A construction of Grothendieck topos from the base

category

Author Thomas CuvillierSupervisor Edmund Robinson

25/09/2013

1 Introduction

They are mainly two different approaches to Grothendieck toposes [PT]. Namely the oneprovided by Grothendieck, where they arise as an ersatz of the category of sheaves on atopological space, and the one provided by Giraud’s theorem. The force of the theoremis that it gives a complete characterisation of such categories in purely categorical terms[Joh]. Among the required properties is the existence of a small generating set.

Therefore, it comes the natural idea to start from this set, and then to build aconstruction of the topos from it, adjoining free more elements to satisfy the requiredproperties. One of the main property of a topos is that it is an exact category. Therefore,it has been widely studied when and how free exact completion turned out to be toposes[Men00] [Vit94] [Men07] .

In this master thesis we focus on the category of sheaves on a category C equippedwith a Grothendieck topology. Can we construct it starting from C ? The sheafifica-tion process is already known, therefore the difficulty is to make arise the category ofpresheaves SetC

Op

. The question of whether it is possible or not to build SetCOp

fromfree constructions starting from C has already been partially answered, and it turns outto work in the case that C has small limits [Car95].

The specificity of this work is that it adopts a purely constructivist point of view (notin the logical sense). Instead of verifying that our construction enforces the Giraud’saxioms to be true, we focus on the structure of the objects of the category of presheaves.Following this idea, instead of using the more general construction of the sheafificationbased on the equivalence between Grothendieck topologies and localisations, we ratherused a direct construction of the sheaf, writing it as a colimit of a construction based onthe original presheaf.

2 Definitions and plans

2.1 The sheaf category

To begin, we will give some basic definitions, reminding what is a presheaf.The idea behind the presheaf is to provide a tool for looking at functions over a

topological space both locally and globally. Let us say we look at the space of functionsf : X 7→ R, where X is a topological space. Then, given the family of open sets of thetopology of X, write O(X) for it, we can define a function Θ : O(X) 7→ Set, which, given

1

U ∈ O(X), gives Θ(U) = f : U 7→ R. The point here is that we got restrictions mapsτu−v : Θ(U) 7→ Θ(V ) whenever v ⊆ u, that behaves nicely according to the followingdiagram.

Θ(U)

U

Θ(V )

V

Θ Θ

⊇

τu−v

By considering the poset (O(X),⊆) as a category, we obtain that Θ is a co-functor fromO(X) 7→ Set. We will write pij : Ui 7→ Uj for the inclusion of an open set Ui into Uj

Definition. A Presheaf from a base category C is a co-functor Cop 7→ Set

In the sequel, we will look at presheaves over a topological space as a category, usingthe following well-known result.

Remark. Given two category C and D, the functors F : C 7→ D form a category, wherethe objects are the functors themselves, whereas the morphisms are the natural transfor-mations between the functors. The composition of morphisms is the usual compositionof natural transformations. It is denoted DC .

So we can now speak about the category of presheaves.

Definition. The precheaf-category SetCop

is the contravariant-functor category from Cto Set.

We will write base category for C, when working with the presheaf category.A sheaf is a refinement of a presheaf. The presheaf regards just local properties, but,

unfortunately, a local property that is true everywhere might not be true globally. Inorder to be able to translate local properties to global ones, we have to define a specialclass of presheaf. A presheaf can be considered as a sheaf if it satisfies new axiomsestablishing equivalence between local and global properties. In the case of a presheafcategory over a topological space, we have the following definition.

Definition. A sheaf is a presheaf that satisfies

• the gluing axiom: if (Ui) is an open covering of a set U , if (si) in F (Ui) is a familyof elements such that proij(si) = proji(sj), then there exists a s in F (U) such that,for all i, projis = si.

• the locality axiom, for every coverage Ui of U , for every s and t in F (U), if projis =projit for all i, then s = t

where projij is defined to be F (Ui ⊇ (Ui ∩ Uj), and proji to be F (U ⊇ Ui)

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However, this definition makes just sense in the category of open sets of a topolog-ical space. In order to define the notion of sheaf precisely, we have to translate somedefinitions, such that open covering, in a categorical language. Starting with a category,C, we look at the object of C as being the open subsets of a more general unknowntopological space. In order for that to make sense, we have to consider an additionalstructure on the category, called as well topology. By doing so, we would later be ableto establish a correspondence (not proven here, see [PT]) between categories with sucha topology and categories of open sets over a topological space. It therefore proves thatthe definition used is the right one.

The topology used over the category C is called the Grothendieck topology. Wewill define it in several steps, the first one being the notion of sieve, which capturesopen-covering.

Definition. A sieve over c, c object of the category C, is a subset S of the commacategory C/c close under precomposition. It means that whenever d 7→ c ∈ S, and e 7→ da map in C, then the morphism e 7→ d 7→ c of C/c belongs in S.

One easy operation on sieve is the pullback. If S is a sieve over c, f : d 7→ c morphism,the we can define the sieve f∗S over d, by considering the arrow of the sieve that factorsthrough f . Note that this exists whether C has pullback or not.

The Grothendieck topology consists in a collection of sieves, called covering sieves,over each object of the category, that are subjects to certain axioms.

Definition. The conditions that the covering sieves must fulfill in order to form aGrothendieck topology are

• For each covering sieve S over c, and each morphism f : d 7→ c, then f∗S is acovering sieve of d.

• Let S be a covering sieve of X, T be any sieve on X. Suppose that for each(f : Y 7→ X) ∈ S, the pullback f∗T is a covering sieve of Y . Then T is a coveringsieve of X.

• id : X 7→ X is a covering sieve.

A category equipped with a Grothendieck topology is called a site. We write J(c)for the collection of covering sieves over an object c of the base category.The next step towards the precise definition of sheaf in the category framework is theone of compatible family (wrt a sieve).

Definition. Let (C, J) be a site, S ∈ J(c) a sieve over c, and finally P a presheaf, that isa functor P : Cop 7→ Set. A compatible family for S of elements in P is a rule assigningto each f : d 7→ c of S an element xf of P (d), that satisfies the following condition:

For all morphisms g : e 7→ d, we have P (g)xf = xgf .

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In topological term, for an open set U and a family of open subsets of it, we mapseach subsets V to a function fV such that, if W subset of V , then fW is the restrictionof fV to W . We now have enough material to settle properly the definition of sheaf.

