Topoi

Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Topoi: Theory and Applications Exponentiation Characteristic maps

The category of sets 1 Oliver Kullmann The of presheaves Monoid operations 1Computer Science, Swansea University, UK Directed graphs http://cs.swan.ac.uk/~csoliver Generalisations Comma categories Categorical logic seminar Properties Swansea, March 19+23, 2012 Meaning Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi I treat “topos theory” as a theory, whose place is similar Exponentiation to, say, group-theory in relation to semigroup-theory: Characteristic maps The category of 1. A “topos” is a special category. sets The topos of 2. Namely a category with “good” “algebraic” structures. presheaves Monoid operations 3. Similar to the operations that can be done with finite Directed graphs Generalisations

sets (or arbitrary sets). Comma categories 4. A “Grothendieck topos” is a special topos, having Properties more “infinitary structure”. 5. It is closer to . Terminology I Topoi Oliver Kullmann

Introduction ◮ “Topos” is Greek, and means “place”. Grothendieck topoi Topoi ◮ Its use in likely is close to “”, Exponentiation Characteristic maps either “” or “set-theoretical space”. The category of ◮ Singular “topos”, plural “topoi” (“” seems to sets The topos of be motivated by a dislike for Greek words). presheaves Monoid operations ◮ Concept invented by Alexander Grothendieck. Directed graphs Generalisations ◮ What was original a “topos”, became later a Comma categories

“Grothendieck topos”. Properties ◮ Grothendieck topoi came from algebraic : “a “topos” as a “topological structure”. ◮ Giraud (student of Grothendieck) characterised categories equivalent to Grothendieck topoi. Terminology II Topoi Oliver Kullmann

Introduction

Grothendieck topoi ◮ Then came “elementary topoi”, perhaps more Topoi Exponentiation motivated from . Characteristic maps The category of ◮ (later together with Myles Tierney) sets developed “elementary” (first-order) axioms for The topos of presheaves Grothendieck topoi. Monoid operations Directed graphs ◮ The “subobject classifier” plays a crucial role here. Generalisations Comma categories ◮ Elementary topoi generalise Grothendieck topoi. Properties ◮ Nowadays it seems “topos” replaces “elementary topos”. Outline Topoi Oliver Kullmann

Introduction Introduction Grothendieck topoi Grothendieck topoi Topoi Exponentiation Topoi Characteristic maps The category of Exponentiation sets Characteristic maps The topos of presheaves Monoid operations The category of sets Directed graphs Generalisations The topos of presheaves Comma categories Monoid operations Properties Directed graphs Generalisations Comma categories Properties Generalised topology Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation From [Borceux, 1994]: Characteristic maps The category of ◮ A Grothendieck topos is a category equivalent to a sets The topos of category of sheaves on a site. presheaves ◮ Monoid operations A Grothendieck topos is complete and cocomplete. Directed graphs Generalisations ◮ Every Grothendieck topos is a topos. Comma categories

◮ A topos in general is only finitely complete and Properties finitely cocomplete. Three operations with sets Topoi Oliver Kullmann

Three related properties of the category SET: Introduction Grothendieck topoi Function sets For sets A, B we have the set BA of all Topoi maps f : A → B. Exponentiation Characteristic maps

Characteristic maps For set A the subsets are in 1-1 The category of correspondence to maps from A to {0, 1}. sets The topos of Powersets For a set A we have the set P(A) of all presheaves Monoid operations subsets of A. Directed graphs Generalisations

Via characteristic maps we get powersets from function Comma categories

spaces: Properties P(A) =∼ {0, 1}A.

