On sheaf cohomology and natural expansions
aa r X i v : . [ m a t h . C T ] S e p On sheaf cohomology and natural expansions ∗ Ana Luiza da Conceição Tenório, IME-USP, [email protected]
Hugo Luiz Mariano, IME-USP, [email protected]
September 17, 2020
Abstract
In this survey paper, we present Čech and sheaf cohomologies – themesthat were presented by Koszul in University of São Paulo ([42]) during hisvisit in the late 1950s – we present expansions for categories of generalizedsheaves (i.e, Grothendieck toposes), with examples of applications in othercohomology theories and other areas of mathematics, besides providingmotivations and historical notes. We conclude explaining the difficultiesin establishing a cohomology theory for elementary toposes, presentingalternative approaches by considering constructions over quantales, thatprovide structures similar to sheaves, and indicating researches relatedto logic: constructive (intuitionistic and linear) logic for toposes, sheavesover quantales, and homological algebra.
Sheaf Theory explicitly began with the work of J. Leray in 1945 [46]. Thenomenclature “sheaf" over a space X , in terms of closed subsets of a topologicalspace X , appears for the first time in 1946, also in a Leray work, according to[21]. He was interested in solving partial differential equations and build up astrong tool to pass local properties to global ones. Nowadays, the definition of asheaf over X is given by a “coherent family" of structures indexed on the latticeof open subsets of X or as étale maps (= local homeomorphisms) into X . Bothformulations emerged at latte 1940s and early 1950s in Cartan’s seminars andthey are intimately related by an equivalence of categories, in modern terms.H. Cartan’s ideas of “coherent family” for ideals [13] was introduced evenbefore Leray’s work on sheaves and is more related to the development of sheaftheory in complex analysis, where certain conditions - as convergence propertiesof power series - if it holds in a point, still holds in the neighborhood of thepoint. On the other hand, the presentation of sheaves as étales spaces (due toLazard) consolidated sheaf theory in algebraic topology, since the sections ofétale maps motivate the construction of the global section functor which give ∗ Ana Luiza Tenorio: Supported by CAPES Grant Number 88882.377949/2019-01 topos in [29], is nowadays known as a
Grothendiecktopos (a category of sheaves over a site , i.e. a pair ( C , J ) , with C a small cate-gory and J a Grothendieck topology); the general notion of topos, or elementarytopos, is due to work of W. Lawvere and M. Tierney in the early 1970s. Theyrealized that a Grothendieck topos have categorical properties which make itclose to the category Set of all sets and functions. For example, sheaves admitexponential objects that are analogs of the set A B of all function from B to A ,and there is an object of truth-values (subobject classifier) that, in the category Set , is the set { true, f alse } . Thus, by only assuming that a category has asubobject classifier and satisfies some finite conditions (as cartesian closed toguarantee the existence of an exponential object) they reached the definition ofa topos, such that any Grothendieck topos is a topos but the converse does not2old.Soon the study of topos theory developed many fronts. For example, thedescription of an internal language (Mitchell–Bénabou language) and it Kripke-Joyal semantic, variations of Cohen’s forcing techniques using toposes, and theestablishment of higher-order logic in terms of categories.In this survey, we present sheaf cohomology and some possible extensions forit, focusing on the notorious and well established Grothendieck topos cohomol-ogy. Section 2 is devoted to preliminaries, where we remember that homologicalalgebra works, mainly, with abelian categories (we will define it, but for now thereader can replace “abelian categories” by abelian groups or R -modules, whatyou prefer) and since most parts of literature - with Grothendieck’s Tôhokupaper [22] the classical exception - toward homological algebra treat of specificabelian categories as modules over rings, we provide preliminaries about abeliancategories and cohomology in this more general setting. Besides that, we ex-plain how to extract abelian categories from a not necessarily abelian category C , requesting only the existence of binary products and terminal objects, whichis the case for sheaves over topological spaces and Grothendieck topos. In thismanner, C is what we could call a “Set-like” category, i.e, a category that hasthe basics properties to not lose constructions we usually made in Set , play thesame role of
Set but could be more general than just
Set . The essential part is:since we use sets to define another structure (topological spaces, groups, rings,manifolds) we can use a
Set - like category C to construct other categories, inthe particular case we will see, categories with abelian group structure.In Section 3, we introduce the basics of sheaf theory, sheaf cohomology,and Čech cohomology, following the work of H. Cartan, J-L. Koszul and R.Godement.In Section 4, we track the ideas of A. Grothendieck and its collaborators butwithout mention a specific site. Instead, we define Grothendieck toposes, passby elementary topos to furnish the internal logical tool of a topos, and apply itto simplify arguments in Grothendieck topos cohomology.We do not show new results or original proofs but pointed main ideas andchoose constructions that allow awareness of how Grothendieck topos coho-mology extends sheaf cohomology. Some demonstrations are omitted becausethey require extremely technical machinery (such as spectral sequences) and sowould be out of our purpose of making this text a gentle introduction to toposcohomology, pointing out how it subsumes sheaf cohomology.In Section 5, we clarify that the current topos cohomology has issues - thedefinition of flabby sheaves, the non-availability of enough injectives in the cat-egory of abelian groups over toposes that are not Grothendieck - and a strongdependence of classic logic that difficult the “internalization" of these notions tothe intrinsic intuitionistic (constructive) character of the toposes. We describesome attempts to address these problems, including extensions of topos coho-mology over “sheaf-like" categories that are internally governed by even moregeneral logics: the linear logics. 3 Preliminaries
Essentially, what is done in a cohomology theory is associate a sequence of ob-jects to a space. The objects can be abelian groups, the space can be any topo-logical space, and we can associate one with the other using chain complexes.However, the reader a bit more familiar with Homological Algebra knows that,instead of abelian groups, we can work with modules over commutative rings,vector bundles over topological spaces, or even abelian sheaves. This is duethe fact that all theses objects form their respective categories, and they areexamples of abelian categories. Summarizing, when we work with cohomologywe are working, to the greatest extent, with abelian categories.In this section we present the basics of abelian categories, state the mainresults of homological algebra in this general setting, and define the notion ofabelian group object - which later will provide a technique to extract abeliancategories from toposes.
We will assume that the reader is familiar with the basic notions of categorytheory: category, functor, natural transformation, product, and equivalence ofcategories.
Let s and t objects in a category C . If for all a object in C there is an uniquemorphism s → a then s is an initial object ; if there is an unique morphism a → t , then t is a terminal object . Thus, an object is initial if we always havearrows leaving the object, and is final if we always have arrows arriving at theobject. The uniqueness property satisfied by a initial (respectively, terminal)object ensures that it is unique up to (unique) isomorphism. In case an object issimultaneously initial and final, it is called a zero object. After the preliminaries,we will change notation: initial and terminal objects will be denoted by and , respectively.In categories with some zero object, we also have a notion of null morphism:a morphism f : A → B that factors through any zero object. The null morphismfrom A to B is unique and will be denote by A,B or just by .Now we can define an important concept to construct cohomology in generalabelian categories.Let f : A → B a morphims in a category C with zero objetct. A morphism k : K → A is said kernel of f if f ◦ k = 0 and, for all morphism h such that f ◦ h = 0 , there is an unique h ′ = k ◦ h ′ . Or simply, if k is the equalizer of f and the null morphism . Diagramatically, K A BK ′ k K,B f A,B h ′ h K,B f is defined dually. Through the text we will denote themorphism k kernel of f by ker ( f ) , and the object K associated to it by Ker ( f ) .Analogously, for the cokernel.We were not imposing additional conditions over our category C early, butaiming at a smooth introduction of abelian categories further we will start todo so.If the set of morphisms Hom ( A, B ) of a category C has structure of anabelian group, and the composition of morphisms is bilinear, then C is an Ab -category (or preaddictive ). Here Ab is the category of abelian groups and weadopted the nomenclature of Ab -category to familiarize the reader with the ideaof an enriched category. In this case, C is enriched over Ab so Hom ( A, B ) ismore than a set, it is an object in Ab , for every A, B objects in C .Examples of Ab -categories are the category Ab , of all abelian groups andits homomorphisms, and R - M od , the category of all left modules over a ring R and homomorphisms. More involved examples came from categories usefulin homological algebra whose objects are complexes of abelian groups, com-plexes of modules over a ring and filtered modules over a ring. Moreover, everytriangulated category is an Ab -category.A biproduct is a quintuple ( P, p A , p B , s A , s B ) such that: p A : P → A , p B : P → B , s A : A → P and s B : B → P satisfying theequations: p A ◦ s A = id A , p B ◦ s B = id B , p A ◦ s B = 0 , p B ◦ s A = 0 and s A ◦ p A + s B ◦ p B = id P We observe that in Ab -categories the existence of biproduct is equivalent tothe existence of product and, also, the existence of coproduct.When a category is an Ab -category and has a zero object, then it is called an additive category. So, with the abelian groups structure in the set of morphisms,we obtain that the zero of Hom ( A, B ) coincides with the null morphism A,B ,see a demonstration of this in [10, Chap. 1.2]. This is an interesting propertysince abelian categories have to indirect handle null morphisms, through kerneland cokernel.An Ab -category C that has null object and biproducts is an abelian cate-gory if it satisfies also the following conditions:AB1 Every morphism has kernel and cokernel.AB2 Every monomorphism is a kernel and every epimorphism is a cokernel.Except for filtered modules over a ring, and, in general, triangulated cat-egories, the examples of Ab -categories are also abelian categories. For us, aimportant example case is the category of abelian sheaves.Given an abelian category C we can add ABn axioms. In this survey, theimportant one is AB .AB3 Given a family { A i } i ∈ I of objects in C , then exists L i ∈ I A i , the direct sumof A i ’s. 5B4 The AB axiom holds and direct sum of family of monomorphisms is alsoa monomorphism.AB5 The AB axiom holds and if { A i } i ∈ I is a direct family of subobjectsof an object A in C , and B any subobject of A , then ( P i ∈ I A i ) ∩ B = P i ∈ I ( A i ∩ B ) , where sum denotes sup of