Definition. Let C be a small category and (C, J) be a site. Then a presheaf P is a sheafif, for every c ∈ C, every covering families S ∈ J(c), every matching families xf for Sof elements in P , there exists a unique element x of P (c) such that for every f : d 7→ c,we have xf = P (f)x

Before ending this section, let us remind some very basic properties about scheaves.First, every scheaf is a presheaf. A morphism of between sheaves P and Q is a morphismof presheaves between them. That means that the scheaves form as well a category, and,furthermore, there is an inclusion functor : Sheaf(C) → Prescheaf(C) that is bothfull and faithfull.

2.2 Grothendieck toposes

A Grothendieck topos is a category that looks like the category of scheaves on a topo-logical space.

Definition. A Grothendieck topos is a category of sheaf over a site.

Equivalently, some other definitions might be given.

2.2.1 As a projective category

An interesting property is that the inclusion functor Sheaf → Presheaf(C) has aleft adjoint called sheafification or sheafing, that is limit preserving. We say that thesubcategory Sheaf is projective. In fact, there is a one-to-one equivalence betweenprojective subcategories of a presheaf categories, and topologies on it. Therefore, we cangive a new characterisation of the Grothendieck topos.

Definition. A category D is a Grothendieck topos over a category C if there is a mor-phism D → Presheaf(C) that is full and faithful, and that admits a finite preservingleft adjoint.

2.2.2 By elementary toposes - Giraud theorem

In this paragraph, we will give a characterization of Grothendieck topos just based on theproperties of the category. Indeed, a Grothendieck topos is a refinement of an elementarytopos, that is a category with some additives properties that force it to behave, in someway, like the Set category.

Definition. An elementary topos is a category which

1. has finite limits,

2. is cartesian closed, and

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3. has a subobject classifier.

We will assume the reader knows the definitions of limits and cartesian closed, andwill therefore focus on sub-object classifier.

Definition. A suboject classifier is an object Ω together with a morphism t : 1 7→ Ω,where 1 is a terminal object, such that, for each object U and mono-morphism f : V 7→ U ,there is an arrow g : U 7→ Ω, such that the below diagram is a pullback square.

V

U Ω

1!V

f

g

t

In order to understand the definition, let us spell it out in Set. In this framework, amono-morphism is just an injective function, and we can identify V with a subset of U .We have Ω = 0, 1, and t : 1 7→ 1. Then g just picks the elements of U that belong to Vand send them to 1, while mapping the other to 0. Basically, we just identify the set ofsubsets of U with 2U .

Claim. Every presheaf category is an elementary topos

This property follows from the fact that Set is itself an elementary topos, and there-fore induces one on the functor category. But we can notice that never Set can not beviewed as a Grothendieck topos, it is far too large.

Definition. A Grothendieck topos is an elementary topos which has

1. all finite colimits

2. a small generating set

This point of view of Grothendieck topos follows from a famous paper of Giraud,where he prooves this theorem. It gives the huge advantage to be able to defineGrothendieck topos in a purely categorical way , where we do not speak anymore aboutpresheaf, base category, or topology.

Definition. An Grothendieck topos is a locally small category with

1. A small generating set

2. All finite limits

3. All small coproducts, which are disjoint, and pullback stable

4. All equivalence relations in C are effective

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2.3 Aim of the thesis

We will prove the following theorem in this paper.

Theorem 1. Every Grothendieck topos over a base category C that has all finite limitscan be built from colimits of the exact completion of the coproduct completion of its basecategory.

Apart from giving a new insight to Grothendieck toposes, one of the main advan-tage of this theorem and this thesis is that it actually gives a way to construct everyGrothendieck toposes over a category C starting from that category. Moreover, the sub-ject of toposes being very close to foundations of mathematics, one would like to avoidresting on the mathematical language, and especially the set theory. Indeed, by addingfew axioms to the elementary topos, capturing both the existence of N (the object ofnatural numbers), and the axiom of choice (every epi splits), one could establish a corre-spondence between such elementary toposes and the model of the Set theory. Therefore,we define, in the sequel, Set to be defined as such a topos.

The idea of defining properly all the foundations of mathematics by the means ofcategory has yet, to the most of the author knowledge, not been achieved. Though, someresearchers such as Benabou have highlighted the possibility to undertake it [Ben85]. Inorder to do so, one must rely on the fibration theory, that is developed in the nextsection.

2.4 Plan

The plan can be sum up in the following diagram.

Fam(C)C SetCOp Sheaf(C, J)

add

coproduct

add

coequalizer ∼ exact completion

force

certain colimits

We use the theory of fibration to define Fam(C) and show it has coproducts. We thendefine the exact completion of it, and ensure carefully that the result is indeed SetC

op

,by looking at the subcategory of projective elements of it. In the last step we enforcethe presheaves to be more ”global” by taking colimits over covering sieves, turning theminto sheaves.

3 Fibrations and coproducts

3.1 Basic definitions

A fibration is a pair E and B of categories, together with a functor p : E 7→ B, suchthat E appears to be indexed over B. The key example to understand the concept is totake B to be Set, and therefore objects of E turn out to be indexed families of elementseii∈I where I is a set. More generally, given a category C, we will speak of fam(C)to be the fibration defined below. The concept of fibration will be defined preciselyafterwards.

6

Definition. The fibration fam(C) over Set has

1. as objects : indexed family of objects cii∈I , where I ∈ Set and each ci ∈ C

2. as morphisms Cii∈I 7→ Djj∈J : a morphism f : I 7→ J in Set, together with afamily of morphisms fi;Ci 7→ Df(i)i∈I .

The composition of morphism is the canonical one.The fibration functor p sends Cii∈I to I, and a morphism Cii∈I 7→ Djj∈J to itsunderlying morphism f in Set.

In general, we wil speak of Ei to denote the elements e of E in the fibre over i, thatis such that p(e) = i

Let us focus on the properties that must have a fibration. First, if we index families,we should be able to re-index them. Given f : I 7→ J in the base category B, givenCjj∈J over J in E, we want to transport Cj along f to create the family Cf(i)i∈Iover I. To do such, we need the family Cf(i)i∈I together with a morphism f∗ :Cf(i)i∈I 7→ Cjj∈J over f . Such a morphism is called cartesian. One of the definitoncould be that we requires each fi to be an isomorphism, but we would rather have apurely categorical definition.

Definition. A morphsim f : A 7→ C in E is a cartesian morphism if, for every mor-phism g : B 7→ C such that the projection Φ = p(g) : J 7→ K factorizes throughΨ = p(f) : I 7→ K, that is there exists χ : J 7→ I such that Φ = Ψ χ, then there existsa unique morphism h over χ that factorizes g through f , that is such that g = fh.