◮ Perhaps in categories which have “map objects” and “characteristic maps”, we have also “power objects”? ◮ Conversely, perhaps from power objects we get, as in set theory, map objects and characteristic maps? Exponentiation: The idea Topoi Oliver Kullmann

Introduction Fix a category C. We consider “exponentiation” with an Grothendieck topoi

object E — easiest to fix E: Topoi Exponentiation pow : B ∈ Obj(C) → BE ∈ Obj(C). Characteristic maps E The category of sets

What could be the ? “Currying”?!: The topos of presheaves E Monoid operations Mor(A × E, B) =∼ Mor(A, B ). Directed graphs Generalisations

Comma categories So we need products in C. Looks like adjoints?! Recall Properties F : C → D, G : D → C yields an adjoint (F, G) iff there is a natural isomorphism

Mor(F(A), B) =∼ Mor(A, G(B)).

So F(A) := A × E and G(B) := powE (B). Exponentiation: The formulation I Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Definition Topoi The category C has exponentiation with power Exponentiation Characteristic maps

E ∈ Obj(C) if the functor A ∈ Obj(C) → A × E ∈ Obj(C) The category of has a right adjoint B ∈ Obj(C) → BE ∈ Obj(C). sets The topos of According to the general theory of adjoints, this is presheaves Monoid operations equivalent to the property that for all B ∈ Obj(C) the Directed graphs Generalisations functor P := (A × E) has a universal arrow A∈Obj(C) Comma categories (“cofree object”, “coreflection”) from P to B, that is, a Properties e : BE × E → B such that for all e′ : A × E → B there exists a unique E ′ f : A → B with e = e ◦ (f × idE ). Exponentiation: The formulation II Topoi Oliver Kullmann

Definition Introduction Grothendieck topoi A category C with binary products has exponentiation, if Topoi all powers admit exponentiation. Exponentiation Characteristic maps ◮ The category of If C has exponentiation, then so does its skeleton, sets

and thus having exponentiation is an invariant under The topos of presheaves equivalence of categories. Monoid operations ◮ Directed graphs For exponentiation we need the existence of binary Generalisations products, however, as usual, a different choice of Comma categories binary products leads to (correspondingly) Properties isomorphic exponentiations. ◮ So the above “with” can be interpreted in the weak sense, just sheer existence is enough (no specific product needs to be provided). Cartesian-closed categories Topoi Oliver Kullmann

Introduction Definition Grothendieck topoi Topoi A category is cartesian-closed, if it has finite limits and Exponentiation exponentiation. Characteristic maps The category of sets ◮ Being cartesian-closed is just a property of The topos of categories, no additional structure is required (“has” presheaves Monoid operations means “exists”). Directed graphs Generalisations ◮ If a category is cartesian-closed, so is its skeleton, Comma categories and thus being cartesian-closed is an invariant under Properties equivalence of categories. The category CAT of all (small) categories is cartesian-closed, with FUN(C, D) as the exponential object of C, D, and so we can write DC := FUN(C, D). Subobject classifier: The idea Topoi Oliver Kullmann

Introduction Now let’s turn to “characteristic functions”. Consider a Grothendieck topoi category with finite products and an object X. Topoi C Exponentiation Characteristic maps

A subobject A of X shall correspond The category of sets to that morphism χA : X → Ω The topos of for some fixed “subobject classifier” Ω ∈ Obj(C), presheaves Monoid operations such that the image of A under χA Directed graphs is the same as t : 1 → Ω Generalisations Comma categories for some fixed t. Properties

◮ If such a pair (Ω, t) exists, it is called a “subobject classifier” of C. .So consider a subobject(-representation) i : A ֒→ X ◮ Subobject classifier: The conditions Topoi Oliver Kullmann