A i , and the intersection denotesinf of subobjects.The AB will be central because of the following Grothendieck’s Theorem:If an abelian category satisfies AB and has generator then it has enoughinjectives [22, Theorem 1.10.1]This Theorem will be used to show that the abelian categories extractedfrom Grothendieck toposes are good enough to develop cohomology.Lets return to our preliminaries.One of the main difficulties in working with an arbitrary abelian categoryis that it is abstract, in the sense we do not know who the objects are, andwhat kind of structure they have. For example, when we consider a concreteabelian category as Ab , we know that the objects are abelian groups, that arejust sets endowed with some extra structure, but if we have to deal with anyabelian category this information is not (directly) available. Accordingly to thisdelicate scenario there are, at least, two techniques to enable means of provingresults regarding a general abelian categories.The simplest technique is apply the Freyd-Mitchell embedding Theorem [18]that guarantee we can fully embed small abelian categories into some category R -Mod. Roughly speaking, it is enough prove things for all small full subcate-gories of the categories of modules over a ring.However, there is non small abelian categories so a stronger but more com-plex technique is construct the notion of pseudoelement , as nominated in [10](or generalized element , as in [48])Once mentioned that, we can argue that the famous snake lemma holds inany abelian category. We will skip even state it here, the important part iswarn that abelian categories were designed to this lemma arise. When Cartan’sand Eilenberg’s book “Homological Algebra" appeared in 1956 [16], developedfor categories of modules over rings it was also known that the theory could bereplicated for other structures, for instance, abelian sheaves. This motivatedA. Grothendieck - and not only him - to define a general concept, and estab-lish Homological Algebra for it in [22]. Nowadays, we have even more generalcategories, for example, the homological categories [12], where the snake lemmaholds (observe that the snake lemma holds for non abelian groups, even so it isnot an additive category [10, Chap 1.2]). Although, abelian categories still arethe most studied when working with cohomology.6 .2 Homological Algebra The reader used to Homological Algebra techniques for a particular abeliancategory (as presented in [64], for example) can skip this subsection. However,if there is a curiosity to see how to construct cohomology in the abstract settingof any abelian category, we exhibit here the modifications that have to be doneto define the basics concepts. We state without proof the results that will beneeded in 3.1 and 4.3.For any abelian category C we define a complex cochain by taking sequences { C q } q ∈ Z of objects in C , and endow it with coboundary morphisms d qC : C q → C q +1 such that d q +1 ◦ d q = 0 , for all q ∈ Z . A complex cochain is a denoted by C • , and we establish morphisms of complexes h • : C • → D • with a colection ofmorphism h q : C q → D q such that h q +1 ◦ d qC = d q +1 D ◦ h q , for all q ∈ Z . Observethat, with coordinatewise composition and identities, this forms a category,called category of complex cochain of C , Ch ( C ) , and it is as abelian categorywhenever C is abelian.Since d q ◦ d q − = 0 , we have that ⊆ Im ( d q − ) ⊆ Ker ( d q ) ⊆ C q . Thismakes possible to define the q -th cohomology object of C • by H q ( C • ) = Ker ( d q ) /Im ( d q − ) = Coker ( Im ( d q − ) → Ker ( d q )) . Do not forget we are working with arbitrary abelian categories so this Ker-nel is the K object in the domain of the morphism ker ( d q ) , and Im ( d q − ) = Ker ( Coker ( d q − )) .Let f • : C • → D • be a complex morphism. Since we are working witharbitrary abelian categories, define a induced morphism H q ( f ) : H q ( C • ) → H q ( D • ) , q ∈ Z is more complicated than usual, but lets describe the idea.The morphism f q : C q → D q restricts to f qK : Ker ( d qC ) → Ker ( d qD ) and to f qI : Im ( d q − C ) → Im ( d q − D ) : this follows directly from diagram chases, by theuniversal properties of kernels and cokenerls. The coboundary morphism alsoprovide a morphism α q : Im ( d q − C ) → Ker ( d qC ) where Coker ( α qC ) = H q ( C • ) .By the universal property of cokernel there is a unique morphism Coker ( α qC ) → Ker ( d qD ) as follows: Im ( d q − C ) Ker ( d qC ) Coker α qC Im ( d q − D ) Ker ( d qD ) Coker α qDf qI α qC f qK α qD Completing the bottom part of this diagram with the cokernel of α qD weobtain a unique morphism H q ( C • ) ∼ = Coker ( α qC ) → Coker ( α qD ) ∼ = H q ( D • ) .This induced morphism is H q ( f ) : H q ( C • ) → H q ( D • ) . Clearly, the mapping f H q ( f ) determines a (covariant) functor H q : Ch ( C ) → C , q ∈ Z .Given complex morphisms f • , g • : C • → D • , then f • and g • are called homotopic if for each q ≥ there is h q : C q → D q − such that f q − g q = d q − D ◦ h q + h q +1 ◦ d qC . Chain homotopies are important because they relate7wo different morphism through their induced maps in the cohomology objects.More precisely, Proposition 2.1 If f is homotopic to g , then H q ( f • ) = H q ( g • ) , q ∈ Z . Now, remember that an object I in an abelian category is injective whenever:for all morphism α : A → I and all monomorphism m : A → B , there isat least one morphism β : B → I such that α = β ◦ m (equivalently, I isinjective whenever the functor Hom ( − , I ) is exact). A resolution of an object A , A → I • , is an exact sequence → A → I → I → ... ; this resolution is a injective resolution if I i in injective for each i ≥ . If an abelian category hasenough injectives , then any of its objects A admits some injective resolution.The concept of enough injectives is central in homological algebra becauseof the following theorem. Theorem 2.2
Let C and C ′ abelian categories, with C having enough injectives,and F : C → C ′ . Then:(i) There are addictive functors R q F : C → C ′ for all q ≥ ;(ii) F ∼ = R F is an isomorphism;(iii) For each exact sequence E : 0 → A → A → A → and each q ≥ ,there is exact morphism δ qE : R q F A → R q +1 F A that makes the followingsequence exact · · · → R q F A → R q F A → R q F A δ qE −−→ R q +1 F A → . . . (iv) The morphisms δ qE are natural in E . These R q F : C → C ′ functors are unique up to natural isomorphisms, theyare called q -th right derived functor of F and R q F ( A ) ∼ = H q F ( I • ) , where I • isa resolution of A .Let F : C → C ′ as in the above theorem. An object A of C is F -acyclic (oracyclic for F ) if R q F ( A ) = 0 for all q > .Notice that this definition can describe an way to measure the failure of asequence to be exact, so we could define derived functors using acyclic objectsinstead of injective ones. If C is a category with binary products and terminal object , we can define the group object in C as a object G in C equipped with morphisms e : 1 G i : G G m : G × G G in C , such the following diagrams commute8 × G × G G × G × G G × G G × G × G G G id G × mm × id G m e × id G ∼ = m id G × e ∼ = m G G × G G × G G ! △ i × id G me G G × G G × G G ! △ id G × i me The morphism △ = ( id G , id G ) : G → G × G is the diagonal morphism. Notethat this diagrams are expressing the axioms of group. If we want to add anabelian condition and form an abelian group object, then we must include G × G G × GG τm m commutative, where τ = ( π , π ) : G × G → G × G is the twist morphism.So an abelian group object is a quadruple ( G, e, i, m ) , where the above fivediagrams commute, and the category Ab ( C ) of abelian groups object in C is thecategory where the objects are abelian groups objects and the morphisms aremorphisms in C that commute with the morphisms e, i , and m. Two notable examples of group objects are topological groups, when C isthe category of topological spaces, and Lie groups, when C is the category ofsmooth manifolds.The case we will be working through this text is C as a topos. Interested in fixed points results applied to the realm of partial differentialequations, Jean Leray published in 1945, while a prisoner in the 2nd world war,the paper [46] that would originate sheaf theory. He published a more refinedand clear paper about sheaf theory and spectral sequences in 1950 [47], withthe original ideas preserved. Meanwhile, Henri Cartan starts the Séminaire atthe École Normale Supérieure, and reformulates sheaf theory. Also in 1950, inthe third year of this seminar, sheaves appear as what is now know as “étaléspaces". Results using sheaf methods were showing up and a lot of new notionswere arising but the terminology was not established. It was Roger Godementwho achieve a standard language for the theory (for example, presheaves arefunctors, sheaves are a special kind of presheaves; the notion of sheaf in Cartan’sseminars is denominated an étalé space) with his book published in 1958 [20].9ess about the history and more about the philosophy of sheaf theory: sincethe beginning, there was some notion that would allow extending local sectionsto global sections. In the work of Godement, the flabby sheaves are responsibleto play this role, while Grothendieck worked more with injective sheaves. Theidea is that the cohomology groups obtained from resolutions of this specialkind of sheaves are trivial, so we do not have obstructions from local to global.The power of sheaf theory is to provide machinery to solve global problems fromlocal, so simpler, ones. This is especially interesting for algebraic geometry andcomplex analysis.Let X be a topological space. We denote by O ( X ) the category associatedto the poset of all open sets of X . A presheaf of sets is a (covariant) functor F : O ( X ) op → Set , and a morphism of presheaves is a natural transformation.Given inclusions U ⊆ V , we use s | VU (or just s | U ) to denote the “restriction map"from F ( V ) to F ( U ) .If U ⊆ X is open and U = S i ∈ I U i is an open cover, a presheaf F is a sheaf (of sets) when we have the following diagram F ( U ) Q i ∈ I F ( U i ) Q ( i,j ) ∈ I × I F ( U i ∩ U j ) e pq is an equalizer in the category Set , where:1. e ( t ) = { t | Ui | i ∈ I } , t ∈ F ( U ) p (( t k ) k ∈ I ) = ( t i | Ui ∩ Uj ) ( i,j ) ∈ I × I q (( t k ) k ∈ I ) = ( t j | Ui ∩ Uj ) ( i,j ) ∈ I × I , ( t i ) i ∈ I ∈ Q i ∈ I F ( U i ) This definition is useful to understand categorical properties and providea simple way to visualize it generalization when we substitute O ( X ) by anarbitrary category. However, there is an equivalent and more concrete form todescribe a sheaf. Instead of presenting an equalizer diagram, we say that thepreasheaf F satisfies two conditions:1. (Gluing) Let s i ∈ F ( U i ) , i ∈ I be a compatible family , i.e. it satisfies s i | Ui ∩ Uj = s j | Ui ∩ Uj for all i, j ∈ I . Then there exists some s ∈ F ( U ) a gluing of this compatible family, i.e. s is such that s | Ui = s i , i ∈ I .2. (Separability) Let s, s ′ ∈ F ( U ) such that s | Ui = s ′| Ui , i ∈ I . Then s = s ′ .If F : O ( X ) op → Set is a presheaf, then F x := lim −→ U ∈ V iz ( x ) F ( U ) is called the stalk of F at the point x ∈ X , where V iz ( x ) = { U ∈ O ( X ) : x ∈ U } is the posetof open neighborhoods of x . A presheaf F satisfies the separabilty conditionabove if and only if the canonical morphisms F ( U ) → Q x ∈ U F x , U ∈ O ( X ) , aremonomorphisms. 