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Cohomology to Logic and Back Toposes in Como 2018

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Cohomology to Logic and Back Toposes in Como 2018 torsors fundamental group cohomology Cohomology to logic and back Toposes in Como 2018 Tibor Beke University of Massachusetts tibor [email protected] June 28, 2018 Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical Spaces have I Euler characteristic I Betti numbers I homology groups I cohomology rings(!) I set of connected components I fundamental group I homotopy groups Which of these generalize to toposes? Why should they even generalize? Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical Tibor Beke Cohomology to logic and back Challenge: can your favorite functor C! Topoi be extended to a 2-functor? For example, can the category of topological spaces (of schemes, etc) be turned into a 2-category compatibly with the functor X 7! Sh(X )(X 7! Sh(Xet´ ) etc)? torsors motivation fundamental group simplicial cohomology classical I the category of Grothendieck toposes and geometric morphisms is excellent for representing categories of `locally structured objects' I functors C! Topoi are seldom full (Topoi is not locally small!) I some of these functors are onto (up to isomorphism) on objects, allowing one to express Topoi as a localization of C I Topoi is a 2-category: given morphisms f ; g : E ! F, a 2-arrow is a natural transformation from f ∗ to g ∗. Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical I the category of Grothendieck toposes and geometric morphisms is excellent for representing categories of `locally structured objects' I functors C! Topoi are seldom full (Topoi is not locally small!) I some of these functors are onto (up to isomorphism) on objects, allowing one to express Topoi as a localization of C I Topoi is a 2-category: given morphisms f ; g : E ! F, a 2-arrow is a natural transformation from f ∗ to g ∗. Challenge: can your favorite functor C! Topoi be extended to a 2-functor? For example, can the category of topological spaces (of schemes, etc) be turned into a 2-category compatibly with the functor X 7! Sh(X )(X 7! Sh(Xet´ ) etc)? Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical C := 0; 1 × [0; 2π] modulo h0; 0i ∼ h0; 2πi and h1; 0i ∼ h1; 2πi M := 0; 1 × [0; 2π] modulo h0; 0i ∼ h1; 2πi and h1; 0i ∼ h0; 2πi natural projection to S1 = [0; 2π] modulo 0 ∼ 1 C is not isomorphic to M in Top =S1. Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical Boundary of n times twisted Moebius strip is isomorphic in Top =S1 to C for n even, and to M for n odd. What are these objects? Why only two isomorphism types? What about bases other than S1? Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical default assumptions All toposes are Grothendieck. Base topos Set is model of ZFC. Additional universes as needed. In topos E, objects X and Y are locally isomorphic if there exists isomorphism X × U ≈ Y × U over U, for some U 1. Canonical geometric morphism γ : Sh(S1) ! Set. What the pictures showed (identifying an object of Et´ =S1 with its sheaf of continuous sections) were isomorphism classes of objects of Sh(S1) that are locally isomorphic to γ∗(f0; 1g). Given a topos E, let TorsZ=2(E) denote the subcategory of E whose objects are locally isomorphic to the two-element constant set, and whose morphisms are the isomorphisms between them. Let 1 H (E; Z=2) denote the set(!) of isomorphism types of objects of TorsZ=2(E). Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical presheaf topos Pre(C) op I T : C ! Set is locally isomorphic to f0; 1g iff T (c) = f0; 1g for all c 2 C, and T (f ) 2 aut(f0; 1g) for all morphisms c −!f d op I this amounts to functor C ! Z=2 (where a group is thought of as one-object category) I isomorphism between two such objects amounts to natural isomorphism between the associated functors I equivalence of categories Fun(C; Z=2) ! TorsZ=2(Pre(C)) Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical presheaf topos Pre(C) I equivalence of categories Fun(C; Z=2) ! TorsZ=2(Pre(C)) I Grpd is a reflective subcategory of Cat: π1 : Cat Grpd : i π1(C) is the fundamental groupoid of the category C, obtained by formally adjoining an inverse to each morphism of C. I equivalence of categories Grpd(π1(C); Z=2) ! TorsZ=2(Pre(C)) I bijection between isomorphism types of objects (i.e. conjugacy classes of homomorphisms) in Grpd(π1(C); Z=2) and 1 H (Pre(C); Z=2). Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical H1(E; G) conj classes in Grpd(π1(C); G) Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical spatial topos Sh(X ) What is the set of Z=2-torsors T that are trivialized by a given open cover fUi j i 2 I g of X ? By assumption, T restricted to Ui is represented by the ´etalespace Ui × f0; 1g. (`0' and `1' are arbitrary labels, not assumed globally defined!) When do these ´etalespaces `glue' together to a two-sheeted cover? To every non-empty intersection Ui \ Uj , associate gij 2 aut(f0; 1g). If the `cocycle conditions' −1 I gii = id and gij = gji I gij gjk = gik for non-empty Ui \ Uj \ Uk are satisfied then the gij define an equivalence relation on ti2I Ui × f0; 1g, and quotient is an ´etale space over X with two-point fibers. Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical spatial topos Sh(X ) 0 I two such cocycles gij , gij result in spaces isomorphic over X iff 0 there exist hi 2 aut(f0; 1g) such that hi gij = gij hj I refinement of cover U = fUi j i 2 I g by V = fVj j j 2 Jg, given by J ! I , induces map from the set of Z=2-torsors trivialized over U to the set of Z=2-torsors trivialized over V I given U and V, induced map does not depend on J ! I I since any torsor must be trivialized over some open cover, 1 ˇ ˇ ˇ 1 H (Sh(X ); Z=2) = colim U Z(U; Z=2)=B(U; Z=2) =: H (Sh(X ); Z=2) Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical H1(E; G) conj classes in Grpd(π1(C); G) Hˇ 1(X ; G) Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical simplicial coding of torsors Let [n] = f0 < 1 < 2 < ··· < ng and ∆ the category with objects [n], n 2 N, and morphisms monotone maps. op SSet is shorthand for Set∆ . I ∆n is the simplicial complex consisting of all (non-empty) subsets of f0; 1;:::; ng I Λn;k is ∆n omitting f0; 1;:::; ng and f0; 1;:::; k − 1; k + 1;:::; ng I these determine simplicial sets, denoted the same way. Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical simplicial coding of torsors X −!f Y 2 SSet is a Kan fibration if it satisfies the lifting property Λ / X n;k > f ∆n / Y for all n 2 N, 0 6 k 6 n. It is an N-exact Kan fibration if, in addition, the dotted lift is unique for all commutative squares with n > N. This is equivalent to the canonical map X (n) ! Λn;k (f ) being epi for all n, and iso for n > N. This is the definition we will use for simplicial objects in a topos. Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical simplicial coding of torsors I for group object G in a topos E, write K(G; 1) for its nerve, considered as a one-object category I for object U of E, U• = cosk0(U) is the standard simplicial 3 2 object ··· U V U ) U Theorem [Duskin 1975] There is a natural bijection between isomorphism classes G-torsors trivialized over U, and simplicial homotopy classes of 1-exact fibrations of the form U• ! K(G; 1). 1 H (E; G) = colim [U•] [−; K(G; 1)] i.e. the filtered colimit, along homotopy classes of simplicial covers, of simplicial homotopy classes of maps into K(G; 1) 2 E∆op . Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical H1(E; G) conj classes in Grpd(π1(C); G) Hˇ 1(X ; G) colim [U•] −; K(G; 1) Tibor Beke Cohomology to logic and back torsors motivation fundamental group simplicial cohomology classical relation to Quillen model categories X −!f Y 2 mor E∆op is a local or internal weak homotopy equivalence if f induces isomorphisms on the homotopy group objects (constructed internally) with respect to arbitrary (local) basepoints. Let W ⊂ mor E∆op denote the class of local weak equivalences. W is the class of models in mor E∆op of a countable geometric sketch. See e.g. [TB 2000] for an axiomatization by sentences of geometric logic. Theorem [Joyal ca. 1984] Setting cofibrations to be the monomorphisms and weak equivalences to be W, E∆op satisfies Quillen's axioms for a homotopy model category.
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