Introduction We get the diagram Grothendieck topoi Topoi 1 Exponentiation A Characteristic maps A / 1 The category of i t sets  The topos of  presheaves X / Ω Monoid operations χA Directed graphs Generalisations What are the conditions? Comma categories Properties For every mono i : A ֒→ X there shall be exactly one .1 χA : X → Ω making the diagram commute, and fulfilling the further conditions. 2. A pushout? 3. No, a pullback! Definition Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Definition Exponentiation Characteristic maps For a category C with a terminal object 1C,a subobject The category of classifier is a pair (Ω, t) with Ω ∈ Obj(C) and t : 1C → Ω, sets such that for all monos i : A ֒→ X there is exactly one The topos of presheaves χA : X → Ω such that (i, 1A) is a pullback of (t,χA). Monoid operations Directed graphs ◮ All subobject classifiers for C are pairwise Generalisations Comma categories isomorphic. Properties ◮ If C has a subobject classifier, then so does its skeleton, and thus having a subobject classifier is an invariant under equivalence of categories. Definition of (“elementary”) topoi Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Definition Exponentiation Characteristic maps

A topos is a cartesian-closed category which has a The category of subobject classifier. sets The topos of ◮ presheaves Being a topos is just a property of categories, no Monoid operations Directed graphs additional structure is required (“has” means Generalisations “exists”). Comma categories ◮ If a category is a topos, so is its skeleton, and thus Properties being a topos is an invariant under equivalence of categories. Equivalent characterisation of topoi Topoi Oliver Kullmann

Introduction

Grothendieck topoi With [Lane and Moerdijk, 1992], Section IV.1: Topoi Exponentiation ◮ The “global” point of view is used, similar to the use Characteristic maps The category of of adjoints in characterising exponential objects. sets ◮ The “power object” operation P is assumed. The topos of presheaves ◮ This is a map P Obj C Obj C such that for all Monoid operations : ( ) → ( ) Directed graphs objects A, B ∈ Obj(C) there are natural isomorphisms Generalisations Comma categories Sub(A × B) =∼ Mor(A, P(B)) Properties

(between sets). Using Ω := P(1) we get the subobject classifier. Reminder: limits Topoi Oliver Kullmann

◮ The category of sets is complete, that is, has all Introduction (small) limits. Grothendieck topoi ◮ It is also cocomplete (has all (small) colimits), but we Topoi Exponentiation do not need this here (finite cocompleteness follows Characteristic maps from being a topos). The category of sets ◮ Completeness is equivalent to having all (small) The topos of products and all (binary) equalisers. presheaves Monoid operations ◮ The canonical terminal object is the empty product, Directed graphs Generalisations i.e., ∅ P 0 1. Q ∅ = ∅ = (∅)= {∅} = { } = Comma categories Properties (Xi )i∈I → Y Xi pr i∈I pr p § 77 q §§ 77 Ó§§ 7 Xp Xq

f f / in / X / Y → {x ∈ X : f (x)= g(x)} / X / Y g g Reminder: pullbacks Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps

The category of C C sets ? → 9 eK ? _ ? ss KK The topos of f  ? g f ss KK g  ?? s K presheaves  ? ss KKK  ?? sss KK Monoid operations  ss K Directed graphs A B A K B Generalisations e KK ss9 KK ss Comma categories KK ss pr KK sspr 1 K ss 2 Properties {(a, b) ∈ A × B : f (a)= g(b)} Exponentiation and subobject classifier Topoi Oliver Kullmann

Introduction

Grothendieck topoi Exponentiation: Topoi Exponentiation Characteristic maps X (X, Y ) → Y := {f : X → Y } The category of sets X e : Y × X → Y , e(f , x) := f (x). The topos of presheaves Monoid operations Subobject classifier: Directed graphs Generalisations Ω := {0, 1} = P(1) Comma categories Properties .t : 1 = {0} ֒→ Ω, 0 → 1

X {P(X) ֒→→ Ω , A → χX (A) := A × {1}∪ (X \ A) × {0 The category of finite sets Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation ◮ The full subcategory of SET given by all finite sets Characteristic maps The category of (in the current universe, of course) is a topos, with sets the same operations. The topos of presheaves ◮ In general, if for a topos T and a full subcategory C Monoid operations Directed graphs ◮ C is closed under the the topos-operations (finite Generalisations product, exponentiation, subobject classifier), Comma categories ◮ C is closed under subobject-formation, Properties then also C is a topos. Monoid operations Topoi Oliver Kullmann