10 morphism of sheaves is a morphism of presheaves, i.e., just a naturaltransformation between functors, and is clear that this define a category, denotedby Sh ( X ) .Note that in the definition of sheaf we could replace Set by any categorywith all small products, for example, the category of abelian groups Ab , and inthis case we change the nomenclature to abelian sheaves . We will return to thisin subsection 3.1.The idea is that sheaves capture global information from the gluing of localproperties. For example, given a open subspace U of a topological space X , andan open cover U = S i ∈ I U i . There is a functor, C R , that takes opens U in X andsends to the set C R ( U ) = { f : U → R | f is a continuous function } . Since therestriction of a continuous function to a subset of its domain is still a continuousfunction, C R is a presheaf. Since, f i ( x ) = f j ( x ) , ∀ x ∈ U i ∩ U j , there is a uniquefunction f such that f | Ui = f i . Besides that, the continuity of the f i ’s impliesthe continuity of the gluing f , so f ∈ C R ( U ) . Analogously, the presheaves ofdifferential, smooth, or analytic functions are sheaves [63].This example may remind the reader of germs and stalks over points in atopological space w.r.t étale bundles (or local homeomorphisms) and this is notonly a coincidence: for any continuous function p : E → X it can be defined Γ p ( U ) = { s : U → E | s is continuous and p ( s ( x )) = x, ∀ x ∈ U } and is possibleto prove that Γ p is a sheaf, called sheaf of sections of the continuous function p . Moreover, if F is sheaf over a topological space X , taking E F := ` x ∈ X F x thedisjoint union of stalks of F for each point x in X , and constructing an ade-quate local homeomorphism p F : E F → X , this leads to a natural isomorphismbetween F and Γ( p F ) . So every sheaf over X is (naturally isomorphic to) thesheaf of sections of a local homeomorphism over X . Sheaf Theory inherits thenomenclature of constructions involving étale bundles because the two notionsare strongly related through the category equivalence between the category ofétale bundles over X and the category of sheaves over X , for each topologicalspace X [49, Chap. II].The spatial-functorial identification process described above is useful to pro-vide the “best sheaf given presheaf": any presheaf F : O ( X ) op → Set , canbe “sheafificated" into a ( F ) := Γ( p F ) : O ( X ) op → Set above F , i.e. a ( F ) isa sheaf over X and there is a natural transformation η F : F → a ( F ) that isinitial among the natural transformations σ : F → S , where S is a sheaf over X ; moreover, the stalk of a ( F ) at a point x ∈ X is isomorphic to the stalk F x . For instance, given a set A , the “constant presheaf" with value A is thecontravariant functor F A ( U ֒ → V ) = ( A id A ← A ) ; its stalk at a point x ∈ X isisomorphic to A and its sheafification, a ( F A ) : O ( X ) op → Set , is isomorphic tothe sheaf of continuous function with value A (viewed as a discrete topologicalspace): C A ( U ) = { f : U → A | f is a continuous function } , U ∈ O ( X ) .Another relevant example of sheaf came from Commutative Algebra and iscentral for the development of modern Algebraic Geometry: for each commu-tative unitary ring R , there is a canonical sheaf, O R , of rings defined over its11rime spectrum space , Spec ( R ) , this sheaf is determined on a (canonical) basisof the (spectral) topology of Spec ( R ) just taking adequate localizations of thering R ; the stalk of this sheaf at a proper prime ideal p ∈ Spec ( R ) is isomorphicthe local ring R p = R [ R \ p ] − . The pair ( Spec ( R ) , O R ) is called the affinescheme associated to R ; will return to this example latter, we in subsection 3.3. In this section we present the subject “Sheaf Cohomology" in the usual way,omitting proofs that can be easily found in the literature, [25, 20], but providingintuition about the associated ideas. Our aim here is list some results of thistheory that will reappear in the next section with the appropriate modifications.Thus we hope it is clear that the (Grothendieck) Topos Cohomology exhibitedis in fact an extension of Sheaf Cohomology.For the comfort of the reader, we start explaining why we can do sheafcohomology in Ab ( Sh ( X )) , i.e., how abelian sheaves are equivalent to abeliangroups objects of Sh ( X ) Note that abelian presheaves O ( X ) op → Ab , form the category of functors Ab O ( X ) op . Then, for every functor F object in Ab O ( X ) op , we have that F ( U ) isan abelian group for every U ∈ O ( X ) . So, for each U ∈ O ( X ) , there are m U :( F × F )( U ) ∼ = F ( U ) × F ( U ) → F ( U ) , i U : F ( U ) → F ( U ) , and e U : 1 → F ( U ) such that they determine natural transformations and the diagrammatic rulesof abelian group object holds, i.e., F is an abelian group object of Set O ( X ) op .On the other hand, if G ∈ Ab ( Set O ( X ) op ) , then G ∈ Set O ( X ) op and we have m, i, and e as in the definition of abelian group object. For every U ∈ O ( X ) we consider m U , i U , and e U such that the diagrammatic rules still hold, then, G ( U ) is an abelian group, i.e., G is a functor of O ( X ) op to Ab . These describethe equivalence of categories Ab ( Set O ( X ) op ) ≃ Ab O ( X ) op . Now, observe that Ab ( Set ) ≃ Ab and consider E : Ab ( Set ) → Set theforgetful functor ( E “forgets" the group operations). An abelian sheaf F : O ( X ) op → Ab where the composition O ( X ) op → Ab → Sets is a sheaf inthe sense of 3. Denote the category of abelian sheaves by Sh Ab ( X ) . Since wehave inclusions Sh ( X ) → Set O ( X ) op and Sh Ab ( X ) → Ab O ( X ) op , the equivalence Ab ( Set O ( X ) op ) ≃ Ab O ( X ) op induces an equivalence Ab ( Sh ( X )) ≃ Sh Ab ( X ) ,since the subcategories of sheaves, over Set and over Ab , is closed under products(in fact, it is closed under all small limits).Therefore, to apply cohomological techniques in Ab ( Sh ( X )) is equivalent toapply it in Sh Ab ( X ) . Many classical books of Sheaf Cohomology prove that Sh Ab ( X ) is an abelian category (see, for instance, [37, Theorem 2.5]). We,alternatively, can show that Ab ( E ) is an abelian category for any topos E , inparticular, Ab ( Sh ( X )) is abelian. We will comment more on this in the subsec-tion 4.3.Besides the fact that Sh Ab ( X ) is an abelian category, to define the coho-mology group of sheaves, we will use right derived functors. Then we need too Spec ( R ) = { p ⊆ R : p is a proper prime ideal of R } , and it is endowed with the so called"Zariski Topology". Sh Ab ( X ) has enough injectives [37, Theorem 3.1].For every sheaf F in Sh Ab ( X ) and U open set of X , we have the abeliangroup of sections of F over U defined by Γ( U, F ) = F ( U ) . Sections over X arecalled global sections , and Γ( X, − ) : Sh Ab ( X ) → Ab is a left exact functor thatsend a abelian sheaf to its global section abelian group, know as global sectionfunctor .Then the q -group cohomology group of X with coefficients in F is, by def-inition, the q -th right derived functor of Γ( X, F ) . In other words, given aninjective resolution F → I • , we have H q ( X, F ) = R q Γ( X, I • ) .A special type of sheaves are the flabby sheaves. As we will see, they areimportant because, like injective objects, they allow the construction of acyclicresolutions. By definition, if the restriction maps s U : F ( X ) → F ( U ) is ontofor every U ⊆ X open, the sheaf F is flabby. Or, equivalently, if F ( V ) → F ( U ) is onto for any pair U ⊆ V of open sets in X. Proposition 3.1
Every injective sheaf is flabby.Proof.
To establish this result, we will need an auxiliary construction.Consider a functor x ∗ : Set → Sh ( X ) , such that ( x ∗ H )( U ) = ( H, x ∈ U {∗} , x / ∈ U where H is set, U an open set in X , and {*} unitary set. This is known asthe skyscraper sheaf . In the abelian sheaf version, we have x ∗ : Ab → Sh Ab ( X ) ,H ( x ∗ H )( U ) , with the difference H is now an abelian group and x ∗ H is afunctor that sends open sets of X to H or in the trivial group.For each x ∈ X , let D x be an injective abelian group. We define a injectivesheaf D := Q x ∈ X x ∗ D x . It is not difficult to see that D ( X ) → D ( U ) is surjective,i.e, D is flabby.Now suppose F is an injective sheaf. We will show that F is flabby. Since F is injective, for each x ∈ X , the stalk F x is an injective abelian group. Considerthe family of injective abelian groups D ( F ) x := F x , x ∈ X . Then D ( F ) := Q x ∈ X x ∗ D ( F ) x is an injective and flabby sheaf and, since F ( U ) → Q x ∈ U F x is amonomorphism, U ∈ O ( X ) , there is a mono i : F → D ( F ) . Since F is aninjective sheaf, we can select a morphism f : D ( F ) → F such that f ◦ i = id F .Since the all components of identity morphism are surjective homomorphism,the same holds for the components of f . Besides that, the following diagramcommutes, by naturality of f : D ( X ) D ( U ) F ( X ) F ( U ) s U,D f ( X ) f ( U ) s U,F It preserves all small limits.
13e already know f ( U ) and s U,D are surjectives, so f ( U ) ◦ s U,D is surjective.By commutative of the diagram, s U,F ◦ f ( X ) is surjective and so also is s U,F .This holds for every open set U of X , then F is a flabby sheaf.Now we show that flabby sheaves can build acyclic resolutions: Proposition 3.2 If F is an flabby sheaf, then H q ( X, F ) = 0 , for all q > .That means F is Γ( X, − ) -acyclic.Proof. Since F is flabby, we can construct → F f −→ G g −→ Q → exactsequence, where G is injective because Sh Ab ( X ) has sufficient injectives. Bythe proposition above, G is flabby.Using the left exactness of the global section functor we immediately obtainthe exact sequence → Γ( X, F ) Γ f −−→ Γ( X, G ) Γ g −→ Γ( X, Q ) .The flabby condition of F implies more: → Γ( X, F ) Γ f −−→ Γ( X, G ) Γ g −→ Γ( X, Q ) → is exact. This is not straightforward and use Zorn’s Lemma to beproved [37, Theorem 3.5].By Theorem 2.1, the derived functors induce a long exact sequence. We willanalyze the following part of the sequence: Γ( X, G ) Γ g −−→ Γ( X, Q ) δ −→ H ( X, F ) f −→ H ( X, G ) Where g = g . Note H ( X, G ) = 0 , since G is injective. Since the sequenceabove is exact sequence, H ( X, F ) ∼ = Γ( X, Q ) /Ker ( g ) , by the IsomorphismTheorem. But g = g is surjective morphism so Ker ( δ ) = Im ( g ) ∼ = Γ( X, Q ) . Then, H ( X, F ) = 0 .To conclude the result uses an induction argument in q and the fact that ifthe first two objects in a short exact sequence are flabby, the third one is alsoflabby. Remark:
All proofs we know of this proposition require Zorn Lemma, sountil now, a constructive proof seems to be not available.With the fact that every sheaf admits a flabby resolution, via "Godementresolution", the Proposition above implies we can define cohomology groups withcoefficient in F using flabby sheaves instead of injective ones. The reason whythis is possible is that what we need to construct cohomology is a procedure thatmeasures the “failure of it right exactness" and the proposition above guaranteessuch procedure for flabby sheaves [20]. The Čech nerve construction came before of the development of sheaf theory,it is an algorithmic form of associate a (simplicial) complex to a topologicalspace [3]. Godement improved in his book the brief discussion about ČechCohomology made in Cartan’s seminars, and it is a fundamental reference onthe subject until today. Additionally, we recommend Kozsul’s note classes [42].14or references in English, there are algebraic geometry books as [35]. Here weintroduce Čech Cohomology as a technique to calculate Sheaf Cohomology bytaking open covers of a fixed topological space, construct a cochain complexfrom it, and so compute the cohomology groups using kernel and image. Ouraim is to use this section to compare it with Čech Cohomology for GrothendieckToposes.Fix F in Sh ( X ) and consider U = ( U i ) i ∈ I an open cover of X ( S i ∈ I U i = X ),where I is an well-ordered set of indices. For each q ∈ N , denote U i ,...,i q = U i ∩ ... ∩ U i q for i , ..., i q ∈ I . The Čech cochain complex is C q ( U , F ) = Q i <...
