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Giraud's Theorem Giraud’s Theorem 25 February and 4 March 2019 The Giraud’s axioms allow us to determine when a category is a topos. Lurie’s version indicates the equivalence between the axioms, the left exact localizations, and Grothendieck topologies, i.e. Proposition 1. [2] Let X be a category. The following conditions are equivalent: 1. The category X is equivalent to the category of sheaves of sets of some Grothendieck site. 2. The category X is equivalent to a left exact localization of the category of presheaves of sets on some small category C. 3. Giraud’s axiom are satisfied: (a) The category X is presentable. (b) Colimits in X are universal. (c) Coproducts in X are disjoint. (d) Equivalence relations in X are effective. 1 Sheaf and Grothendieck topologies This section is based in [4]. The continuity of a reald-valued function on topological space can be determined locally. Let (X;t) be a topological space and U an open subset of X. If U is S covered by open subsets Ui, i.e. U = i2I Ui and fi : Ui ! R are continuos functions then there exist a continuos function f : U ! R if and only if the fi match on all the overlaps Ui \Uj and f jUi = fi. The previuos paragraph is so technical, let us see a more enjoyable example. Example 1. Let (X;t) be a topological space such that X = U1 [U2 then X can be seen as the following diagram: X s O e U1 tU1\U2 U2 o U1 O O U2 o U1 \U2 If we form the category OX (the objects are the open sets and the morphisms are given by the op inclusion), the previous paragraph defined a functor between OX and Set i.e it sends an open set U to the set of all continuos functions of domain U and codomain the real numbers, F : op OX ! Set where F(U) = f f j f : U ! Rg, so the initial diagram can be seen as the following: / (?) FX / FU1 ×F(U1\U2) FU2 / F(U1 \U2) then the condition of the paragraph states that the map e : FX ! FU1 ×F(U1\U2) FU2 given by f 7! f jUi is the equalizer of the previous diagram. 1 S If now we consider all the coverings U = i2I Ui and it not just the open set X, then the diagram (?) becomes: p e Q Q FU / FUi / F(U1 \U2) q/ where p(ti) = tijUi\Uj and q(ti) = t jjUi\Uj for ti 2 F(Ui). S In case the morphism e is an equalizer for all open coverings U = i2I Ui then we say F is a sheaf, if e is just a monomorphism then F is called a separated presheaf. Example 2. [1] Let (X;t);(Y;h) be a topological spaces. For every open subset U of X, define F(U) = f f : U ! Yj f is continuosg and choose for restriction mappings the usual set theorical restriction of a function to a subset. Then F is a sheaf since given continuos mappings fi : Ui ! Y such that fi and f j coincide on Ui \Uj the unique set theorical gluing: [ f : Ui ! Y; x 7! fi(x) i f x 2 Ui i2I is continuos since it is continuos on a neighborhood Ui of each point. We can say a sheaf if a presheaf satisfying good gluing properties with respect to the coverings of a open subsets. So if we want to generalize the notion of sheaf over a presheaf category, it’s necessary to create a good notion of open covering and this happen if we generalize the notion of topology for a small category, that motivated the definition of Grothendieck topology. Example 3. Let (X;t) a topological space and we consider again the category OX , a sieve on U is simply a family S of open subsets of U with the property that if V 0 ⊆ V 2 S then V 0 2 S. So a sieve in the previous example is formate by a collection of open subsets, but another way to see, it’s to consider a sieve to be a function that associates to each U a family S of morphisms with codomain U, under this point of view, we can generalize the notion of sieve in a category. Definition 1. Let C be a small category and C an object of C. A sieve S on C is a subpresheaf of the representable presheaf hC. In example 3, we take a special case of subsets, the open subsets are determined for the topology over the set, so it is necessary generalize the notion of topology for a small category and again the previous example gives us the tools of this new notion. Definition 2. A (Grothendieck) topology on a category C is a presheaf J wich assigns to each object C of P a collection J(C) of sieves on C, that satisfy the following conditions: 1. (The maximal sieve) tC = f f j cod( f ) = Cg 2 J(C). 