Consider a fixed monoid M =(M, , 1). Introduction Grothendieck topoi An operation of M on X is given by a map Topoi Exponentiation ∗ : M × X → X Characteristic maps

The category of such that for all a, b ∈ M and x ∈ X we have sets

The topos of 1 ∗ x = x presheaves Monoid operations a ∗ (b ∗ x)=(a b) ∗ x. Directed graphs Generalisations ◮ (M, ∗) is also called an M-set. Comma categories ◮ Isomorphically, we have the point of view of a Properties “representation via transformations”: a morphism from M into the transformation monoid X T(X)=(X , ◦, idX ). See Section 4.6 in [Goldblatt, 2006] for basic information on the topos of M-sets. The category of M-sets Topoi Oliver Kullmann

Introduction

For M-sets X, Y , a morphism f : X → Y is a map fulfilling Grothendieck topoi

Topoi ∀ a ∈ M ∀x ∈ X : f (a ∗ x)= a ∗ f (x). Exponentiation Characteristic maps

The category of The category of M-sets is denoted by OPRM (SET). sets ◮ The topos of I use the terminological distinction between “action” presheaves and “operation”, where for the former structure on the Monoid operations Directed graphs object acted upon is involved (e.g., the action of a set Generalisations on a group via automorphisms), and for the latter Comma categories structure on the side of acting object (the operation Properties of a group on a set). ◮ OPRM (SET) is a concrete category.

OPRM (SET) is canonically isomorphic to the functor category SETM , considering M as a one-object category. Some remarks Topoi Oliver Kullmann

1. The functor O : MON → CAT′, mapping monoid M Introduction Grothendieck topoi to category OPRM (SET), is a contravariant functor. ′ Topoi 2. Here CAT is the category of “large” categories (in Exponentiation the parameter-universe). Characteristic maps The category of 3. More generally, the functor sets ′ ′ M The topos of O : MON × CAT → CAT , given by (M, C) → C , presheaves Monoid operations mapping a monoid M and a category C to the Directed graphs category of operations of M on C, is a bifunctor, Generalisations contravariant in the first argument. Comma categories Properties 4. More generally, the mapping FUN : CAT × CAT′ → CAT′, (C, D) → DC,isa bifunctor, contravariant in the first argument. 5. More generally, for a cartesian-closed category C, the mapping C2 → C, (X, Y ) → Y X , is a bifunctor, contravariant in the first argument. Limits Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi The forgetful functor V : OPR (SET) has a left-adjoint, Exponentiation M Characteristic maps

the formation of free operations. So V preserves limits. The category of sets ◮ I.e., if limits exist, they must have the underlying sets The topos of as given by the limits in SET. presheaves Monoid operations ◮ It is easy to see, as in all algebraic categories, that Directed graphs Generalisations

the operations of M defined in the obvious ways for Comma categories the SET-limits, yield limits in OPRM (SET). Properties

It also follows that the monomorphisms of OPRM (SET) are precisely the injective . Representable functors Topoi Oliver Kullmann

Introduction

Grothendieck topoi The representable functors of a category C are those Topoi Exponentiation functors F : C → SET which are isomorphic to a Characteristic maps The category of Hom-functor X ∈ Obj C → Mor(A, X) ∈ Obj(SET) for sets

