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Topoi: Theory and Applications Exponentiation Characteristic Maps Topoi Oliver Kullmann Introduction Grothendieck topoi Topoi Topoi: Theory and Applications Exponentiation Characteristic maps The category of sets 1 Oliver Kullmann The topos 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 topology. Terminology I Topoi Oliver Kullmann Introduction ◮ “Topos” is Greek, and means “place”. Grothendieck topoi Topoi ◮ Its use in mathematics likely is close to “space”, Exponentiation Characteristic maps either “topological space” or “set-theoretical space”. The category of ◮ Singular “topos”, plural “topoi” (“toposes” 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 geometry: “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 set theory. Characteristic maps The category of ◮ William Lawvere (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 universal property? “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 morphism 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
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