Let F sheaf in Sh Ab ( X ) , and U = ( U i ) i ∈ I an well-orderedcovering of X . There is a canonical map k q U : ˇH q ( U , F ) → H q ( X, F ) naturaland functorial in F for each q ∈ N . Now we will briefly examine the behavior of the Čech cohomology groupsunder the dynamic of refinement of coverings. We will return to this point latter,in the subsection 4.4.Let V = ( V j ) j ∈ J be another (well-ordered) covering of X . Suppose that U is a refinement of V , i.e., for each i ∈ I , there is j ∈ J such that U i ⊆ V j .Select any function c : I → J such that U i ⊆ V c ( i ) , i ∈ I ; then there is ainduced morphism of cochain complexes m c : C • ( V , F ) → C • ( U , F ) and, thusa corresponding morphism of Čech cohomology groups w.r.t. the coverings U and V , ˇ m c : ˇH • ( V , F ) → ˇH • ( U , F ) . Moreover, if d : I → J is anotherselection function w.r.t. the refinement of V by U , then the induced morphismsof complexes m c , m d are homotopic, thus by Proposition 2.1, there is a uniqueinduced morphism of cohomology groups ˇ m U , V : ˇH • ( V , F ) → ˇH • ( U , F ) .Note that the class Ref ( X ) of all well-ordered coverings of X is partiallyordered under the refinement relation, this is in fact a directed ordering relation.The construction above is functorial in the following sense: ˇ m U , V = id :ˇH • ( U , F ) → ˇH • ( U , F ) and, if W = ( W k ) k ∈ K is an well-ordered covering of X such that V is a refinement of W , then ˇ m U , W = ˇ m U , V ◦ ˇ m V , W : ˇH • ( W , F ) → ˇH • ( U , F ) . 15he (absolute) Čech cohomology group is, by definition, the directed (co)limit : ˇH • ( X, F ) := lim −→ U∈ Ref ( X ) ˇH • ( U , F ) . The main result concerning Čech cohomology is the following:
Theorem 3.4
The canonical maps k q U : ˇH q ( U , F ) → H q ( X, F ) , q ∈ N , accord-ing notation in Proposition 3.3, are compatible under coverings refinement. Theinduced morphism on colimit k q : ˇH q ( X, F ) → H q ( X, F ) , q ∈ N , is an isomorphism if q ≤ and a monomorphism if q = 2 . Most part of mathematician will not be interested in abstract sheaf theory alone,but in its applications for specific sheaves. For example, if ( X, O X ) is a ringedspace, i.e., X is a topological space and O X is a ring-valued sheaf, we can definea coherent sheaf F on ( X, O X ) that will look like a vector bundle with theadvantage of forming an abelian category. Thus, we can study coherent sheafcohomology.In this context, we have an analog of Poincaré Duality of algebraic topology,and the Serre Duality, that relates cohomology groups at level n - q with Ext groups at level q , where n is the dimension of the particular scheme we arestudying, by [35, Theorem III 7.6]). Coherent sheaf cohomology also provides acharacterization for Euler Characteristic by an alternated sum of the dimensionof the cohomology groups of a scheme with coefficient in a coherent sheaf.The affine (locally) ringed space ( Spec ( R ) , O R ) where Zariski Topology isused to construct O R and give topological structure to the spectrum of a com-mutative ring, Spec ( R ) , form a quasi-coherent sheaf and eventually leads to thedefinition of schemes, essential in modern algebraic geometry.Additionally, we can apply Čech cohomology in de Rham cohomology asfollows: Given a topological space X , and a set A , the constant presheaf withvalues in A , F A , is such F A ( U ) = A and F ( U ֒ → V ) = id A , where U, V ∈ O ( X ) ;by transforming this presheaf into a sheaf through a standard “sheafification”process, we obtain the “constant sheaf" with values in A . In particular, theset A can be R , the set of real numbers, and the topological space, a compactmanifold M of dimension m and class at least C m +1 . So, there is an isomor-phism H qdR ( M ) ∼ = ˇ H q ( M, R ) , for all q ≤ m , where H qdR denotes the de Rhamcohomology groups [57, Appendix].More recently, sheaf and Čech cohomologies had been used in quantum me-chanics because of the general idea of measure the obstruction between local andglobal properties. For example, in [1], the Čech cohomology groups are defined This (co)limit has to be taken with some set-theoretical care, we will not detail this pointhere. contextuality .In the next section we will generalize the categories of sheaves over sometopological space, defining the notion of Grothendieck topos, and will mentionspecific Grothendieck topos that are used in other areas of Mathematics.
Cohomology groups often provide good invariants to classify objects: if twoRiemann surfaces (with some additional conditions) agree in it level of the co-homology groups, then they are the same from a topological point of view. Inthe 1950s, this kind of problem was well understood for algebraic curves overthe field of complex numbers but not much was know for algebraic curves overother fields. In 1954, Jean-Pierre Serre introduced sheaf theory in AlgebraicGeometry with coherent sheaves [60], and one year later, in [61], he showed thatwith coherent sheaves in hand there are cases such that the cohomology groupsof complex and non-complex algebraic varieties coincide, by using the Zariskitopology.However, in most cases, the Zariski topology does not have “enough” opensets. So, motivated to prove the Weil’s Conjectures, A. Grothendieck had theidea of stop trying to find open sets, in the usual sense, and defined an analogousversion of inclusion of open sets using more general morphisms in small cate-gories. This gave birth to Grothendieck topologies and to Grothendieck toposes,particularly, the étale topos of a scheme X - the category of all étale sheaves ona scheme X - and so to Étale Cohomology. A. Grothendieck, M. Artin, and J-L.Verdier proved three of the four Weil’s Conjectures, and the remaining one wasproved by Deligne in 1974 [15]. The main references to see the development ofthis program aiming the proof of Weil’s Conjectures passes through Bourbakiseminars [28], “Eléments de Geométrie Algébrique" [24, 26, 27, 30, 31, 32, 33, 34],and “Séminaire de Géométrie Algébrique" (SGA). We highlight SGA4 [29], asthe one dedicated to topos theory and étale cohomology.We gave an intuition of how it is possible to generalize sheaves over topo-logical spaces in settings that we do not have a proper notion of open sets.Now, remember that a locale ( L, ≤ ) is a complete lattice such that a ∧ ( W i ∈ I b i ) = W i ∈ I ( a ∧ b i ) , ∀ a, b i ∈ L .The poset of all open sets of a topological space X is a locale. Locales areprecisely the complete Heyting algebras .Now note that in the definition of a sheaf over a topological space we did notuse anything about the points of the space, that is, only their locale structure The class of all Heyting algebras provides the natural algebraic semantics for the intu-itionistic propositional logic, that is the “constructive fragment" of the classical propositionallogic. F : L op → Set ,where L is the category associated to a locale L , since it is a poset. This is onesimple case where the notion of sheaves still is available in a category differentfrom O ( X ) . There are others? Yes, introducing an abstract idea of open coverwe can define sheaves for any small category C .First, we will be a bit less general. Suppose C is a small category with finitelimits (or just with pullbacks). A Grothendieck pretopology on C associates toeach object U of C a set P ( U ) of families of morphisms { U i → U } i ∈ I satisfyingsome simple rules. They are:1. The singleton family { U ′ f −→ U } formed by an isomorphism f : U ′ ∼ = → U isin P ( U ) ;2. If { U i → U } i ∈ I is in P ( U ) , and V → U is any morphism in C , then thefamily of pullbacks { V × U U → V } is in P ( V ) ;3. If { U i f i −→ U } i ∈ I is in P ( U ) and { V ij g ij −−→ U i } j ∈ J i is in P ( U i ) for all i ∈ I ,then { V ij f i ◦ g ij −−−−→ U } i ∈ I,j ∈ J i is in P ( U ) .The families in P ( U ) are called covering families of the object U in C .Note that the “concrete" notion of covering in topological spaces providesan example of Grotendieck pretopology: recall that an object in O ( X ) is justan open set U in X and the morphisms in O ( X ) are inclusions of open subsetsof X , this category has all finite limits (they are given by finite intersection ofopen subsets). Thus is natural to define a Grothendieck pretopology P in O ( X ) by { U i f i ֒ → U } i ∈ I ∈ P ( U ) iff U = S i ∈ I U i . This can be carry out analogously foreach locale L instead the spatial locales O ( X ) .Now let P be a pretopology on the category with pullbacks C and let F : C op → Set be a presheaf. F is s sheaf for the Grothendieck pretopology P if thefollowing diagram is an equalizer in the category Set : F ( U ) Q F ( U i ) Q F ( U i × U U j ) However, different pretopologies can provide the same class of sheaves. Toimprove the above definition we will use the notion of covering sieve .Let C be an object in C (the assumption of existence of pullbacks over C canbe dropped now), a sieve on C is a collection S of morphisms f with codomain C such that f ◦ g ∈ S, for all morphism g with dom ( f ) = cod ( g ) . Given h : D → C ,define h ∗ ( S ) = { g | cod ( g ) = D, h ◦ g ∈ S } . Then, a Grothendieck Topology in C associates each object C of C to a collection J ( C ) of sieves on C such that:1. The largest sieve on C , { f | cod ( f ) = C } , is in J ( C ) ;2. If S is in J ( C ) , then h ∗ ( S ) is in J ( D ) for all h : D → C ;3. If R and S are sieves on C , S is in J ( C ) and h ∗ ( R ) is in J ( D ) for all h : D → C in S , then R is in J ( C ) .18he collection of sieves in J ( C ) are the covering sieves (or J -covers). Thepair ( C , J ) formed by a small category C and a Grothendieck Topology J iscalled site . To each pretopology P on a category with pullbacks C , it can beassociated a Grothendieck topology J P : a covering sieve S ∈ J P ( U ) is just asieve on the object U that contains some family in P ( U ) .We can also define sheaves for the Grothendieck topology J , but more con-cepts would be introduced and we can be satisfied with what we have because itsdefinition is equivalent to the definition of sheaves for Grothendieck pretopolo-gies [38]. Morphisms of sheaves are natural transformations, and so we obtain Sh ( C , J ) , the category of sheaves over this site.Finally, a Grothendieck Topos is a category that is equivalent to Sh ( C , J ) , forsome site. Note Sh ( X ) = Sh ( C , J P ) is a Grothendieck topos where C = O ( X ) and J P is the Grothendieck topology described above.Grothendieck toposes also are characterized with pure categorical axioms, byGiraud’s Theorem [38, Theorem 0.45]. If a category has some specific properties,it is a Grothendieck topos. Conversely, every Grothendieck topos satisfies thesesame properties. We provide below a list of properties we will need to sketchthe proof that Ab ( E ) is AB and has generators: Lemma 4.1
