We want to know what follows from what?.This is formulated using formal or natural derivations

Γσ

where Γ is a collection of sequents, and σ is a sequent. This isinterpreted as Γ collectively implies σ.There are, depending on what kind of logic we are working with,certain structural rules and axioms which are assumed to be given.Many of them may seem like trivialities, but have to be specifiedon such an elementary level. For example,

φ ` ψ χ ` σCut rule, or

φ ` ψ



φ ` ψ φ ` χRule for ∧.

φ ` ψ ∧ χ

An important example of an axiom is the Law of ExcludedMiddle

> `x1,...,xn (φ ∨ ¬φ),

which is not derivable in intuitionistic logic. It gives the basis forproofs by contradiction in classical logic.Example for natural deduction: Implication is associative:

A ⇒ B, B ⇒ C , A ` B ⇒ C

A ⇒ B, B ⇒ C , A ` A ⇒ B A ⇒ B, B ⇒ C , A ` AElimination

A ⇒ B, B ⇒ C , A ` BElimination

A ⇒ B, B ⇒ C , A ` CRule for ⇒

A ⇒ B, B ⇒ C ` A ⇒ C


Toposes

Toposes are “almost as nice as sets”. Let’s see what makes Setspecial. For sets, we have the well-known adjunction

Hom(Z × X ,Y ) ∼= Hom(Z ,Y X ),

where Y X = f : X → Y . That is, (−)× X ` (−)X as functors.We want to generalize this idea.Definition: Let C be a category with finite products. We say thatA ∈ Ob C is exponentiable if the functor (−)× A has a rightadjoint, denoted by (−)A.A category C is called cartesian closed if it has all finite productsand exponentials. Cartesian functors are those preserving thesestructures.


Toposes

Remark: The counit εB : BA × A→ B gives an evaluation map,i.e.

C × Ah

- B

BA × A

∃!g × IdA

?ε B

-

commutes for any such morphism h. In Set, εB(f , a) = f (a), and

g(c) = (a 7→ h(c , a)).


Toposes

Example: We have exponentials in the functor category[O(X )op,Set] := Fun(O(X )op,Set) for a topological space X (thisis called a presheaf topos). They can be found using the Yonedalemma:

RQ(C )Yoneda∼= Hom[O(X )op,Set](yC ,R

Q)

assuming exp. exist∼= Hom[O(X )op,Set](yC × Q,R)︸︷︷︸use as definition

.

for presheaves Q,R and an open set C (yC is the image of Cunder the Yoneda embedding).If Q,R are sheaves, then so is RQ , i.e. Sh(X ) has the sameexponentials as [O(X )op,Set].


Toposes

In Set, we can describe subset by classifying arrows, i.e for S ⊂ Xwe have

χS : X → Ω := 0, 1, x 7→

1 if x ∈ S

0 if x /∈ S .

This map contains all the information about S . Denote by

true : ∗ = 1Set → Ω

the function ∗ 7→ 1. Then

S1

- ∗

X

ι

?

∨

χS - Ω

true

?

∨

is a pullback square, Ω = 0, 1 is the set of truth values.Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011

Toposes

We can categorify this:Definition: Let C be a category with all finite limits. A subobjectclassifier is an object Ω with a map true : 1C → Ω s.t. for eachmonomorphism m : A′ A there is a unique morphismχm : A→ Ω (called classifying arrow of m) s.t.

A′1

- 1C

A

m

?

∨

χm - Ω

true

?

∨

is a pullback square.


Toposes

Remark: We denote by SubC the lattice of subobjects of A. For asubobject, the classifying arrow is unique only up to isomorphismof morphisms. For a morphism f : A→ B, pullback induces amapping SubC B → SubC A. Thus we have a functor

SubC : Cop → Set

which is, provided C is locally small, represented by the pair(Ω, true).


