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,τ) be a topological space and U an open subset of X. If U is S covered by open subsets Ui, i.e. U = i∈I 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 |Ui = fi.

The previuos paragraph is so technical, let us see a more enjoyable example.

Example 1. Let (X,τ) 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 : U → R}, 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 |Ui is the equalizer of the previous diagram.

1 S If now we consider all the coverings U = i∈I 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) = ti|Ui∩Uj and q(ti) = t j|Ui∩Uj for ti ∈ F(Ui).

S In case the morphism e is an equalizer for all open coverings U = i∈I 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,τ),(Y,η) be a topological spaces. For every open subset U of X, deﬁne

F(U) = { f : U → Y| f is continuos} 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 ∈ Ui i∈I 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,τ) 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 ∈ S then V 0 ∈ 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 | cod( f ) = C} ∈ J(C). 2. (Stability axiom) If S ∈ J(C) then for any h : D → C we have h∗(S) = {g |cod(g) = D,h ◦ g ∈ S} ∈ J(D).

3. (Transitivity) If S ∈ J(C) and R is any sieve on C such that h∗(R) ∈ J(D) for all h : D → C in S then R ∈ 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 τ : S → F wich assigns each element f : D → C of S to element τ( f ) ∈ F(D) such that:

F(g)[τ( f )] = τ( f ◦ g) f or all g : E → D in C

An amalgamation of a matching family is a single object h ∈ F(C) with h ◦ f = τ( f ) for all f ∈ S.

So a sheaf 1 is a presheaf such that for each matching familiy there exists a unique amalgamation, a equivalent deﬁnition: a presheaf F is a sheaf when for each natural transformation τ : S → F exist a unique extension to the representable functor of C, i.e.

S(C) / F(C) τ ;

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 = η ◦ η.

Where η : P(C) → P(C) the functor defined in objects as follows, let S a covering sieve of C and η(F)(C) = colim Match(R,F) R∈J(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 η(F) is a separated presheaf and applying the following lemma, then it’s natural defined a = η ◦ η. Lemma 1. If F is a separeted presheaf, then η(F) is a sheaf. Additionally the functor η 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 { fi : Ci → C | i ∈ I} ∈ 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.

η(limFi)(C) = colim Match(R,limFi) i∈I R∈J(C) i∈I

= colim limMatch(R,Fi) R∈J(C) i∈I

= colim limHomP(C)(R,Fi) R∈J(C) i∈I

= lim colim HomP(C)(R,Fi) i∈I R∈J(C)

= lim colim Match(R,Fi) i∈I R∈J(C)

= limη(Fi)(C) i∈I 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 λ be a regular cardinal strictly bigger than the number of arrows of C, then the aim is to prove that X is λ-presentable.

Let a λ-ﬁltred colimit in X, we can calculated this colimit in P(C) applying the right adjoint of L. The category P(C) is λ-presentable and therefore the colimit exists and applying L and the fact the colimits are preserved by left adjoints, the λ-ﬁltred colimit exists in C.

The λ-presentable objects of X are the image under L of the λ-presentable objects P(X) and their are the representable presheaf Hom(−,C) induced by the objects of C, i.e. let F : I → X be a diagram

HomX(C,colimF) = HomP(C)(y(C),colimF) X/F X/F

= colimHomP(C)(y(C),F) X/F

= colimHomX(L(C),L(F)) X/F

= colimHomX(C,F) X/F

So if we take C the full subcategory formatted with the previous λ-presentable objects then X is genereted by C. The previous proposition allows us to work in the context of presentable categories, this will help us to formulatted the Giraud’s axioms (b) and (c) in a simpler way.

3 Giraud’s axioms

For formulate the Giraud’s axiom (b), we need the functor given by pullback along f : X → Y.

Deﬁnition 5. Let C be a category with pullbacks and small colimits. Let f : X → Y be a ∗ morphism in C, the pullback functor f : C/Y → C/X associate to an object h in CY/ the object

4 f ∗(h) deﬁne by the following diagram:

f ∗(h) ∗ ×Y X / X f ∗ / Y h

In case the functor f ∗ preserves colimits then we will say that the colimits in C are universal 3. Proposition 3. [1] Let L : P(C) → X be a left exact localization then the colimits in X are universal.

∗ Proof. The colimits in P(C) are universal, so if we apply the right adjoint of L to f : X/Y → X/X and use the fact the left exact localization is stable under slice constructions, the functor ∗ ∗ ∗ f : P(C)/Y → P(C)/X preserves colimits, and applying L to f , we deduced f : X/Y → X/X preserve colimits. The coproducts in the category of Sets are constructed as disjoint unions of sets, this property is caracteristic of a topos, and in the lenguage of categories motivated the following deﬁnition. Deﬁnition 6. Let C be a category with coproducts and initial object. The coproducts in C are disjoint, if the following diagram is a pullback in C:

/0 / Y

X / X qY

Proposition 4. [1] Let L : P(C) → X a left exact localization then the coproducts in X are disjoint. Proof. Let X,Y objects of X applying the right adjoint functor R of L and use the fact the coproducts in P(C) are disjoint, the following diagram is a pullback.