A Grothendieck topos E satisfies the following conditions:1. all colimits are universal (i.e, preserved by pullback);2. has all small coproducts;3. has a set of generators (i.e., exists a small family { G i } i ∈ I of objects in E where given distinct morphisms f, g : X → Y in E there are i ∈ I and h : G i → X such that f ◦ h = g ◦ h );4. filtered colimits commute with finite limits. In this list, only the last property is not part of Giraud’s Theorem, but wewill use it and it follows, not immediately, from the fact the same holds for
Set . An elementary topos is a category that is cartesian closed, has all finite limits(or, equivalently, has all finite products and equalizers, or even has pullbacksand a terminal object), and a subobject classifier.A category is cartesian closed if it has binary products and it is possible todefine an exponential object for every two objects as follows: given B and C objects, there is a C B object endowed with an evaluation map ev : C B × B → C such that for any other object A , endowed with an arrow f : A × B → C , thereis an unique morphism ¯ f : A → C B where ev ◦ ( ¯ f × id B ) = f . An importantproperty that arises from this definition is the isomorphism Hom ( B × A, C ) ∼ = Hom ( A, C B ) .The subobject classifier of a locally small category that has all finite limits,and as terminal object consists of an object Ω of truth values and a truth orphism t : 1 → Ω such that given any object E , and any "subobject" r : U E , there is a unique morphism χ r : E Ω that makes thefollowing diagram a pullback: U E Ω r ! tχ r This χ r : E Ω is called characteristic morphism of r . It can look toabstract but when the category is Set , we have
Ω = { , } and for each subset U of a fixed set E , the morphism χ U is the well known characteristic function.In fact, Set is an example of elementary toposes [11, Example 5.2.1] and, moregenerally, every Grothendieck topos is an elementary topos [11, Example 5.2.9].Any elementary topos E enjoys some categorical properties that holds in thecategory Set , e.g.: a morphism in E is an isomorphism iff it a monomorphismand a epimorphism; every epimorphism in E is a coequalizer; any morphism in E has a (essentially unique) factorization as the composition of a monomorphismwith an epimorphism.An important type of morphism between toposes f : F → E is called geo-metric morphism . It consists of a pair of functors, f ∗ : F → E , the direct image ,and f ∗ : E → F , the inverse image , such that:1. f ∗ is left adjoint of f ∗ ;2. f ∗ preserves finite limits, i.e, it is left exact.The reader does not need to know the definition of adjoint pair of functorsto understand the ideas covered in this survey and can think in adjointness asan abstraction of the notion of free construction in Algebra; the sheafificationprocess is an instance of adjointness. We recommend [48] if there is a curiosityto understand better the few proofs we will explicitly use the adjoint propertyof geometric morphisms.Since, in general, each side of an adjoint pair of functors determines theother side, up to isomorphism, it is not difficult to see that Set is the terminalGrothendieck topos, concerning the geometric morphisms. This motivates todefine the notion of point in a topos E : it is just a geometric morphism f : Set → E . Another distinguished example of geometric morphism in the realmof Grothendieck toposes is i = ( i ∗ , i ∗ ) : Sh ( C , J ) → Set C op , here the direct imagepart is just the (full) inclusion i ∗ : Sh ( C , J ) ֒ → Set C op and the inverse imagepart is the "sheafification" functor, i ∗ : Set C op → Sh ( C , J ) .Every topos naturally encodes a “local set theory" [5]. In fact, each toposhas a internal language, known as Mitchell-Bénabou language , and a canonical interpretation - i.e. a procedure to give a meaning for the symbols introducedin this canonical language. In the next section we will use these notions in theproof of Theorem 4.3. These data are unique up to unique isomorphisms. E , the Mitchell-Bénabou language L ( E ) consists of three parts:sorts (or types), terms, and formulas. For each object A in E , there is an asso-ciated sort s A (they are distinct from each other). The terms τ of L ( E ) have avalue sort s ( τ ) and are inductively constructed from the basic terms by apply-ing certain natural constructors; the basic terms of sort s A are the constants ofvalue sort s A , that corresponds to morphisms → A in E , and a denumerableset of variables { x Ai : i ∈ N } of sort s A . The formulas are inductively con-structed from the basic (or atomic) formulas by applying (higher-order) logicalconstructors; the atomic formulas are defined “to abbreviate relations betweenterms". As a simple example of (atomic) formula we have τ = A σ , where τ and σ are terms with same value sort s A .Now, for the canonical interpretation of the language L ( E ) in the topos E ,the main idea is establish, for each term τ of type s A with variables x , ..., x n ofrespective types s X , ..., s X n , a realization by an arrow in E , [ τ ] : X × ... × X n → A , and, for each formula ϕ with (free) variables x , ..., x n of types s X , ..., s X n a truth table by an arrow in E , [ ϕ ] : X × ... × X n → Ω , where Ω is thesubobject classifier of E . For instance: (i) if x is a variable of type s A , then [ x ] def = id A : A → A ; (ii) we have considered above the formula τ = A σ , where s ( τ ) = s ( σ ) = s A , continuing this, we state that the truth table of τ = A σ isthe morphism X × ... × X n ([ τ ] , [ σ ])) −−−−−→ A × A δ A −−→ Ω , where the free variables in τ and σ have types among s X , ..., s X n and δ A is the characteristic morphism of △ A def = ( id A , id A ) : A → A × A , the diagonal morphism.In this setting, a formula ϕ is valid if this canonical interpretation if X × .... × X n ! −→ t −→ Ω is the truth table of ϕ , where x , ..., x n are free variables oftypes s X , ..., s X n . To denote that ϕ is valid we use E | = ϕ .As a simple example, we will show that E | = x = A x , where x is a variable oftype s A , in other words, the formula x = A x is valid in E . By the commentariesabove we know that the truth table of x = A x is A ([ x ] , [ x ])) −−−−−→ A × A δ A −−→ Ω . Since [ x ] = id A : A → A , the morphism ([ x ] , [ x ]) is precisely the diagonal morphism △ A . By definition, δ A is the characteristic morphism of △ A , then, the followingdiagram is a pullback A A × A Ω !([ x ] , [ x ]) tδ A In particular, the diagram commutes, so A ! −→ t −→ Ω , is the truth table of x = A x . Therefore, x = A x is valid in E .21e saw x = A x is valid for any topos, but the equality sign carries a lotof information - in fact, it is a specific characteristic morphism - and, at first,any property regarding = A should work explicitly with characteristic morphismand other tools presented by the internal language. However, when we workwith toposes is usual to omit the internal language machinery and pretend thatobjects are sets, monomorphism are injective functions, epimorphisms are sur-jective functions, isomorphisms are bijective functions, and so on. Basically,we pretend that a topos is the specific topos Set . This is possible due to the
Soundness Theorem [50, Chapter 15], but we only can replicate a constructionin
Set to an arbitrary topos if we restrain ourselves to “constructive aspects"presented in intuitionistic logic, because, in general, the law of excluded middle (i.e., ϕ ∨ ¬ ϕ ) does not hold for all toposes. In the same vein, it should be avoidedthe use of the axiom of choice, a “non-constructive" set-theoretical axiom. Wewill apply this procedure in the next section. Now we will replicate the cohomology construction above in the more generalsetting of Grothendieck Toposes, however, new techniques are necessary to provethe toposes versions of the results introduced in the previously section . Herewe fix E ≃ Sh ( C , J ) a Grothendieck Topos.We begin with a useful but simple concept: a parallel pair of morphisms f, g : A → B is reflexive if exist a common section s : B → A of f and g . Inparticular, a reflexive coequalizer is a coeaqualizer of a reflexive pair. Lemma 4.2
1. The forgetful functor E : Ab ( E ) → E creates reflexives co-equalizers and finite limits [38, Lemma 6.42];2. For abelian categories, the AB condition is equivalent to the category hasall small colimits and all filtered colimits being universal [22]. We use the above lemma to sketch the proofs of the main results in thissubsection.
Theorem 4.3
The category Ab ( E ) is an abelian category for any elementarytopos E . Proof.
Show that Ab ( E ) is Ab -category follows by straightforward calculation.To see it is an additive category we will use the internal language of E . Thus weneed to prove that there exists in Ab ( E ) : terminal and initial objects, and binaryproducts and binary coproducts; moreover they should coincide in Ab ( E ) . Wealready know that terminal objects and binary products exists in Ab ( E ) because E has finite limits and, by Lemma 4.2.1 above, the forgetful functor creates finitelimits, so Ab ( E ) has finite limits. If E = Set, then Ab ( E ) ≃ Ab.