Toposes

Definition (Lawvere-Tierney): A topos is category C with allfinite limits, exponentials, and a subobject classifier.Remark: This implies that C also has all finite colimits.Examples: For any topological space X , the category of sheaveson X , Sh(X ), is a topos. More generally, one can define Sh(C, J)for any small category C and a Grothendieck topology J, this is ageneralization of the notion of covering used to define sheaves.Such a topos is called a Grothendieck topos. It has theadvantage that it is cocomplete and all colimits commute withfinite limits, hence it is categorically “as good as” Set, which isalso a topos.


Toposes

Toposes may be seen as mathematical universes (i.e. we canperform logic in them) because of the following fact:Theorem: For each topos, the subobject sets SubC A have thestructure of a Heyting algebra (complete for Grothendiecktoposes).We call the logical operations in these subobject lattices theexternal Heyting algebra of the topos. The Yoneda lemma givesthat these operations are described by a model structure on Ω, theso-called internal Heyting algebra of the topos. By this, wemean here that for each logical operation, we have maps frompowers of Ω to Ω, e.g.

∧ : Ω× Ω→ Ω,

induced by conjunction on the subobject lattices.


Toposes

Thus toposes give a way of “doing logic in them”, more specifically

Toposes ←→ First order logicGrothendieck toposes ←→ Geometric logic, having infinitary

conjunction and disjunction.


Categorical Semantics

We want to interpret theories in categories. First, we look atrealizations of signatures:Definition: Let C be a category with finite limits and Σ asignature. A Σ-structure M in C consists of

I An object MA for each type A of Σ, s.t.M(A1 × . . .× An) = MA1 × . . .×MAn.

I A morphism Mf : MA1 × . . .×MAn → MB for each functionsymbol f : A1 × . . .× An → B of Σ.

I A subobject MR MA1 × . . .×MAn for each relationsymbol R A1 × . . .× An of Σ.



A Σ-structure homomorphism h : N → M between twoΣ-structures M and N is a collection of morphisms

hA : MA→ NA, for each type A in Σ, s.t.

I For any f : A1 × . . .× An → B in Σ,

MA1 × . . .×MAnMf - MB

NA1 × . . .× NAn

hA1 × . . .× hAn

? Nf - NB

hB

?

commutes.



I For each relation symbol R A1 × . . .× An in Σ,

MR > - MA1 × . . .×MAn

NR

∃

?> - NA1 × . . .× NAn

hA1 × . . .× hAn

?

commutes.

Any finite limit preserving functor F : C → D transfers Σ-structuresin C into Σ-structures in D.F



We can recursively define how to interpret terms in context ~x .t,where ~x = x1, . . . , xn is a list of variables containing thoseoccurring in t, as morphisms

[[~x .t]]M : MA1 × . . .×MAn → MB.

(I) If t is a variable of type Ai , then

[[~x .t]]M = πi : MA1 × . . .×MAn → MAi

is the projection on the i-th component.

(II) If t = f (t1, . . . , tn), ti : Ci , then

[[~x .t]]M = Mf ([[~x .t1]]M , . . . , [[~x .t1]]M).



To interpret formulae in C, we need to assume that C has enoughstructure. Formulae in context ~x .φ will be interpreted assubobjects

[[~x .φ]]M MA1 × . . .×MAn ∈ SubCMA1 × . . .×MAn.

These subobject lattice are all Heyting algebras in a topos. Thus,for a topos, we have enough structure to interpret ∧,∨,>,⊥, ∃, ∀of formulae in it. One just uses the corresponding operations in thesubobject lattices.Note that formulae build up using relation symbols areinterpreted using the pullback, i.e. if R B1 × . . .× Bm andti : Ai are terms with a list of variable ~x = x1, . . . , xn, xi : Ai ,suitable for all of them, define [[~x .R(t1, . . . , tm)]]M by thesubobject in the pullback square



[[~x .R(t1, . . . , tm)]]M - MR

MA1 × . . .×MAn

?

∨

([[~x .t1]]M , . . . , [[~x .tm]]M)- MB1 × . . .×MBm.

?