some A ∈ Obj(C) (the representing object). The topos of presheaves ◮ We consider the objects of OPRM (SET) as functors Monoid operations Directed graphs (“covariant presheafs”). Generalisations ◮ There is then only one object, thus only one Comma categories Hom-functor. Properties ◮ This is the canonical operation of M on itself, via multiplication. Exponentiation Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi For M-sets B E the exponential BE is defined as having Exponentiation , Characteristic maps ◮ base set Mor(M × E, B) The category of sets ◮ operation (for m ∈ M and a morphism f : M × E → B) The topos of presheaves Monoid operations (m ∗ f )(a, e) := f (m a, e) Directed graphs Generalisations ◮ evaluation e : BE × E → B given by Comma categories Properties e(f , e) := f (1, e). Alternative for groups Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi If M is a group, then we have a simple (of course, Exponentiation isomorphic) possibility to define the exponential BE : Characteristic maps The category of ◮ sets base set MorSET(E, B) ◮ The topos of operation (for g ∈ M and a map f : E → B) presheaves Monoid operations −1 Directed graphs (g ∗ f )(e) := g ∗ f (g ∗ e) Generalisations Comma categories E ◮ evaluation e : B × E → B given by Properties

e(f , e) := f (e). Reminder: Topoi Oliver Kullmann

Introduction Lemma Grothendieck topoi Topoi Consider a category C, an object A ∈ Obj(C) and a Exponentiation Characteristic maps functor T : C → SET. The Yoneda map The category of sets YA,T : NAT(MorC(A, −), T ) → T (A) The topos of presheaves Monoid operations is a bijection. Directed graphs Generalisations So, for C = OPRM (SET), for every M-set X we have a Comma categories natural bijection Properties Mor(M, X) =∼ X. This is also easy to see directly, since a morphism from M to X is uniquely determined by the image of 1 (M is the free operation generated by one element). Subobject classifier Topoi Oliver Kullmann

What shall be Ω ?! Introduction Let’s consider the M-set M and its subobjects: Grothendieck topoi Topoi 1. Subobjects are the left ideals of the semigroup M Exponentiation Characteristic maps

(subsets stable under left multiplication). The category of sets 2. As we have seen, Mor(M, Ω) =∼ Ω holds. The topos of So we should take as the base set of the set of left presheaves Ω Monoid operations ideals of M: Directed graphs Generalisations Ω := {I ⊆ M |∀ a ∈ M ∀ x ∈ I : a x ∈ I}. Comma categories Properties (Thus |Ω|≥ 2.) It is natural to choose t := M ∈ Ω. What is now the operation of M on Ω ?

a ∗ ω := {b ∈ M : b a ∈ ω}

for ω ∈ Ω and a ∈ M. Determining morphisms into Ω Topoi Oliver Kullmann

Introduction Lemma Grothendieck topoi For an M-set X and a morphism f : X → Ω we have Topoi Exponentiation ∀ x ∈ X : f (x)= {a ∈ M : f (a ∗ x)= M}. Characteristic maps The category of For a subset A ⊆ X there exists a morphism f : X → Ω sets −1 with f ({M})= A iff A is closed (i.e., is a subspace), in The topos of presheaves which case f is unique, namely f = χA with Monoid operations Directed graphs Generalisations χA(x) := {a ∈ M : a ∗ x ∈ A}. Comma categories

Proof: For a morphism f : X → Ω we have: Properties f (a ∗ x)= M ⇔ a ∗ f (x)= M ⇔ {b ∈ M : b a ∈ f (x)} = M ⇔ a ∈ f (x). M operates trivially on M ∈ Ω, so f −1({M}) is closed. Finally a ∗ χA(x)= {b ∈ M : b a ∈ χA(x)} = {b ∈ M : (b a) ∗ x ∈ A} = {b ∈ M : b ∗ (a ∗ x) ∈ A} = χA(a ∗ x). General directed graphs Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps

The category of sets

The topos of presheaves Monoid operations Directed graphs Generalisations

Comma categories

Properties Generalisation to functor categories Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi With Remark B:2.3.19 in [Johnstone, 2002]: Exponentiation Characteristic maps ◮ C If C is a finite category and D is a topos, then D is a The category of topos. sets The topos of ◮ If C is a small category and D is a cocomplete topos, presheaves C Monoid operations then D is a topos. Directed graphs Generalisations

Comma categories For a small category C t Properties the category SETC of presheaves is a topos. Reminder: Comma categories Topoi Oliver Kullmann