B

A C

J

I K

p

g

f

Φ

Ψ

∃χ : J 7→ I

∃!h : B 7→ A

Let us first notice that two cartesian morphisms over Φ with same image are iso-morphics. Therefore, given f : I 7→ J in B, C over J in E, there is a unique, up toisomorphism, D in E such that there is a cartesian morphism f∗ : D 7→ C. We call sucha f∗ a cartesian lifting of f .

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For Ep−→ B to be a fibration, we have to ensure that, for every object in the category

E, and every morphisms f : I 7→ J in B, we are able to reindex along f . That willdefine precisely when such a structure is a fibration.

Definition. Two categories E and B along with a morphism p : E 7→ B form a fibrationif, for every I in B, for every CI in E belonging over I, for every f : J 7→ I in B, thereis a Dj in E over J together with a cartesian lifting f∗ : DJ 7→ CI over f .

∃DJ CI

I J

p

f∗

f

In a fibration, for every CJ in EJ , the data of f : I 7→ J gives, up to isomorphism,a unique DI in EI , by taking the domain of the cartesian lifting of f . Furthermore,because of the definition of cartesian, one can see that f carries also the morphismsof EJ in a canonical way to EI . For every I in B, by considering the objects of EI

together with the morphisms between them that lie over idI , one can see that we obtaina category. If we look at Ej this way, we can see that each f : I 7→ J in B defines aunique (up to isomorphism) functor F ∗ : EJ 7→ EI . Therefore, there actually exists another definition of fibration that rests on a functor p : Bop 7→ Cat. Fibrations definedthat way are called Grothendieck fibrations. They require a choice, for each f ∈ B, of aparticular cartesian lifting.

Respectively, one can also define the co-cartesian morphisms, as in the followingdiagram ;

A

C

B

JI

K

p

g

f

Ψ

Φ

∃χ : J 7→ K

∃!h : B 7→ C

and the definition of co-fibration follows in a straightforward way. A cofibration p :E 7→ B is a functor between two categories such that for every C ∈ E, for every

8

f : I = p(C) 7→ J in B, there is a f∗ : C 7→ D lying over f such that f∗ is cocartesian.We have the properties that come logically from those of the cartesian morphisms; if aco-cartesian morphism exists it is unique up to isomorphism, and hence, in the case of aco-fibration, every f : I 7→ J defines a functor F∗ : EI 7→ EJ . In fam(C), given Cii∈Ifamily over I, f : I 7→ J in Set, the co-cartesian lifting of f starting from Cii∈I willhave, as co-domain,

∐i∈I,f(i)=j Cij .

We see in this formula the apparition of the coproduct, indexed over Set. Therefore,the co-cartesian morphisms will be the cornerstone to define indexed coproducts, in thecase of a fibred category. A quick analysis of F ∗ and F∗ shows that there are adjoint.And this is exactly what we need to define a coproduct.

3.2 Coproducts and Beck-Chevalley conditions

Before giving the final definition of a fibred category with co-products, a careful analysisis needed. Indeed, for a fibration which is also a co-fibration, or, such that every f inthe base category gives rises both to a cartesian and a co-cartesian lifting, we need toensure that these both liftings behave nicely with each others. Indeed, as one can see inthe equations below (where every elements lie in fam(C)) (that comes from [McL]), weshould expect the results of these two operations to be equal (or, at least, isomorphic).

Let πI,J0 : I × J 7→ I be the first projection in Set. Let∐

I,J beπI,J∗ : fam(C)I×J 7→ fam(C)I . Let α : K 7→ J .

α∗∐

I,J :Xi,ji∈I,j∈J 7→ ∐

iXi,jj∈J 7→ ∐Xi,α(k)k∈K∐

I,K(1I × α)∗ : Xi,ji∈I,j∈J 7→ Xi, α(k)i∈I,k∈K 7→ ∐Xi,α(k)k∈K

Therefore, some conditions appear in order to enforce such diagrams to commute;The Beck-Chevalley conditions. A morphism p : E 7→ B that is both a fibration and aco-fibration is called a bifibration.

Definition. [RR90] We say that a bifibration has indexed co-products if the Beck-Chevalley condition holds, that is : whenever f, g, k, h is a pullback in the base cate-gory, for f∗, g∗, k∗, h

∗ lying over in the category E, whenever f∗, k∗ is cartesian, g∗ isco-cartesian then h∗ is co-cartesian.

f

gh

k

f∗

g∗h∗

k∗

As in Set the following diagramm is a pullback square, the required equality followsfrom the fact that every two co-cartesian lifting with same domain are isomorphic.Particularly the one from the conditions, and the one desired.

9

I × J I ×K

I K

(1I × α)

πI,KπI,J

α

3.3 Properties of fam(C)

Theorem 2. [Car95]

1. fam(C) has small coproducts

2. if C is left exact, then so is fam (C), moreover the sum are disjoint

3. fam(C) is the coproduct completion of C

Proof. We will just prove (3). The two other are easy and follow from simple compu-tation. The coproduct completion is a universal property which characterizes fam(C)up to equivalences. For every functor F : C 7→ D, where D has coproducts, thereis a unique coproduct preserving functor F ′ : fam(C) 7→ D that makes the followingdiagram commutes.

C D

fam(C)

F

i F ′

It suffices to define F ′(∐

i∈I Ui) =∐

i∈I F (Ui) The uniqueness of F′ on objects follow the

fact that each object is a coproducts of objects of C, whose image are already definedby F , and whose coproducts have to respect this property as F ′ is coproduct preserving.On morphism, the definition of F ′ follows in a straightforward way.

4 Construction of the Presheaf category

We look at the Grothendieck topos as a sheaf category, that is as a subcategory ofthe contravariant functor category from a given category C to Set. We will write basecategory for C. From now on, we will only consider small category C, that is such thatits objects form a set, and such that the hom functor has Set as codomain. The centraltool in the following will be the Yoneda Lemma.

Theorem 3. Let F be a presheaf. Then hom(hom( , A), F ) = F (A)

Definition. We say that an object of a Grothendieck topos is reprensentable if it isrepresentable as a functor. X ∈ SetC

Opis representable, if there is x ∈ D, such that

X( ) ≃ hom( , x)

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The representable elements are the central link between the Grothendieck topos andthe underlying C category, from whom we will construct it. There is a evident full andfaithfull functor Y : C 7→ SetC

Opthat sends x to hom( , x) (defined up to isomorphism).

This is called the Yoneda inclusion.