It is know that Ab is an additive category and the demonstration of this fact only uses constructive On the other hand, if a Grothendieck topos has “enough points", then its cohomologycoincides with some spatial sheaf cohomology, see [54]. Ab ( E ) with E an arbitrary topos. For instance, everytopos E satisfies the formula ϕ , that describe that any terminal object is a(internal) singleton, and satisfies also the formula ψ , that asserts that the everysingleton supports a structure of (internal) abelian group and this is an initialabelian group. Thus Ab ( E ) is an additive category.It is not difficult to see that any morphism f : A → B in Ab ( E ) has akernel: since the forgetful functor creates finite limits (Lemma 4.2.1) ker ( f ) = equal (0 , f ) .Now let f be an epimorphism in Ab ( E ) , then it is also an epi in E . Butany epi in E is a coequalizer, then f = coeq ( g, h ) for some g, h ∈ E and, since f ∈ Ab ( E ) , it can be rewritten as f = coeq ( g ′ , h ′ ) for some g ′ , h ′ ∈ Ab ( E ) . Thus f = coeq ( g ′ , h ′ ) = coeq (0 , h ′ − g ′ ) = coker ( h ′ − g ′ ) .To conclude Ab ( E ) is an abelian category we have to construct a cokernelof an arbitrary morphism f : A → B in Ab ( E ) and that any monomorphism in Ab ( E ) is a kernel in Ab ( E ) .We take a coequalizer in E of the pair m ◦ ( f × id B ) and p , where m : B × B → B is the morphism m introduced at the definition of group object, p : A × B → B is the projection in the second coordinate, and f : A → B is amorphism in Ab ( E ) .Let q = coeq ( m ◦ ( f × id B ) , p ) . First, note that A × B B m ◦ ( f × id B ) p is areflexive pair with section s = (0 , id B ) : B → A × B . Considering parts of thediagram of coequalizer, and of the cartesian product of morphisms, we have: B A × B B × B B CA B s p f × id B p mp qf id B With a lot of diagram calculations and the coequalizer universal property,is possible to show that q , a coequalizer in Ab ( E ) , is the cokernel of f, for any f in Ab ( E ) . Now, let f be a monomorphism in Ab ( E ) . Denote coker ( f ) = q , then q ◦ f =0 . Since q ◦ ker ( q ) = 0 , then, by the universal property of ker ( q ) , there existsa unique t ∈ Ab ( E ) such that f = ker ( q ) ◦ t and this t is a mono, since f is a mono. Until now, all the information were obtained from very generalcategorical arguments, however E is an elementary topos and we can simulatein E the proof, made in Set with elements, that establishes that t is “surjective"(i.e. an epimorphism) in E thus, as we already mentioned before, it follows that t it is an isomorphism in the topos E . Since t ∈ Ab ( E ) , t is an isomorphism in Ab ( E ) . Summing up, we have shown that any mono in Ab ( E ) is a kernel, in fact,it is the kernel of its own cokernel. 23y the Grothendieck Theorem 2.1, if an abelian category satisfies AB andhas a generator then it has enough injectives. Thus, to state Ab ( E ) has enoughinjectives, we only need to prove this two conditions.Lets see that Ab ( E ) satisfies AB . By Lemma 4.2.2, we need to prove Ab ( E ) has all small colimits with allfiltered colimits being universal. We known E has all small coproducts and,since the forgetful functor E : Ab ( E ) → E creates reflexive coequalizers, it canbe shown that Ab ( E ) has all small colimits.In a Grothenciek topos, filtered colimits and finite limits commutes. Since E creates finite limits, E creates filtered colimits and pullbacks. Besides that, allcolimits are universal in a topos, in the sense they are preserved by pullbacks,thus filtered colimits are universal in Ab ( E ) . See 4.1 to remember Grothendiecktoposes’s properties.Now we prove that Ab ( E ) has a set of generator.By Giraud Theorem, E has a set of generators { G i } i ∈ I . Let f, g : X → Y in Ab ( E ) so f and g are morphisms in E . If f = g , since { G i } i ∈ I is a generator of E , there is h i : G i → E ( X ) , for some i ∈ I and E : Ab ( E ) → E forgetful functor,such that E ( f ) ◦ h i = E ( g ) ◦ h i . Consider the coproduct universal morphism h : ` i ∈ I G i → E ( X ) and the canonical morphism α i : G i → ` i ∈ I G i . We have h i = h ◦ α i so E ( g ) ◦ h ◦ α i = E ( g ) ◦ h i = E ( f ) ◦ h i = E ( f ) ◦ h ◦ α i Then E ( g ) ◦ h = E ( f ) ◦ h .Now we use the fact that the forgetful functor has a right adjoint functor, Z : E → Ab ( E ) , this is a generalization of the "free abelian group" constructionfrom the topos Set to any Grothendieck topos , and apply it in h , obtaining ˜ h : Z ( ` i ∈ I G i ) → X, its associated morphism in Ab ( E ) . The adjointness of Z and E guarantees that f ◦ ˜ h = g ◦ ˜ h so Z ( ` i ∈ I G i ) is generator of Ab ( E ) . Then, by the Grothendieck Theorem:
Theorem 4.4
The abelian category Ab ( E ) has enough injectives. In the case of Grothendieck toposes, the global section functor is Γ Ab = Hom E (1 , − ) : Ab ( E ) → Ab ( Set ) , inducted by the unique geometric morphism Γ :
E →
Set , with terminal object. Since Γ Ab is induced by the direct imagepart of Γ , we conclude Γ Ab preserves injectives.Prove that for any geometric morphism their direct image preserve injectivesis not difficult. We will do this here to introduce a usual and simple manipulationwith direct and inverse images using adjoint properties:Let f : F → E geometric morphism, and I injective object in Ab ( F ) . Con-sider the following diagram in Ab ( E ) X Yf ∗ ( I ) mh Because these toposes have the internal "set of all natural numbers". Ab ( F ) f ∗ ( X ) f ∗ ( Y ) I f ∗ ( m )˜ h Now we use the injectiviness of I to complete the diagram with an Ab ( F ) -arrow g : f ∗ ( Y ) → I that makes it commutative. Then we transpose, byadjoint property, one last time, and find a commutative diagram in Ab ( E ) thatguarantees that f ∗ ( I ) is an injective object in Ab ( E ) X Yf ∗ ( I ) We define the q -th cohomology group of E with coefficientes in F , object in Ab ( E ) as the q -th right derived functor of Γ Ab ( F ) . In other words, H q ( E , F ) = R q (Γ Ab )( F ) We can define cohomology for objects different from the terminal: Let B object of E , since Hom E ( B, − ) is a left exact functor we can consider rightderived functor for it, denoted by H q ( E , B ; F ) . The problem is how to describe H q ( E , B ; F ) in terms of H q ( E , F ) . The idea is that the funtor B ∗ : E → E ↓ B ,which sends an object A in E into p : A × B → B in E ↓ B , induces an exactfunctor B ∗ Ab : Ab ( E ) → Ab ( E ↓ B ) that preserves injectives, and is possible toestablish an isomorphism H q ( E , B ; F ) ∼ = H q ( E ↓
B, B ∗ Ab ( F )) [38, page 262].There is also a notion of flabby object. We say that F in Ab ( E ) is flabby if H q ( E , B ; F ) = 0 , for all q > and all B object in E . Proposition 4.5
Every injective object in Ab ( E ) is a flabby object in Ab ( E ) .Proof. More generally, for any F injective object in an abelian category wehave an injective resolution → F id F −−→ F → → → ... of F . Thus,applying a left exact functor Γ and taking its right derived functors will furnish R q Γ( F ) ∼ = H q (Γ( F • )) . Translating for our scenario, F is an injectiveobject in Ab ( E ) with the above injective resolution. For each object B in E ,we construct a left exact functor B ∗ Ab : Ab ( E ) → Ab ( E ↓ B ) , as previouslymentioned. Then H q ( E ↓
B, B ∗ Ab ( F )) ∼ = R q ( B ∗ Ab ( F )) = 0 . Therefore, F isflabby.The following lemma is useful to prove the analogous version of Proposition3.2. We exhibit a proof because it uses manipulations with geometric morphismsthat show up every time we are working with Grothendieck Toposes. Lemma 4.6
Let f : F → E a geometric morphism, with E = Sh ( C , J ) , F objectin Ab ( F ) , and l : C → Sh ( C , J ) canonical functor ( U i ∗ ( Hom ( − , U )) ). Then R q f ∗ ( F ) is the J -sheaf associated to the pre-sheaf U H q ( F , f ∗ l ( U ); F ) in C . roof. We separate the proof of this lemma in two parts. First we consider J as minimal topology, and after J will be an arbitrary Grothendieck Topology.The Grothedieck topology J be minimal means J ( C ) = { maximal sieve in C } ,where C is an object in C . The minimal topology implies E = Set C op . Since f is a geometric morphism, f ∗ preservers limits and is left adjoint of f ∗ . So f ∗ preserves small limits, f ∗ ( − )( U ) is a left exact functor, and we can obtain theright derived functor f ∗ ( − )( U ) . Besides that, by group cohomology definitionand adjoint property of geometric morphism: R f ∗ ( − )( U ) ∼ = f ∗ ( − )( U ) ∼ = Hom E ( Hom ( − , U ) , f ∗ ( − )) ∼ = Hom F ( f ∗ ( Hom ( − , U )) , − ) ∼ = H ( F , f ∗ ( Hom ( − , U )) , − ) : Ab ( F ) → Ab ( Sets ) So the lemma holds for J minimal.Suppose J is an arbitrary Grothendieck Topology in C , let i = ( i ∗ , i ∗ ) : E →
Set C op the inclusion geometric morphism, and define g = i ◦ f : F →
Set C op .The adjoint properties guarantees that i ∗ g ∗ = ( i ∗ i ∗ ) f ∗ ∼ = f ∗ . Since i ∗ is an exactfunctor, i ∗ R q g ∗ ∼ = R q ( i ∗ g ∗ ) ∼ = R q ( f ∗ ) . By the fact l is canonical functor and i ∗ is the associated sheaf functor [49, Chapter III.5], we have g ∗ ( Hom ( − , U )) = f ∗ i ∗ ( Hom ( − , U )) = f ∗ l ( U ) . Then, we apply this in the calculations for J minimal and conclude the desired result. Proposition 4.7 If F is a flabby sheaf, then R q f ∗ ( F ) = 0 , for all q > . Inother words, F is f ∗ -acyclic.Proof. We have R q f ∗ ( F ) is the J -sheaf associated to U H q ( F , f ∗ l ( U ); F ) ,by the above Lemma. Since F is flabby, H q ( F , f ∗ l ( U ); A ) = 0 for all q > so R q f ∗ ( F ) = 0 , for all q > .Since the notion of flabby sheaf implies an acyclicity, and there is a Godementresolution in this context [38, page 265], we can use it to define cohomologygroups using flabby sheaves instead of injective ones, by the discussion at theend of section 3.1. This approach is particularly interesting for cohomology ina topos because injectives resolutions depend on the axiom of choice to worksproperly and general toposes rely on intuitionistic logic. However, we observethat this definition of flabby does not coincide with the flabby definition forsheaves over topological spaces when Sh ( C , J ) = Sh ( X ) . How to constructivelygeneralize the flabby definition in Sh ( X ) to Sh ( C , J ) ? We do not know a definiteanswer to that but we will explain more about it in the last section. As expected, Čech Cohomoloy in the Grothendieck Topos case is more compli-cated. We will be more careful now than before, and use some lemmas withoutproofs to not exceed in technicalities.We fix E = Sh ( C , J ) , consider P = Set C op it correspondent presheaves cate-gory, and i : E → P the canonical inclusion.26uppose that C has pullbacks. For sheaves over topological spaces, whenconstructing the Čech Cohomology, we considered U i ,...,i q as a intersection offinite subfamily of open sets that covers an open U . Now we need to find ananalogous of this. Let U = ( U i f i → U ) i ∈ I be an well-ordered family of morphismsin C , define U i ,...,i q := U i × U ... × U U i q . Applying morphisms U i ,...,i q δ k −→ U i ,..., b i k ,...i q that “forgets i k ", we have a diagram in P as follows: . . . ` i