2. (Stability axiom) If S 2 J(C) then for any h : D ! C we have h∗(S) = fg jcod(g) = D;h ◦ g 2 Sg 2 J(D). 3. (Transitivity) If S 2 J(C) and R is any sieve on C such that h∗(R) 2 J(D) for all h : D ! C in S then R 2 J(C). and we define a site as a pair (C;J) where C is a small category and J is a Grothendieck topology. 2 The next step defines the notion of "good gluing" for a sieves. Definition 3. Let C be a object of C and S a sieve of C. A matching family for S of elements of a presheaf F is a natural transformation t : S ! F wich assigns each element f : D ! C of S to element t( f ) 2 F(D) such that: F(g)[t( f )] = t( f ◦ g) f or all g : E ! D in C An amalgamation of a matching family is a single object h 2 F(C) with h ◦ f = t( f ) for all f 2 S. So a sheaf 1 is a presheaf such that for each matching familiy there exists a unique amalga- mation, a equivalent definition: a presheaf F is a sheaf when for each natural transformation t : S ! F exist a unique extension to the representable functor of C, i.e. S(C) / F(C) t ; y(C) Theorem 1. The category Sh(C;J) is a left exact localization 2 of P(C). Proof. In the context of presheaves of sets the theorem is equivalent to prove the inclusion functor ı : Sh(C;J) ! P(C) has a left adjoint which preserves finite limits. This construccion is known as sheafification and its denoted by a where a = h ◦ h. Where h : P(C) ! P(C) the functor defined in objects as follows, let S a covering sieve of C and h(F)(C) = colim Match(R;F) R2J(C) with R a cover of C, S is a refinament of R (in other words S is a subpresheaf of R) and Match(R;F) the set of matching families for the cover R of C, under this construction the object h(F) is a separated presheaf and applying the following lemma, then it’s natural defined a = h ◦ h. Lemma 1. If F is a separeted presheaf, then h(F) is a sheaf. Additionally the functor h preserves small limits. To prove this we need to note the functor Hom(R;−) preserves limits, the filtred colimits commutes with the finite limits in the category of sets and the limits in P(C) are calculated point wise, so we can write the following equality 1We can defined a sheaf with a basis for a Grothendieck topology K and it will give us a diagram analogous to the diagram (?) Definition 4. A presheaf F on C is a sheaf for J Iff for any cover f fi : Ci ! C j i 2 Ig 2 K(C) in the basis, the diagram p e Q Q FC / FCi / F(Ci ×C Ci) q / is an equalizer. 2A functor F : X ! Y is a localization if F has a fully faithful right adjoint, additionally if the left adjoint preserves small limits then we say F is a left exact localization. 3 for a finite limit. h(limFi)(C) = colim Match(R;limFi) i2I R2J(C) i2I = colim limMatch(R;Fi) R2J(C) i2I = colim limHomP(C)(R;Fi) R2J(C) i2I = lim colim HomP(C)(R;Fi) i2I R2J(C) = lim colim Match(R;Fi) i2I R2J(C) = limh(Fi)(C) i2I So a preserves finite limits and therefore this functor is a left exact localization. 2 Left exact localizations of presheaves of sets Proposition 2. Let L : P(C) ! X be a left exact localization then X is presentable. Proof. Let l be a regular cardinal strictly bigger than the number of arrows of C, then the aim is to prove that X is l-presentable. Let a l-filtred colimit in X, we can calculated this colimit in P(C) applying the right adjoint of L. The category P(C) is l-presentable and therefore the colimit exists and applying L and the fact the colimits are preserved by left adjoints, the l-filtred colimit exists in C. The l-presentable objects of X are the image under L of the l-presentable objects P(X) and their are the representable presheaf Hom(−;C) induced by the objects of C, i.e.
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