4.1 Presheaf as colimits

Theorem 4. Every presheaf is a colimit of representable.

Proof. Let F ∈ SetCop, and let us consider the arrow category Y (C)F . Such category

has as objects arrows D 7→ F , where D ∈ Y (C), and as maps commutative triangles asshown below.

Y(A) Y(B)

F

The point is that F is the colimit of the previous diagram. More precisely, let p :Y (C)F 7→ SetC

Op

the forgetfull functor. Then F is the colimit of p. The statement canbe proven in a purely formal manner, but we will prefer here a more convenient fashion.Let us consider (G,µA) an other presheaf that makes the diagramm commutes. Thenif we prove that there exists an unique f from F to G, such that the following diagramcommutes

A B

G

FµA µB

we obtain the desired result.By abuse of notation let write A for both the object of C and the presheaf Y (A).

The Yoneda lemma tells us that C(A,F ) = F (A) and equally C(A,G) = G(A). So letψA ∈ F (A) and µA ∈ G(A) be the representant of the maps in the upper diagramm. Tosay that the upper triangle commutes means that for every maps χ : A 7→ B, ψA is sentto ψB via the functor F (χ) : F (A) 7→ F (B). We respectively has the same propertywith G and µ Therefore, the families of map fA : F (A) 7→ G(A) that would send ψA toµA for every A have the following property.

F(A) F(B)

G(A) G(B)

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that is this is a morphism of presheaf and there is a map F 7→ G in SetCOp. So F is the

colimit of the diagram.

Respectively, as the presheaf category is a topos, every colimit of presheaves is apresheaf. Particularly, every colimits of representables is a presheaf, and we thereforehave a bijection between them.We therefore need to construct the colimit completion of the category C. We will dothat in two distinctive steps. In the first one, we will add the coproducts, and then wewill build the exact completion.

4.2 Coproduct of representables

In SetCop, we would like to construct the subcategory of coproducts of representables.

These coproducts are obtained by the set-indexed completion of C. One can see that fromthe family (Fi)i∈I , we can associate a coproduct

∐i Fi. Extending this, we can create

a functor fam(C) 7→ coprod(C). However, it is rather unclear if from a coproduct wecan associate the associated family of indexed objects. Indeed, for i, i′, we might thinkthat there may exist j such that F (i)

∐F (i′) ≃ F (j), and therefore the map would

be ill defined. Actually in SetCOp

there is no such issues, and these two categories areequivalent.

Definition. We write coprod(C) for the full subcategory of SetCop

of representable andtheir small coproducts.

Theorem 5. There is an equivalence fam(C) ≃ CopropRep(C).

Before prooving the theorem, let us denote why establishing an isomorphism (ifthere is one) might be more complicated. For every families of elements, there mightexists several coproducts of it in SetC

op

, as well as several families representing it infam(C) (depending on on which set the family is actually indexed). Actually, in SetC

Op

these coproducts are isomorphic, and the isomorphisms maps lie in Set. Two of thesecoproducts will send each A in C to sets that are in bijection. So we might think thereis a much larger class of coproducts than fam(C). However, one might notice as wellthat two sets that are in bijection I ≃ J defined as well two different but isomorphicfamily of fam(C). So, to sum up, for each families of representables FI over I, theirclass of coproducts in fam(C) is, in bijection (of class) with the class of isomorphic sets

of I together with the isomorphism, whereas, in SetCOp, their class of coproducts is, this

time, in correspondance with the class of isomorphic sets of∐

I F (I) times all I in C.This actually does not make a lot of sense, and therefore, in order to study it properly,one has to rely on a upper level of category theory, namely 2-category. The proof of thetheorem rests almost only on Yoneda, and the limit and colimit preservation of the homfunctor.

Proof. We are gonna look at the morphisms between coproducts of representables.

• hom(∐

αC( , Aα),∐

β C( , Bβ))

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• ≃∏

α hom(C( , Aα),∐

β C( , Bβ))

• ≃∏

α(∐

β C(Aα, Bβ)) (Yoneda)

So for each α we pick some β and an element in C(Aα, Bβ). Other way to say, amorphism between

∐αC( , Aα) and

∐β C( , Bβ) can be seen as the data of a func-

tion f : α 7→ β together with a family of maps ρa : Aa 7→ Bf(a)a∈α. Therefore,

there is a bijection SetCop(∐

α C( , Aα),∐

β C( , Bβ)) ≃ fam(C)(Aα, Bβ). The functorF : fam(C) 7→ Coprod(C) carries these bijection and is hence full and faithfull, andis, moreover dense. So using the well known lemma characterizing an equivalence ofcategory, we can conclude that these two categories are equivalent.

4.3 Towards the colimit completion

Once we have coproducts, only the missing coequalizers prevents us from having allcolimits. Indeed, we have the following theorem

Theorem 6. A category with pullback and equalizers have all small limits

See [MES] for a proof. Therefore, as a corollary, a category which has both co-products and co-equalizers have all co-limits.

We will rest on the results of the following paper [CM80], that explores the free exactcompletion of a left exact category, that is a category with all finite limits. The wholepoint consists in looking at equivalence relations, as a coequalizer can be seen as theimage object quotiented by an equivalence relation. Indeed in set we have :

A B C = BR

= Bx∼y⇔∃z,f(z)=x∧g(z)=y

f

g

.

Definition. An exact category is one such that

1. every morphism have a regular epi-mono factorization

2. regular epis are pullback stable

3. every equivalence relation is a kernel pair

Before going on any further with the equivalence relations, we will shortly investigatewhat it means for an epimorphism to be regular.

Definition. An epimorphism is regular if it is a co-equalizer of a pair of arrows.

In any topos, every epimorphism is regular.

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4.4 The equivalences relations

The definition is a categorical reformulation of the equivalence relations on Set.

Definition. An equivalence relation on a object A consists in an object R together withtwo arrows r1, r2 : R 7→ A, such that

• r1, r2 are jointly monic

• (S¯) there is a s : R 7→ R such that r1 s = r2 and r2 s = r1

• (R¯) ∃i : A 7→ R such that r1 i = r2 i = idA.

• (T¯) Let T be the pullback of R

r1−→ A along to Rr2−→ A. T is the set of pairs (γ1, γ2)

of elements of R such that r2γ1 = r1γ2. Then there is a map t : T 7→ R such thatr2 p2 = r2 t and r1 p1 = r1 t

T R

R A

p1

p2 r1

r2

Unfortunately, to add all the equivalence relations would not be enough. We have toconsider a slightly larger class, the one of pseudo equivalence relation. The main changeis that we do not impose r1, r2 to be jointly monic anymore. Therefore, the equivalencerelation is not directly induced on A, but on C(U,A), the set of morphims U 7→ A,whatever U . Formally;

Definition. A pseudo equivalence relation on A is an object R together with two mor-phims r1, r2 : R 7→ A such that it induces an equivalence relation C(U,R) 7→ C(U,A) forevery object U .