Given F sheaf in Ab ( P ) , the Čech cochain complex is C q ( U , F ) = Hom Ab ( P ) ( N q ( U ) , F ) , with coboundary morphisms d q = − ◦ d q . Since ( − ◦ d q +1 ) ◦ ( − ◦ d q ) = − ◦ ( d q ◦ d q +1 ) = − ◦ , we define the q -th Čech cohomology group of U withcoefficients in F by H q ( U , A ) = Ker ( d q ) /Im ( d q − ) .Considering another well-ordered family of morphisms in E , V = ( V j g j → U | j ∈ J ) that refines the family U = ( U i f i → U | i ∈ I ) , we select a refinementmap r : V → U , i.e. a pair formed by a function r : J → I and a family offactorisations ( V j U r ( j ) U r j g j f r ( j ) : j ∈ J ) . If R is the sieve of U generated by the family U (i.e., for any morphism α in R , α = f i ◦ h i , for some i ∈ I and some h i ), then the inclusion map U → R determines a refinement map. I.e., we select a specific pullback for each subfamily. roposition 4.8 Given r, s : V → U refinement maps, r • and s • are chainhomotopic.Proof. We have to find a sequence of morphisms N q ( V ) → N q +1 ( U ) that makes r • and s • chain homotopics.Consider σ = ( i , ..., i q ) where i , ..., i q ∈ I . For each l ∈ { , , ..., q } wedefine a morphism over U as follows: t lσ = ( r i , ..., r i l , s i l , ..., s i q ) : V σ → U ( r i ,...,r il ,s il ,...,s iq ) This morphism induces a group homomorphism t lq : N q ( V ) → N q +1 ( U ) . Wewill exhibit the homotopy chain construction for the case q = 1 by an alternatedsum of t l . So we have σ = ( i , i ) , t σ = ( r i , s i , s i ) , t σ = ( r i , r i , s i ) , anddefine t ( l )1 = X l =0 ( − l +1 t l = − t q + t q = − ( r i , s i , s i ) + ( r i , r i , s i ) . Since r and s are refinement maps we will extract indices j , ..., j q ∈ J from i , ..., i q ∈ I . The (non commutative) diagram we must have in mind is: . . . ` i
Let U = ( U i → U | i ∈ I ) be a family of morphisms and R the sieve generated by U . Then H q ( U , F ) ∼ = H q ( R, F ) is an isomorphism for any F sheaf in Ab ( E ) .Proof. Since R is generated by U , there is a refinement map h : R → U . Onthe other hand, we also have that the inclusion i : U → R is a refinement map.By the previously proposition, this refinement is unique up to homotopy, thus h.i and i.h are homotopic to the corresponding identity refinements. Thus, byProposition 2.1, i induces a map in the cohomology group that is invertible. Inother words, H q ( U , F ) ∼ = H q ( R, F ) , canonically.If C has pullbacks we can define Čech cohomology groups of an object U of C with coefficient in F an abelian presheaf in C as the filtered colimit below ˇH q ( U, F ) := lim −→ R ∈ J ( U ) H q ( R, F ) Note that previously we defined Čech Cohomology for a family of morphismswith a commom codomain instead of for a sieve, but both cases are relatedsince we can switch cover sieves with the family that generates it, by the aboveProposition. We introduce this definition to obtain an analogous version ofTheorem 3.3 for Grothendieck topos:
Theorem 4.10
Let U object in C and F sheaf in Ab ( E ) . There is a homomor-phism k q : ˇH q ( U, F ) → H q ( E , l ( U ); F ) , q ∈ N , where l : C → Sh ( C , J ) is thecanonical functor. Moreover, k q is a isomorphism if q = 0 or , and it is amonomorphism if q = 2 . To have an isomorphism in other cases we need to impose conditions onsubsets of the set of objects in C as follows: Proposition 4.11
Let E = Sh ( C , J ) , F sheaf in Ab ( E ) . If there is a subset K of the set of objects in C such that:(i) ˇH q ( V, F ) = 0 , ∀ q > , for each V ∈ K ; ii) For each object U in C , there is a J -cover { V j g j → U | j ∈ J } with V j ∈ K, ∀ j ∈ J ;(iii) Every pullback of the form V × U W is in K , whenever V and W are in K .Then the homomorphism map ˇH q ( U ; F ) → H q ( E , l ( U ); F ) is an isomorphismfor any object U in C and for all q ∈ N The proofs for both the above results use spectral sequences and can befound at [38, Chapter 8].
We already mentioned that Grothendieck topos cohomology was constructedto prove Weil’s conjectures. However, for this propose, Étale Cohomology isenough: there is no need to work with an arbitrary site ( C , J ) . If C is the slicecategory of schemes over a scheme X , where the objects are étale morphisms Spec ( R ) f −→ X , and the morphism f ϕ −→ g are, by abuse of notation, morphismsof schemes Spec ( R ) ϕ −→ Spec ( R ’ ) such that g ◦ ϕ = f .Étale cohomology has good properties, e.g, can be related to singular coho-mology, and has a Künneth formula, and Poincaré Duality with an adequateformulation. Furthermore, it has applications in number theory, K -theory, andrepresentation theory of finite groups, besides its original use in algebraic ge-ometry for fields different of C and R .For other sites, we obtain other cohomologies such as crystalline, Deligne,and flat cohomologies. They also are instances of the Grothendieck topos coho-mology we presented.There are other kinds of applications of Grothendieck topos cohomology. If C is a small category, and F is an abelian presheaf in Ab ( Set C op ) , we can definea cochain complex C q ( C , F ) = Q c ← ... ← c q F ( c q ) with an appropriate coboundary d q : C q ( C , F ) → C q +1 ( C , F ) , to obtain H q ( C , F ) = Ker ( d q ) /Im ( d q − ) as thecohomology groups of the category C with coefficients in F . Then, we have anisomorphism H • ( C , F ) ∼ = H • ( Set C op , F ) . For a proof of this and an explicitdescription of the coboundary maps, consult [53, Chap. II.6]The most simple example of how this works is when C is the category withonly one object (a group G can be seen as such a category). Thus the categoryof abelian group objects in Set C op is equivalent to the category of right modulesover the group ring Z G . So, by the above result, the cohomology groups of G , obtained from group cohomology, are isomorphic to the sheaf cohomologygroups of Set C op . This is better know in the form H • ( BG, M ) ∼ = H • ( G, M ) ,where BG is the classifying space of G and M is a G -module. Consult [2, Chap.II] to see the usual approach.Hence, Grothendieck topos cohomology also is related to non-sheaf cohomol-ogy, and not only with cohomology for specific sites. We will provide furtherapplications in the next section. 30 Remarks and New Frontiers
Topos are excellent environments for internalizing mathematical objects, and wecan write formulas for a language (type theory) like arrows hitting the subobjectclassifier. For example, to each formula φ ( x ) with a free variable x of type X is associated with the subobject of X that classically corresponds to “ { x ∈ X | φ ( x ) } ”. In this way, we can interpret a high order type theory in a topos viathe so-called semantics of Kripke-Joyal. Results on elementary topos includethat they are finitely co-complete, represent the idea of “parts of an object" andthat its internal logic is intuitionist and, in particular, the parts of an objectdefine an internal Heyting algebra. So, a topos is an environment for higherorder intuitionist mathematics — evidently not all the topos are equivalent, sothere is a diversity of environments.Daily mathematics makes use of set theories to represent higher-order aspectsof mathematical theories: this can be understood as the use of the higher-orderinternal logic of the Set topos. Since the 1970s, mathematical applications ofhigher-order intuitionist internal logic approaches have been applied to topos:(i) an internal approach to the Serre-Swan duality, through a simple theorem,was described in [56] (essentially) of Linear Algebra, Kaplansky’s Theorem ;(ii) in model constructions, via Grothendieck topos, of synthetic differential ge-ometry ([55], [50]): for instance, in [55], there is an internal version of de RhamTheorem (a deep connection between de Rham cohomology and singular homol-ogy);(iii) to represent results of quantum mechanics as results of classical mechanicsinternal to a topos [17];(iv) in algebraic geometry, although the origin of Grothendieck’s notion of toposcame from specific needs of algebraic geometry, more systematic explorations ofthe internal language of topos for this area are very recent: e.g., [8] contains adictionary between the external and the internal point of view (for example, ob-jects in a topos, are, internally, just sets; monomorphism are injections; sheavesof rings are rings), works with the big and small Zariski Topos associated toa scheme to exhibit simpler definitions and proofs by using the internal lan-guage provided by these toposes, and in [9] explores a proposal of a constructiveversion of the main homological tools (flabby and injective objects).Attempts to develop constructive approaches to homological algebra, with-out the aid of axiom of choice (as in the usual injective resolution construction),are different from “cohomology in topos", although they can be related. In thelatter case, we usually are interested in a Grothendieck topos, and constructingcohomology groups with coefficient in Ab ( Sh ( C , J )) , for some site ( C , J ) . Thatis exactly what we exposed in the previous section, using P. Johnstone’s book“Topos Theory" [38], as the main reference. However, similar to the extensionof sheaf cohomology to Grothendieck topos cohomology, how could we extendGrothendieck topos cohomology to (elementary) topos cohomology? The firstproblem is that for an elementary topos E we can not guarantee that Ab ( E ) Every module on a local ring that is projective and finitely generated is a free module. Ab ( E ) isnoetherian abelian. Still in the topic of “topos cohomology" we could try toswitch Ab ( E ) for E . In this direction, we have the work of I. Blechschmidt thatis closely related to develop a constructive version of homological algebra: hereintroduces the concepts of injective object and flabby sheaf, as objects in anelementary topos, and replaces injective resolutions with flabby resolutions toavoid the use of the axiom of choice. However, in the final chapter of his article,he calls attention to the open problem of how to embed an arbitrary sheaf ofmodules into a flabby sheaf in intuitionistic logic. We understand that this wayof proceeding (defining objects inside a topos) was successfully adopted beforein another context, by A. Grothendieck, when he defined the fundamental groupon a topos and originated the “Grothendieck’s Galois Theory" [25]. The theorywas later extended by A. Joyal and M. Tierney in [41]. The results of this latterarticle are constantly used in nowadays works, which indicates that studies inthe same direction for homological algebra would provide important