∨

We say that a sequent φ `~x ψ is valid in M if [[~x .φ]]M ≤ [[~x .ψ]]Mas subobjects. A model of a theory T is a Σ-structure M in whichall axioms of T are satisfied.F



Let X ,Y be sets. For S ⊂ X × Y , consider p : X × Y → Y whichinduces a functor

p∗ : P(Y )→ P(X × Y ), T 7→ p−1(T ) = X × T .

Now ∀pS := y ∈ Y | ∀x ∈ X , (x , s) ∈ S, and∃pS := y ∈ Y | ∃x ∈ X , (x , y) ∈ S define functors∃p, ∀p : P(X × Y )→ P(Y ) satisfying the chain of adjunctions

∃p ` p∗ ` ∀p, i.e. ∃pS ⊆ T ⇔ S ⊆ X × T

X × T ⊆ S ⇔ T ⊆ ∀pS .



More generally, in a topos C f : A→ B induces

f ∗ : SubC B → SubC A

preserving the lattice structures. This functor has adjoints

∃f ` f ∗ ` ∀f .

These adjunctions give a way of interpreting the quantifiers ∃ and∀ in C.



An important feature of categorical semantics is that it gives a wayof arguing in a pointless category as if there where elements. In asignature, variables x : A may be interpreted as “x ∈ A”. Hencewe can still interpret formulae using variables in categories in whichthe objects are no sets.One can also proceed conversely. Given a category C one candefine the canonical signature ΣC of C as having a type pAq foreach object, a function symbol pf q : pAq→ pBq for eachmorphism f : A→ B in C, and a relation symbol pRq pAq foreach subobject R A in C. In addition, all the compositionalidentities in C are formulated as axioms in ΣC . If C is a topos onecan for example proof

f is mono in C ⇔ f (x) = f (y) `x ,y x = y is derivable in ΣC .

Thus, ΣC gives a way of arguing in a topos using elements.


Soundness and Completeness

Soundness is the property one naturally expects from awell-defined way of interpreting logic in a category.Theorem: Let T be a first-order theory over Σ. If a sequent σ isderivable in T, then σ is valid in any T-model in any topos.Proof (Sketch): One needs to verify that all the structural rulesΓω hold in any model M satisfying the axioms of T in the

following way: If all the sequents of Γ hold in M, then so does ω.For most rules, this is trivial, e.g. for the Cut rule

φ `~x ψ χ `~x λφ `~x ψ

this just means that [[~x .φ]]M ≤ [[~x .ψ]]M , and [[~x .χ]]M ≤ [[~x .λ]]Mimplies [[~x .φ]]M ≤ [[~x .ψ]]M , i.e. φ `~x ψ holds in M.



One is also interested in the converse of the Soundness Theorem,the so-called Completeness Theorem. Such a theorem was firstfound for classical logic in Set:Classical Completeness (Godel, 1929): Let T be a coherenttheory (this is basically a first order theory only using ∧,∨,>,⊥, ∃with some extra axioms) then a sequent σ is derivable in T usingclassical logic if and only if it is valid in each model of T in Set.To prove completeness, one constructs a model MT with theproperty that a sequent is valid in MT if and only if it is derivablein T.



Toposes satisfy completeness with respect to all first order theories:Theorem (Completeness): Let Σ be a signature. Then a firstorder formula ~x .φ over Σ is provable in intuitionistic first orderlogic if it is valid in every Σ-structure in any elementary topos.The proof of this uses Kripke-Joyal semantics. Starting with apropositional theory P, one can find a Kripke frame which is aHeyting algebra and contains all the information about validity ofpropositions in P. For a more general intuitionistic theory T, oneobtains a collection of Kripke frames Upp∈P related by a posetP. Then these Kripke models Up correspond to a model U∗ in[P,Set] which is also a Kripke frame, and sequents are valid in allUp if and only if they are valid in U∗.

This completeness justifies the view of toposes as mathematicaluniverses.


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Categorical Semantics and Topos Theory Homotopy type theory Seminar University … · 2011-12-15 · Categorical Semantics and Topos Theory Homotopy type theory Seminar University