0 / R(Y)

R(X) / RX qRY now applying L. L(0) / LR(Y)

R(X) / L(RX qRY) and for the left exact conditions of the functor L (L preserves initial object since left adjoint and preseves pullbacks since left exact), so we can rewrite the previuos diagram as:

/0 / Y

X / X qY and therefore the coproducts in X are disjoint.

3If C is a presentable category then the functor f ∗ preserves colimits if and only if f ∗ has right adjoint.

5 To prove the Giraud’s axiom fourth for a topos, we need to understand the meaning of "effective" for a equivalence relation. A relation R on an object X is a pair (R,m) where R is a object of X and m : R X × X, with this we can define two arrows d0,d1 : R → X, where d0 = π1 ◦ m and d1 = π2 ◦ m with π1,π2 : X × X → X, the projections. Definition 7. A relation R of a object X is called of equivalence if satisfies the following conditions:

1. (Reﬂective) R contains the diagonal.

∃ R o X 4 m " X × X

2. (Symmetric) Exists a morphism s : R → R such that the following diagrams are commutative: s s R o R R o R

d1 d0 d d 0 1 X X 3. (Transitivity) It is possible construct the following diagram and is a pullback:

R ∗ R / R

d1 R / X d0

Given a equivalence relation R, if the category admits colimits it is possible to deﬁne the co- equaliazer q of d0 and d1. We can construct the following diagram:

$ X ×X/R X / X q + X q / X/R

If the dotted arrow is an equality then R is an effective equivalence relation.

Proposition 5. Let X be a left exact localization for P(C) then the equivalence relations in X are effective.

Proof. To prove this, we need the following results:

1. The equivalence relations in the category of Sets are effective (this happen because a function induces a equivalence relation over the domain of the function).

2. The equivalence relations in a presheaf category over Sets are effective (for proof this we need use the fact the equivalence relation in the category of Sets are effective and the limits and colimits in a presheaf category P(C) are calculated pointwise).

6 So let R be an equivalence relation, applying the right adjoint functor R of the left exact localization, we can construct the following diagram in P(C).

RR = & ! RX ×RX/RR RX / RX q , RX q / RX/RR

Applying the functor L since L is left adjoint preserves colimits, and since it is left exact it preserves pullbacks, so we have the following.

R =

& ! X ×L(RX/RR) X / X Lq , X / L(RX/RR) Lq

Let f : X → Y be a morphism in X, for the complete and cocomplete conditions that satisﬁes X we can construct the following diagram:

X ×Y X / X

f X / X qX×Y X X

% 4 Y f

In case the dotted line give us a equality then we say f is a effective epimorphism. This point of view of the "effective" will be used for the proof of the preservation of small limits of the localization.

4 Giraud’s theorem

Theorem 2. Let X be a category that satisﬁes Giraud’s axiom. Then X is a left exact localization of P(C) for some small category C.

Proof. If X satisﬁes the Giraud’s axioms then X is accesible, so there exists a small category C of λ-compact objects which generates X.

The natural candidate to construct the left exact localization is the presheaf category of C, the existence of a inclusion functor between C and X is evident and we will denote as j : C → X. To construct the localization we need to use the next theorem:

7 Theorem 3. [4] Let j : C −→ X be a functor from a small category C to a cocomplete category X, the left Kan extension of j along y exists, and it is always given by the formula:

R(X) : C 7→ HomX(j(C),X)

The left adjoint is L : P(C) → X deﬁned for each presheaf F in P(C) as the colimit:

R πF j L(F) = colim( F / C / X )

Applying that and thanks to X is cocomplete then we can deﬁned the following adjoint:

o a : P(C) / X : ı where ı is deﬁne as ı(X) = HomX(j(−),X).

For the left exact conditions, we ﬁrst prove that ı full and faithful conditions and after the preservation of small limits.

Proposition 6. (Density Lemma [3]) Let X be a category and C a subcategory of X . The subcategory C is dense iff the functor ı : X → P(C) is full and faithful. The previous proposition indicates a new concept dense subcategory which means that every object in X is a colimit of objects in C. Since X is accesible, we know we can even use ﬁltered colimits then ı is full and faithful.

Using the construction of the functor a of the theorem 2. We can use the category C for formed a category of sheaf for a special topology.

We will be deﬁne a covering sieve for a object C to be a family of morphisms with codomain C such that the following morphism is a epimorphism. a (>) D → C g:D→C g∈S

Deﬁne J(C) to be the set of all covering sieves S of C, then we have a presheaf J. Proposition 7. The presheaf J deﬁnes a Grothendieck topology over C.