See Introduction http://en.wikipedia.org/wiki/Comma_category Grothendieck topoi Topoi for more information. Exponentiation Characteristic maps

Consider functors F : A → C, G : B → C. The comma The category of category (F ↓ G) is defined as follows: sets The topos of 1. objects are triples (a, b,ϕ), where a ∈ A, b ∈ B, and presheaves Monoid operations ϕ : F(a) → G(b) Directed graphs 2. morphisms f :(a, b,ϕ) → (a′, b′,ϕ′) are pairs Generalisations Comma categories f =(α,β), where α : a → a′, β : b → b′, and Properties ϕ′ ◦ F(α)= G(β) ◦ ϕ. Special cases: ◮ An object X of a category C stands for 1 → X. ◮ A category C stands for idC. ◮ (C ↓ X) also written as “C/X” (“slice category”). ◮ (C ↓ G) also written as “C/G” (“Artin glueing”). Comma categories yielding topoi Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Consider topoi B, C and a functor G : B → C. Characteristic maps The category of Theorem [Wraith 1974, [Carboni and Johnstone, 1995]]: sets The topos of presheaves If G preserves pullbacks, then (C ↓ G) is also a topos. Monoid operations Directed graphs Special cases: Generalisations Comma categories

1. The product of two topoi is a topos. Properties 2. Slices of a topos are topoi. The topos of (labelled, generalised) Topoi Oliver Kullmann clause-sets Introduction Consider a fixed monoid M. Grothendieck topoi Consider the (forward) powerset functor Topoi Exponentiation P Characteristic maps f : OPRM (SET) → SET which The category of sets ◮ P maps an M-set X to f(X), The topos of ◮ P P P presheaves maps f : X → Y to f(f ): f(X) → f(Y ), where Monoid operations P Directed graphs f(f )(S) := f (S). Generalisations Now let Comma categories P LCLSM := (SET ↓ f) Properties

Pf does not preserve pullbacks, nevertheless these categories are topoi. ◮ LCLS{1} is the category of labelled hypergraphs ◮ LCLSZ2 is the category of labelled clause-sets (allowing degenerated and non-polarised literals!). Without the labelling, we obtain quasi-topoi. A little problem Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps Actually, [Carboni and Johnstone, 1995] show for topoi The category of sets

B, C and a functor G : B → C: The topos of presheaves Monoid operations G preserves pullbacks if and only if (C ↓ G) is a topos. Directed graphs Generalisations Now G clearly does not preserve pullbacks, however we Comma categories have a topos ... Properties Factorisation Topoi Oliver Kullmann

Consider a category C and a morphism f : A → B. Introduction ◮ f has an epi-mono factorisation if Grothendieck topoi

Topoi f = i ◦ π Exponentiation Characteristic maps

for some epimorphism π : A → C and some The category of monomorphism i : C → B. sets ◮ The topos of Such a factorisation is unique if for every other presheaves epi-mono factorisation f i′ ′, i′ A C′, Monoid operations = ◦ π : → Directed graphs π′ : C′ → B, there is an isomorphism ϕ : C → C′ with Generalisations commutative Comma categories Properties C > @ ~~ @@ i ~~ @@π ~~ @@ ~~ @ A ϕ B @@ ~> @@ ~~ ′ @@ ~~ ′ i @  ~~ π C′ Topoi have unique factorisations Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation ◮ A category has epi-mono factorisation (also “epi- Characteristic maps mono decomposition”, or just “factorisation” or The category of sets

“decomposition”) if every morphism has an epi-mono The topos of factorisation. presheaves Monoid operations ◮ Directed graphs And similarly one says a category has unique Generalisations epi-mono factorisation. Comma categories Properties Lemma A topos has unique epi-mono factorisation. Topoi are balanced Topoi Oliver Kullmann