If the category is left-exact, that is has all small limits, then the definition of pseudois almost equivalent to the original one, except that we do not require the maps r, s, t tobe unique anymore.

Following the requirement to add all equalizer we define Cex, the category of equiv-alence relation on a category C. That will turn out to be the free exact completion ofone of its subcategory.

Definition. The category Cex has,

• pseudo equivalence of C as objects, that is triple (A,R, (r = (r1, r2) : R 7→ A×A).

• [f ] : (X,R, r) 7→ (Y,Q, s) classes of morphisms of C compatible with the pseudoequivalence relation as morphisms. That is, class of morphisms f such that, forevery u, u′ : U 7→ X, we have u ∼X,R u′ ⇔ f u ∼Y,S f u′, and, such that forevery morphisms f, g in this class f u ∼Y,S g u. Equivalently, for every f, g inthe class [f ], there exists a morphism F : X 7→ Q, such that s0F = f ∧ s1 F = g.

14

.We interest ourselves in this construction as it allows to create a free exact category

from a left exact one. The following theorem comes from [Car95], [RR90].

Theorem 7. • Aex is the free exact category on the left exact category A.

• An object of Aex is projective (for regular epis) iff it is isomorphic to an object inthe image of A → Aex

• if C is an exact category, the full category of whose regular projectives is left exact,and if for all object C of C there is a regular epi P 7→ C from a projective onto C,then Cex is the exact completion of its subcategory of projectives.

We interest therefore ourselves in the subcategory of projectives ( that will be defined

in below) of SetCOp

, and hope that it will turn out to be coprod(C), so that we can getits free exact completion.

4.5 Projective elements

Definition. An object X of C is called projective if, for every morphism f : X 7→ Z,for every epimorphism g : Y 7→ Z, f factors through g.

X Z

Y

f

gh

One of the point is that it will turn out that we can covered elements by projective.We therefore interest ourselves in the relation between projectives and representableelements.

Claim. In a presheaf category, every representable element is projective.

Proof. Let X be representable, x the element of D such that X( ) ≃ hom( , x). Then wehave a bijection hom(X,Z) ≃ Z(x). So a f can just be seem as an element f∗ of Y (x).The natural transformation Y 7→ Z is an epimorphism, and, as we are in a topos, is epion each element of the category. Namely, the morphisms Y (d) 7→ Z(d) are epimorphismsfor every d ∈ D. This is true in particular for x, so Y (x) 7→ Z(x) is onto. Thereforethere is an y in Y (X) such that gx(y) = f∗.Once again, there is a canonical isomorphism between hom(X,Y ) and Y (X). By pickingthe morphism h of hom(X,Y ) that matches y we obtain the required morphism.

This proof rests on a subtle point that holds because we are working in a topos; everyepimorphism is locally epimorphic. Let us proove that.

15

Claim. In a presheaf category, every epimorphisms is locally onto.

Proof. A presheaf category is a topos, and has, therefore, a subobject classifier.Let f : X 7→ Y be an epimorphism, and let t, tX : Y 7→ Ω be respectively the constanttrue morphism and the classifying map of f . Then, as we have t f = tX f = X 7→true, and f epimorphism, t = tX . Therefore, for every x in the base category D, wehave X(x) ≃ Y (x) as the classifying arrow tX(x) would distinguish them otherwise.

Just for the interested reader, such large property for the representable element isnot required in the following. A more simple property, that will be enough, and thatdoes not rely on the topos structure of the presheaf category is the following.

Claim. Every representable is projective with respect to co-equalizer. Explicitly, forevery representable A, for every morphism f : A 7→ B, every coequalizer g : C 7→ B,there is a map A 7→ C that makes the triangle commutes.

A B

C

•

f

g

In order to prove the claim we need a characterisation of co-equalizer in the categoryof presheaves.

Remark. Let H be a coequalizer of α, β : F 7→ G,

F G Hα

β

γ

then each for each A in the base category, G(A) has an equivalence relation ∼A generatedby

y ∼A y′ if ∃x ∈ F (A), αx = y ∧ βx = y′

such that G(A)/ ∼≃ H(A)

And there is the proof of the claim.

16

Proof. Let x be the representant of A in the base category. By Yoneda, we can considerf as a element f∗ of the set B(x), and respectively, the morphism h as an element ofC(A). So we are looking for an element h∗ of C(A) such that g(h∗) = f∗. By the remarkbelow B(A) ≃ C(A)/ ∼A ( where the projection morphism is g) , and therefore for allelements f∗ we could pick an element h∗ that is a representant of the class, that is suchthat g(h∗) = f∗.

Claim. Every coproduct of representable is projective.(wrt co-equalizer)

Proof. The proof is the same as above exept that this time A is not representable buta coproduct of representable. A =

∐I C( , AI), and therefore hom(

∐I C( , AI), B) =∏

hom(C( , Ai), B) =∏B(Ai). Then, to find an element h consists actually in finding

for each i ∈ I an element hi in C(Ai) as in the first case, considering fi the projectionof f in B(Ai).

This is not true for arbitrary colimits. For example, this does not work with co-equalizer. Indeed let p1, p2 : P 7→ Q, P and Q representable, such that r : Q 7→ R isthe coequalizer of f, g. For every morphism f : R 7→ B and epimorphism g : C 7→ B,we are looking for an h : R 7→ C that makes the triangle commutes, explicitly such thatg h = f . As P representable, there is a map j : P 7→ C that makes the wider trianglef r p1 = f r p2 = g j. For the same reason, we can define a map k from Q to C.By the definition of the co-equalizer, if k p1 = k p2, then k : Q 7→ C coequalizes p1and p2 and therefore there is a map h : R 7→ C with the required property. But we areunable to enforce this equality, and therefore the diagram might not commute. In thiscase, one is not able to conclude about the existence of such an h.

4.5.1 Characterisation of projective elements

The following section is inspired by the paper [MO07].

Lemma. P is projective if and only if every regular epimorphism with codomain P hasa section.

Proof. If P is projective, and g a epimorphism with codomain P let just take the followingdiagram

P P

Q

id

gf

then the f is a section of g.Let consider the following diagramm.