Proof. 1. (The maximal sieve) Let tC = { f |cod( f )` = C} be the maximal sieve over C, in special the identity arrow of C is in tC, therefore f :D→C D → C is epi. f ∈TC 2. (Stability axiom) If S ∈ J(C) then for any h : D → C, we have

h∗(S) = {g | cod(g), h ◦ g ∈ S} ∈ J(D)

Let S be a covering sieve of C and h : D → C, we can deﬁne the following pullback in X.

` p0 Ci ×C D / D

h ` Ci p / / C

8 where p is epi, (Hypotesis of S is a covering sieve) and we need to remember for the 0 second Giraud’s axiom, the epimorphisms` are stable under pullback in X then p is epi. So if we have the following morphism p : g:D j→D D j → D, this can view as de composition g∈h∗(S) of the following diagram: ` D j p

u `% p0 Ci ×C D / D

h ` * Ci p / C ` Where u exist because` Ci ×C D is a pullback and is epi for the surjective of the index ∗ sets of h (S) and Ci ×C D. Therefore p is epi. 3. (Transitivity) If S ∈ J(C) and R is any sieve on C such that h∗ ∈ J(D) for all h : D → C in S then R ∈ J(C). For the hypotesis of h∗ ∈ J(D), we can construct the following diagram `

Ci j pk

u `% p0 ! Ci ×C Ck / Ck

fk ` * Ci p / C

We can do this construction for each morphism in R then this indicate a Ci → C fi:Ci→C fi∈R

can descomposed has sums of coproduct of subcoverings sieves of each Ck, additionally each pk is epi then p also it is.

If we form the category of sheaves over this topology we have the following diagram:

C / Sh(C,J) // P(C ) : j ı # X

Applying the condition of effective epimorphism of the category X to each presheaf a(X) it is possible prove the next statement.

Lemma 2. [4] For each object X of X the functor HomX(−,X) is a sheaf for the Grothendieck topology J on C.

9 Proof. Let C be an object of C and S a open covering of C, i.e. a p : D → C g:D→C g∈S is an epimorphism, in X every epi is the coequalizer of its kernel pair (this happend for the condition of effective epimorphism in X) so we can deﬁne the following diagram: ` ` ` D ×C D / D p ` D p / C ` ` ` as D ×C D = D ×C D (in X the pullback construction preserves coproducts), then we have the following coequalizer diagram. ` ` / g:D→C D ×C D / g:D→C D p / C (1) g∈S g∈S

On the other hand each D×C D can seen as colimit of λ-compact objects of X, so if we take the coproduct over all posible B such that the following diagram is commutative:

B / D

D / C ` then the sets of B form a covering for C, if we precomposed B → C with diagram 1 then we have the following coequalizer: ` ` / B / D / C (2) And applying the hom functor: ` ` o HomX(B, X) o HomX( D,X) o HomX(C,X) (3) but the previous equation can seen as the functor ı (4), so we have the following equalizer diagram. Q o Q HomX(B,X) o HomX(D,X) o HomX(C,X) (4)

And therefore we can restrict the functor a : P(C) → X to the following adjunction:

o C / Sh(C,J) / P(C ) O a : a ı j ı z # X and applying the following lemma the equivalence between Sh(C,J) and X is true. Lemma 3. [4] The following is satisﬁes for the functors a and ı:

1. The counit εX is an isomorphism for every object X of X.

2. The unit ηF : F 7→ HomX(ı,a(F)) is an isomorphism.

10 Proof. 1. The counit εX is an isomoprhism applying the following theorem and use the fact the functor ı is full and faithful.

o Theorem 4. [3] Let L : X / Y : R be an adjunction. Then R is full and faithful iff the morphism εX is a isomorphism.

2. Let F be a sheaf, for the presentability of P(C), this functor can be seen as the colimit of representable functors y(C) with C a object of C, if we apply the functor a and ı to y(C), we have the isomorphism y(C) =∼ ı(a(y(C))) based in the following theorem

Theorem 5. [3] Let K : X → Y and T : X → Z be functors. If the functor K is full and faithful then the unit of the left Kan extension of T along K is an isomorphism.

T X / Z ? K lanKT Y

And use the fact the functor a and ı preserves colimits for sheaves then F =∼ ı(a(F)).

This implies X is equivalent to Sh(C,J) and remember the fact the functor a : P(C) → Sh(C,J) preserves ﬁnite limits and therefore a : P(C) → X also it does.

References

[1] F. Borceux, Handbook of categorical algebra, Vol III, Part of Encyclopedia of Mathemat- ics and its Applications, Universite Catholique de Louvain, Belgium, 1994.

[2] J. Lurie, Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press, Princeton and Oxford, 2009.

[3] S. Mac Lane, Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998.

[4] S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic, A ﬁrst introduction to topos theory, Springer-Verlag, New York, 1992.