Recall: Introduction Grothendieck topoi ◮ A bimorphism is a morphism which is epi and mono. Topoi ◮ Exponentiation A category is balanced if every bimorphism is iso. Characteristic maps

The category of Lemma sets The topos of Every category with unique factorisation is balanced. presheaves Monoid operations Proof: Consider a bimorphism f : A → B. Directed graphs Generalisations A Comma categories ? ? id  ?? Properties A  ??f  ??  ? A ϕ B ?? ? ??  ??  f ? id ?   B B Topoi have power objects Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps

The category of sets

The topos of presheaves The (global) “power object map” can also be localised. Monoid operations Directed graphs Generalisations

Comma categories

Properties Topoi are finitely cocomplete Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps

The category of sets Lemma The topos of presheaves Every topos is finitely cocomplete. Monoid operations Directed graphs Proof: (not completely trivial) Generalisations Comma categories

Properties External Heyting algebra Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps

The category of With [Lane and Moerdijk, 1992], Section IV.8: sets The topos of presheaves Lemma Monoid operations For every object A in a topos, the partial order Sub(A) of Directed graphs Generalisations

subobjects is a Heyting lattice. Comma categories

Properties Internal Heyting algebras Topoi Oliver Kullmann

Introduction

Grothendieck topoi With [Lane and Moerdijk, 1992], Section IV.8: Topoi Exponentiation Lemma Characteristic maps For every object A in a topos, the power object P(A) can The category of sets be given the structure of an Heyting algebra object (an The topos of “internal Heyting algebra”). In particular, this applies for presheaves Monoid operations the subobject classifier Ω= P(1). Directed graphs Generalisations For each object X the internal structure of P(A) makes Comma categories Mor X P A a Heyting algebra. Now the canonical ( , ( )) Properties bijection between Sub(X × A) and Mor(X, P(A)) becomes an isomorphism of Heyting algebras. Proof: Conjunction ∧ : Ω × Ω → Ω is the characteristic morphism of 1 → Ω × Ω. ... Summary Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps

The category of I The notion of a “topos” has been defined. sets The topos of II Examples via categories of presheafs and comma presheaves Monoid operations categories have been discussed. Directed graphs III Basic elementary properties of topoi have been Generalisations Comma categories presented. Properties Bibliography I Topoi Oliver Kullmann

Introduction

Grothendieck topoi Borceux, F. (1994). Topoi Exponentiation Handbook of Categorical Algebra 3: Categories of Characteristic maps The category of Sheaves, volume 52 of Encyclopedia of Mathematics sets

and its Applications. The topos of presheaves Cambridge University Press. Monoid operations Directed graphs ISBN 0 521 44180 3. Generalisations Carboni, A. and Johnstone, P. (1995). Comma categories Connected limits, familial representability and Artin Properties glueing. Mathematical Structures in Computer Science, 5(04):441–459. Bibliography II Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Goldblatt, R. (2006). Topoi Topoi: The categorial analysis of logic. Exponentiation Characteristic maps

Dover Publications, Inc. The category of ISBN 0-486-45026-0; slightly corrected unabridged sets The topos of republication of the revised (second) edition originally presheaves Monoid operations published in 1984. Directed graphs Generalisations

Johnstone, P. T. (2002). Comma categories

Sketches of an Elephant: A Topos Theory Properties Compendium (Volume 1), volume 43 of Oxford Logic Guides. Oxford University Press. ISBN 0-19-853425-6. Bibliography III Topoi Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps

The category of Lane, S. M. and Moerdijk, I. (1992). sets The topos of Sheaves in Geometry and Logic: A First Introduction presheaves Monoid operations to Topos Theory. Directed graphs Universitext. Springer. Generalisations ISBN 0-387-97710-4; QA169.M335. Comma categories Properties Topoi

Oliver Kullmann

Introduction

Grothendieck topoi

Topoi Exponentiation Characteristic maps

The category of sets

The topos of End presheaves Monoid operations Directed graphs Generalisations

Comma categories

Properties