17

A B

CA×B C

•

p1

g′ g∗

f

g

We are looking for a h from A to C. We take the pullback of g along f, that results againin a regular epimorphism g′ : C ×B A 7→ A. We therefore use the hypothesis to exhibita section of g∗ : A 7→ C ×B A of it , and composing it with the first projection of thepullback, we obtain the desired h.

We would like to give a characterization of projective elements that are not co-products, that is that can not be presented as co-products of other elements.

Definition. A element A is indecomposable if whenever A =∐

iAi then there is aunique i such that Ai is not initial, that is such that Ai is not the empty presheaf.

Theorem 8. In SetCOp

an element X is projective and indecomposable if and only ifit is the retract of an indecomposable representable presheaf; There is a element c ∈ Csuch that X → iY (c) 7→ rX and r i = idX .

The theorem gives us the desired characterisation for projective. Therefore, if onewants to ensure that coproduct of representable spans every projective, one has to ensurethat any retract of representable is still a representable.

Proof. 1. Let say X is projective and indecomposable. As every element is a colimitof representable, there is a coproduct of representable

∐iAi together with a regular

epimorphism f :∐

iAi 7→ X. As X is projective, f has a section f ′ : X 7→∐

iAi.It just remains to show that there is a unique i such that Ai not initial. For agiven i, let consider the pullback

Xi

Ai

∐iAi

Xp0

ii

p1 f ′

We can see that we got that X ≃∐

iXi. But as X is indecomposable, there isjust one i such that Xi is not the empty element. Then pick the correspondingAi. Then fAi

is still a regular epimorphism from Ai to X, and we got the desiredresult.

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2. Now we take an X that is a retract of an indecomposable representable. We willshow that X is projective and indecomposable. First, projective. Let f : W 7→ Xa regular epimorphism. Then, consider the pullback of f along r : Y (c) 7→ X,called g. It is a regular epimorphism with codomain a projective element, so itsplits, and we obtain a morphism g′ : Y (c) 7→ Y (c) ×X W . By composing it withthe first projection of the pullback on the left, and by i : X 7→ Y (c) on the right,we obtain a morphism X 7→W that is the required section.

W ×X Y (c)

Y (c) X

Wp0

g

r

g′

i

f

Equivalently, say that there exists Xi such that∐

iXi ≃ X. Then by pulling back alongr : Y (c) 7→ X, we got a presentation of Y (c) as a coproduct Y (c) ≃

∐Yi. As Y (c)

indecomposable, there is only one i such that Y (i) non initial. Let say that Y (j) initial,then, consider the diagram below, that is still a pullback;

Yj

Y (c)

Xj

X

p0

p1

ij

r

then as r is an epimorphism, p1 is an epimorphism as well. But Yj is the empty presheaf,and every epimorphism from the empty presheaf that are epimorphisms are actuallyisomorphisms. So Xj is the initial object as well. And therefore X is indecomposable.

The most famous result in the literature currently [Car95] if that if C has finite col-

imits, then fam(C) is the subcategory of projective elements of SetCOp. We present here

a slightly better one. Indeed, it is enough to state that every retract of a representableis a representable in order to get the result. We will see how to enforce this to be true.Let us look at u = i r : Y (c) 7→ Y (c). Then u is idempotent, that is u u = u, as,when composing, the morphism r i in the middle collapse to identity. Since the Yonedaembedding is full and faithfull, this u actually corresponds to a morphism in C that isas well idempotent.

Definition. We say that a category is Cauchy complete if the idempotent splits, that isif u : C 7→ C idempotent, then there exists D, i : D 7→ C and ir : C 7→ D such thatr i = idD amd i r = u.

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In the case of C Cauchy complete, one can see that every retract of a representableelement is still in C. Therefore, the coproducts of C and the projective elements are thesame.We end this section by an easy lemma

Lemma. If C has all equalizers, then C is Cauchy complete

Proof. Let u : X 7→ X be an idempotent. Let us consider the coequalizer e : Y 7→ X ofu, idX : X ⇒ X. Then, as u : X 7→ X equalizes the two arrows ( u u = id u), thereis an arrow f : X 7→ Y such that e f = u. Now, since e f e = u e = e, and e is amonomorphism, f e = ide.

4.6 Presheaf, the final step

In order to finish, we have to prove that for every presheaf G, there is a coproduct ofrepresentable

∐i Fi together with a regular epimorphism

∐i Fi 7→ G. As C is small, by

considering the object coprodc∈CY (C), we shall get the desired result. Indeed, as one cansee in the previous section, every presheaf is a colimit over all representable presheaf.Therefore, it is construct as

∐i Fi + coequalizers, because of the characterization of

colimits. Therefore, there is a epimorphic map∐

i Fi 7→ G that is the coequalizer fromthe construction.

So to sum up, we have :

• fam(C) ≃ Coprop(C)

• Every coproduct of representable elements is projective. Therefore all elements ofCoprop(C) is projective,

• If C is Cauchy complete, then the coproducts of representables are exactly theprojectives (at least up to isomorphism)

• For every presheaf, there is a regular epimorphism from a coproducts of repre-sentable onto a this presheaf.

• if C is left exact, then the category fam(C) = coprod(C) is left exact

• The category SetCOp

is a topos and is therefore exact

• if C is an exact category, the full category of whose regular projectives is left exact,and if for all object C of C there is a regular epi P 7→ C from a projective onto C,then Cex is the exact completion of its subcategory of projectives.

Therefore, following the theorem, if the category of coprop(C) is the same as the

category of projective, and if this category is left exact, then SetCOp

is the free exactcompletion of it.

Theorem 9. if C is left exact then Fam(C)eq ≃ SetCOp

.

20

In order to make this section clear, we would like to know what would it mean thefree exact completion for a fibred category.

Claim. [RR90] Given a left exact fibration p : E 7→ B, its free exact completion as afibration is otabined by taking the free exact completion in each fibre. Moreover, if Eas B indexed coproducts, then so does its exact completion, and the inclusion F → Fex

preserves them.

5 From presheaf to sheaf

The sheafification functor is the left adjoint of the inclusion functor from the sheafcategory to the presheaf one. It turns a pre-sheaf into a sheaf, and that with respect tothe topology of the category. We will explore in this section how.

This section is mainly inspired by [06]. For a more general construction of sheafifica-tion based on the left adjunction, topologies.. see [PT]. Indeed, in this section, we willnot prove that the described functor is left adjoint to the inclusion one. The aim is moreto show how we can construct sheaves effectively as colimits.

5.1 Topologies in a Lex category

In a Lex category, the existence of small limits, and especially pullbacks, allows us todescribe the topological axioms in a more handleable way.

Definition. A Grothendieck pretopology consists in, for each object X in C, a collectionof sieves such that

1. For all objects X of C, all morphisms f : Y 7→ X, and all covering familiesXα 7→ X, the family Xα ×X Y 7→ Y is a covering family of Y .

2. If Xα 7→ X is a covering family, and if ∀α, Xβα 7→ Xαβ is a covering family,then the family of composites Xβα 7→ Xα 7→ X is a covering family.

3. If f : Y 7→ X is an isomorphism, then f is a covering family of X.

In categories with enough fibred coproducts, Grothendieck topologies are equivalentto pretopologies. In a more subtle manner, as we can see SetC

Opas a completion

of C, there is an equivalence between pretopologies on the category of presheaf andGrothendieck topologies on the base category C.

5.2 Sheaves as coequalizers

This small section aims to show how sheaves can be seen as coequalizers. It serves as ashort introduction to the next big steps which consists in create them using colimits ofpresheaves.

The first step consists in establishing a bijection between sieve and subfunctor. Then,we could observe sieves as presheaves. Given a sieve s over c, one can define the subfunc-tor S( ) of hom( , c) by S(a) = f : a 7→ c, f ∈ s. The functor is defined canonically

21

for morphisms. Respectively, for a subfunctor S of hom(,c), one could define the sieves as the union ∪d∈CF (d). An other description of the sieve from the covering family isgiven in the following theorem.

Theorem 10. given s a covering family over U , the subfonctor s of hom( , U) can bedescribe as the following equalizer

∐i,j(Ui)×u (Uj) ⇒

∐i(Ui) 7→ s(Ui)

Where the Ui are seen in the presheaf category by the Yoneda inclusion, allowing thereforeus to speak about limits and colimits.

We have seen that a presheaf is a sheaf if it can be recovered globally by its localdefinitions. More concretely, there is an equivalence between his definition on a sieve bya matching family and his definition on the whole open set U. By considering U as apresheaf by the Yoneda inclusion, and the covering sieve s to be a functor, the fact thatP is a sheaf can be rephrased by:

hom(U,P ) = hom(s, P ).

And that is true for every covering sieve s over U . Using the Yoneda Lemma we get

P (U) = P (s)

if s is representable. Finally, combining this equality with the formula of the theoremabove, we get .

P (Ui) 7→∏

i P (Ui) ⇒∏

i,j P (Ui ×U Uj)

is a coequalizer.

5.3 The category of covering sieves

The sheafification is the act to transfrom a presheaf into a sheaf in a coherent way.Formally, it is a functor S : Presheaf(S) 7→ sheaf(S). We would like to exhibit thisfunctor. For that, our principal tool will be the next function H. For any U ∈ C,Υ = Ui

υi−→ Ui∈I covering sieve of U , and finally F presheaf, let H(Υ, F ) be :

H(Υ, F ) = (si)i∈I ∈∏

i∈I F (Ui), st si|Ui×UUj= sj|Ui×UUj

, ∀i, j ∈ I

H is, roughly, a sheashification of F relative to a given covering sieve. What we lookfor is to have a sheafification of F for all sieves. Therefore, we would like to take thecolimit of H along every sieve. In order for that to make sense, we have to define thecategory of sieve, and, as a start, a morphism between sieves.

Definition. A morphism between two covering sieves Υ = Uiυi−→ U, i ∈ I and Σ =

Vjσi−→ V, j ∈ J of the open set (object) U and V consists of

22

• a morphism f : U 7→ V

• a morphism α : I 7→ J

• a family of morphism fi : Ui 7→ Vα(i) such that the following diagram commutes

Ui

U

Vα(i)

V

υi

f

fi

σα(i)

This can be presented as a kind of fibration, as a restriction of the fibration fam(Mor(C))over Set, where the families have a common codomain, and the morphisms between ob-jects must behave coherently. For a category C with a Grothendieck topology, let us callS(C) the category of its covering sieves. It is the full subcategory of the sieve categoryon the covering sieves.

Given a morphism of sieves f : Υ 7→ Σ, does it induce a map H(Σ, F ) 7→ H(Υ, F ) ?In order to answer, that, we observe that the following diagram is commutative

Ui

Uj

Ui ×U Uj

Vα(i)

Vα(j)

Vα(i) ×V Vα(j)

Therefore, by mapping (si) in F (Vα(i)) to F (fi)(si), that belongs then to F (Ui),the family obtained satisfies indeed the sheaf condition relative to the covering Υ. Ittherefore create a morphism H(Σ, F ) 7→ H(Υ, F ).

5.4 The colimit is a presheaf

Instead of creating the desired limit globally, we will create it for each object U. Let usconsider JΥ the category of covering sieve of U , where the morphisms are restricted tothose whose underlying morphism in C is idU . We define the following map

F+(U) = colimJOpΥ

H( , F )

The first thing we want to ensure before going any further in that the map defined is,at least, a presheaf. Therefore, let f : V 7→ U a morphism of C. By the axioms of site,there is a functor JΥ 7→ JΣ that sends Ui 7→ U to Ui×U V 7→ V . By composing withthe first projection map, we have a morphism of covering Ui ×U V 7→ V 7→ Ui 7→ U,

23

and this is functorial. To sum up, there is a functor : JΥ 7→ (RJΣ 7→ JΥ) where RJΣis the restriction of Jσ to the coverings coming from the pullbacks. Therefore, from thiswe can create a map H(Ui 7→U , F ) 7→ H(Ui ×U V 7→ V, F ), and this one is functorial.By passing to colimit, we obtain a canonical map from F+(U) 7→ F+(V ).

5.5 Description as a filtered colimit

In this paragraph, we will focus on F+(U), and describe it in such a way we can easilyhandle it and proove its properties. The key remark to achieve that is that the categoryJU is filtered. This basically means that for every two elements, (respectively, twoparallel arrows) there is a common greater element. The colimits of filtered category inSets can be computed in a nice way, following the claim.

Claim. Suppose that M : I 7→ Sets is a filtered diagram. In this case

colimIM =∐

i∈I Mi/ ∼

we may describe the equivalence relation in the formula simply as follows

(mi) ∼ (mi′) if ∃i′′, φ : i 7→ i′′, φ′ : i′ 7→ i”, such that M(φ)mi =M(φ′)mi′

We will not prove it, but the interested reader will find a proof in [15]. It now remainsto prove that H : JU 7→ Set is filtered.

Definition. A diagram M : I 7→ C is filtered if

• for any two elements a and b of I there is an element c together with morphismsa 7→ c and b 7→ c.

• for any two parallel arrows f, g : a 7→ b there exists an object c together with a maph : b 7→ c such that M(h f) =M(h g).

We say that the category C is filtered if any diagram on it is filtered. Basically,this is the same definition as above, but without the M . There is also the equivalentdefinition that every diagram has a cocone. One can easily see the links between thisequivalence and the fact that a category has colimits if and only if it has coproducts andcoequalizers.

Theorem 11. H : JU 7→ Set as defined above is a filtered diagram.

Following the definition, the proof falls into two parts

1. Let us consider two covering Υ and Σ of U . For each Ui 7→ U and Vi 7→ U, letWij be the pullback Ui ×U Vj. Then Wij 7→ U forms a covering sieve. Indeed,for each Ui, the family Ui ×U Vjj is a covering of Ui, and as Ui 7→ U covers U ,we obtain that Wij covers U . There is straightforwardly a morphism of coveringΥ 7→Wij by considering Ui 7→ Ui ×U U .

24

2. Taking Υ and Σ as above, and f, g covering morphisms between them, we arelooking for a a third covering Ξ together with a morphism h to Υ such that H(h f, F ) = H(h g, F ). Actually, we are about to prove a stronger result, that is,H(f, F ) = H(g, F ) = H(idU , F ). The key is to observe that the morphisms f andg have the same underlying morphism in C, namely idU , as they belong in JU .Therefore, the following diagram commutes

Ui Vα(i)

Vβ(i) U

fi

vα(i)gi

vβ(i)

and therefore, we have the following factorization, where the map from Ui toVα(i) ×V Vβ(i) comes from the defintion of pullback and the diagram above.

Ui Vα(i) ×U Vβ(i)

Vα(i)

Vβ(i)

χ

pr1

pr2

fi

gi

Therefore, for s = (si) in H(Σ, F) we have

f∗(s)|i = f∗(sα(i)) = f∗i (sα(i)) = χ∗pr∗1(sα(i)) = χ∗pr∗2(sβ(i)) = g∗(s)|i.

which proves the claim.

So we know the how does F+(U) looks like. Let us remark that there is a canonicalmap θ : F (U) 7→ F+(U), following the fact that for each covering there is a canonicalmap F (U) 7→ H(Υ, F ).

5.6 Sheafification, final proofs

At the end of the section, it will finally be proven that F 7→ F++ is the researchedfunctor. The results will follow from two steps; first we look at the properties of F+,and then at those of F ++.

Claim. F+(U) 7→∏

i F+(Ui)/ ∼, where the ∼ corresponds to the sheaf condition,, is

injective. We say that F+ is separated.

25

The following lemma will be needed.

Lemma. For every (s) in F+(U), there exists a covering Υ = Ui 7→ U such that s|Ui

belongs to the image of θ : F (Ui) 7→ F+(Ui).

Proof. Let Ui 7→ U a coverage such that s comes from H(Ui 7→ U, F ). Then wewant to ensure that ∀i, if we compute si ∈ F+(Ui) by considering his image alongθUi

: F (Ui) 7→ F+(Ui), we would have the same result that by restraning F+(U) toF+(Ui). Let consider the map of covering Ui 7→ Ui 7→ Ui 7→ U. Then H(Ui 7→U, F ) 7→ H(Ui 7→ Ui, F ) and we can use this map to compute the pullback of s inF+(Ui). And by doing so, we exactly get θ(si).

With this material, we prove the claim.

Proof. Let (s), (s′) in F+(U) such that (si) = (s′i) in∏F+(Ui). We use the lemma to

obtain a coverage Vi 7→ U such that s|Vicomes from a ti in F (Vi), and a coverage

V ′i 7→ U such that s′|V ′

icomes from a t′i We will now consider a fourth coverage V ′

i ×Vj ×Uki,j,k of U . As, for a fixed k, this is a coverage of Uk, we must have s′i|V ′

i ×Vj×Uk=

sj|V ′

i ×Vj×UkAnd as this is a coverage of Vi, we have sj |V ′

i ×Vj×Uk= tj |V ′

i ×Vj×Uk. And

samely for t′ and s′. And therefore, by gluing everything together, we obtain that(s) = (s′).

It just remains the surjection property to obtain a sheaf.

Theorem 12. If F is separated, then F+ is a sheaf. Therefore, for all F ,F++ is asheaf.

Proof. Let Ui 7→ U be a covering, and Let (si) ∈∏F+(Ui) satisfying the sheaf

condition, that is for all i, j, (si)|Ui×uUj= (sj)|Ui×uUj

. We are looking for a s in F+(U),such that, for all i, s|Ui

= θ(si). For each (si), let just take for each Ui the coveringUik coming from the lemma, so such that (si)|Uik = θ(sik), sik ∈ F (Uik) unique as F+

separated. From the sheaf condition, we have that for all i, k, i′, k′, (sik)|Uik×UUi′k′=

(si′k′)|Uik×UUi′k′, as it maps to Ui ×U U ′

i , and that there is equality there. Hence, thefamily (sik) ∈ H(Uik 7→ U, F ) gives rises to an element (s) in F+(U).

6 Conclusion

This concludes the master thesis, by providing the way to construct sheaves from thebase category C, in the simple condition that C is left exact. If not, then the lightercondition that C has coequalizers allows us to know that fam(C) is the category ofprojectives. However, as it is not left exact, then the free exact completion is unclear.In order to better the result here obtained, one should investigate the structure of thefam(C)ex in this case.

On the other hand, an interesting topic could also to use some constructions like thatto define Set from category without never relying on Set theory. This was initiate byBenabou in [Ben85] but was left unfinished.

26

References

[06] Sites and sheaves. 2013.

[15] Categories. url: http://stacks.math.columbia.edu/download/categories.pdf.

[Ben85] J. Benabou. “Fibered Categories and the Foundations of Naive Category The-ory”. In: 50.1 (1985), pp. 10 –37. url: http://www.jstor.org/stable/2273784.

[Car95] A. Carboni. “Some free constructions in realizability and proof theory”. In:Journal of pure and applied Algebra 103 (1995), pp. 117 –148.

[CM80] A. CARBONI and R. CELIA MAGNO. “The free exact category on a leftexact one”. In: 1980 Mathematics subject classification 18 E 10. (1980).

[Joh] P.T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium, Vol-ume 1.

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