discoveries.Pertinent to this discussion, we can cite Blass’s work that shows cohomologycan detect the failure of the axiom of choice [7]. He demonstrates the axiomof choice is equivalent to H ( X, G ) = 0 , for all discrete set X , and all group G . Also, the triviality of H ( X, G ) for all G is equivalent to the projectivityof X . This strengthens the relation between logic and geometry that we havebeen pointing through toposes. Note Blass’s results indicate a justification forthe fact that Ab ( E ) does not have enough projectives, in general, because oftoposes’ intuitionistic logic.We believe the subject of “topos cohomology" is far from maturity. One ofthe main references into the subject, SGA4, only addresses the case Ab ( Sh ( C , J )) .P. Johnstone, one of the most prominent topos theorists of our days, had notpublished the third volume of “Sketches of an Elephant: A Topos Theory Com-pendium" that would contain the subject of homotopy and cohomology intoposes (besides chapters about toposes as mathematical universes) and thefirst two volumes were released in 2002 [39, 40].Regarding constructive methods for homological algebra, there also are in-vestigations not involving toposes. For example, in [58], S. Posur’s providesconstructive methods in the context of abelian categories using generalized mor-phisms (we highlight it is not the same definition given by S. MacLane in [48]).He proves the Snake Lemma, establish what are generalized cochain complexand generalized homological groups, and present a notion of homological groupin a concrete way, i.e., he displays explicitly the connecting morphism, and notonly states it exists by universal properties. More than that, he applies thetheoretical definitions to create an algorithm capable of computing spectral se-quences for a certain abelian category, and use it to calculate cohomology groupsof (specific) equivariant sheaves.In recent work ([62]), M. Schulman defends that “Linear Logic" can clarifysome constructive methods better than intuitionistic logic. We highlight Schul-man’s state that it provide constructivist definitions (and proofs) of concepts32laborated in classical logic. Then a “linear approach" could also be useful forthe problems we mentioned concerning constructive cohomology. Furthermore,generalized metric spaces (or quasi-psudo-metric space, or Lawvere metric space[44]) can be redefined using linear logic [62].Linear Logic is a weakening of intuitionistic logic: it is a “sub-structural"logic, i.e., the usual demonstrability rules do not apply in general, with onlyrestricted versions of the contraction and weakening rules available. In linearlogic, the intuitionist conjunction splits into two binary operators: ∧ , the binaryinfimum of the lattice, which is not necessarily distributes over the supreme; & ,another operation that does distribute with arbitrary supreme, but it doesn’thave to be idempotent or commutative.The study of linear logic was initially developed by Jean-Yves Girard [19] inthe context of polymorphic λ calculus, but its nature matches - through splits -somewhat irreconcilable elements, and their many interpretations have profoundmeaning. Pure intuitionistic contexts cannot prove the excluded middle lawand, in classical logic, this is nothing less than an axiom. Linear logic has twocandidates for disjunction ∨ , one for which it is impossible to prove the excludedmiddle, and another for which the evidence is trivial.The presence of “duplication” of operators is natural, as these represent use-ful fragments of the usual logical operations. The result interesting is relatedto the famous correspondence of Curry-Howard: in the same way that intu-itionistic logic is related with type theory and λ calculation simply typed (theimplication can be interpreted as the type of functions, conjunction with productand disjunction with co-product) giving rise to “proof-relevance”, linear logic in-troduces, via non-idempotency or non-commutativity, the relationship of linearimplication to processes that are “Resource-relevant”.Categorical semantics for various forms of linear logics have long been ex-plored (e.g. [59], [36]). Roughly speaking, we can say that closed monoidalcategories have (some form of) internal linear logic.Something very different occurs when we focus on possible conjunctistic orhigher-order aspects ([43], [5]) that are internal to a special type of categorygoverned by some form of linear logic.A natural, and relatively simple, way to expand the notion of (categories of)sheaves with internal logic that is no longer intuitionistic is through appropriateadaptations of the sheaf notion defined over a complete Heyting algebra ( H, ≤ , ∧ ) to other algebras that are also complete lattices.These set-theoretical aspects of the sheaves on “good" complete lattices canalso be approached in an alternative, but often “equivalent" way, through thenotion of expansion of the universe of all sets, V , by an algebra, A , which is acomplete lattice, V ( A ) : in the (traditional) case where A is a Boolean algebraor Heyting algebra this is presented in [6].The complete lattices that have natural relationship with linear logics arethe quantales (see [67]). A quantale ( Q, ≤ , ⊗ , ⊤ ) is a structure where: ( Q, ≤ ) is a complete lattice where ⊤ is the top element, ( Q, ⊗ ) is a semigroup and thedistributive laws are valid: a ⊗ W i ∈ I b i = W i ∈ I a ⊗ b i , ( W i ∈ I b i ) ⊗ a = W i ∈ I b i ⊗ a .There are some early explorations of the strategy of considering “general-33zed sheaves", with applications in Mathematics. In [14], is established a no-tion of category of “sheaves" over a quantale ( Q, ≤ , ⊗ , ⊤ ) , which is right-sided( a ⊗ ⊤ = a, a ∈ Q ) and idempotent ( a ⊗ a = a, a ∈ Q ), and is explored theabove mentioned Kaplansky’s Theorem, now reformulated in the internal linearlogic of this category of sheaves. In the work [52], categories of sheaves areconsidered over quantales ( Q, ≤ , ⊗ , ⊤ ) satisfying a different balance: they arecommutative and semicartesian (or two-sided ). It is important to emphasizethat the two-sided, commutative, and idempotent quantales coincides with thecomplete Heyting algebras.In [52], given a commutative semicartesian quantale ( Q, ≤ , ⊙ , , we canconstruct what is called a Q - set , in the same spirit of the construction of sheavesover complete Heyting algebras. These Q - set will not provide a sheaf, butwill preserve a significant part of a sheaf structure, which had motivated theauthors to called it a “Sheaf-Like category", besides that, pseudo-metric spacesare examples of a Q - set (when Q = ([0 , ∞ ] , ≥ , + , ), and also of an enrichedcategory over Q . This approach seems to expand the development of modeltheory of Continuous Logic, useful in Functional Analysis.In [45], M. Schulman and T. Leinster use semicartesian monoidal categories V to define magnitude homology of V -categories (enriched categories over V ). Inparticular, if V is the extended non-negative real numbers [0 , ∞ ] , that admitsa natural structure of a commutative semicartesian quantale ([0 , ∞ ] , ≥ , + , ,then the correspondent V -category is a generalized metric space. Magnitudehomology describes a general notion of “size". Depending on the case, it coin-cides with the cardinality of a set, the Euler Characteristic of topological space,or of an associate algebra. For the metric space context, magnitude machineryprovide interesting geometric properties as area [66], volume [4], and Minkowskidimension [51].This conjuncture motivates the authors of this survey to wonder about in-ternal cohomological aspects to other categories governed by other logics. Inparticular: (i) if developments in continuous model theory -for instance withapplications to the theory of Banach algebras- could have internal cohomolog-ical aspects better represented in the linear logic style; (ii) if exploring metricspaces as enriched categories over a quantale, it is natural to consider possibleconnections between magnitude (co)homology with an adapted sheaf-like coho-mology by some appropriate version of sheaves over quantales. Acknowledgements:
The comments of Peter Arndt and Walter de SiqueiraPedra, members of the judging committee in the master dissertation (supportedby CNPq) of the first author, and supervised by the second author, which guidedfor paths of study we would not perceive alone, improving this text with moreapplications and related lines of research. We are thankful for it. Since it is already commutative, this is the same that require to be right-sided. eferences [1] Abramsky, S., Barbosa, R.S., Kishida, K., Lal, R., Mansfield, S.: Contex-tuality, cohomology and paradox. arXiv preprint arXiv:1502.03097 (2015)[2] Adem, A., Milgram, R.J.: Cohomology of Finite Groups, 2nd edn.Grundlehren der mathematischen Wissenschaften. Springer (2013)[3] Alexandroff, P.: Über den allgemeinen dimensionsbegriff und seinebeziehungen zur elementaren geometrischen anschauung. MathematischeAnnalen (1), 617–635 (1928)[4] Barceló, J.A., Carbery, A.: On the magnitudes of compact sets in euclideanspaces. American Journal of Mathematics (2), 449–494 (2018)[5] Bell, J.L.: Toposes and local set theories: an introduction. Oxford LogicGuides. Oxford University Press, Oxford and New York (1988)[6] Bell, J.L.: Set theory: Boolean-valued models and independence proofs,3nd edn. Oxford Logic Guides. Oxford University Press (2005)[7] Blass, A.: Cohomology detects failures of the axiom of choice. Transactionsof the American Mathematical Society (1), 257–269 (1983)[8] Blechschmidt, I.: Using the internal language of toposes in algebraic geom-etry. Ph.D. thesis, Universität Augsburg (2017)[9] Blechschmidt, I.: Flabby and injective objects in toposes. arXiv preprintarXiv:1810.12708 (2018)[10] Borceux, F.: Handbook of Categorical Algebra: Volume 2, Categories andStructures. Encyclopedia of Mathematics and its Applications. CambridgeUniversity Press, Cambridge (1994)[11] Borceux, F.: Handbook of Categorical Algebra: Volume 3, Sheaf Theory.Encyclopedia of Mathematics and its Applications. Cambridge UniversityPress, Cambridge (1994)[12] Borceux, F., Bourn, D.: Mal’cev, protomodular, homological and semi-abelian categories. Mathematics and Its Applications. Springer (2004)[13] Cartan, H.: Idéaux de fonctions analytiques de n variables complexes. In:Annales scientifiques de l’École Normale Supérieure, vol. 61, pp. 149–197(1944)[14] Coniglio, M.: The logic for sheaves over right-sided and idempotent quan-tales. Ph.D. thesis, University of São Paulo (1997)[15] Deligne, P.: La conjecture de weil. i. Publications Mathématiques del’Institut des Hautes Études Scientifiques (1), 273–307 (1974)3516] Eilenberg, S., Cartan, H.: Homological algebra. Princeton MathematicalSeries. Princeton University Press, Princeton (1956)[17] Flori, C.: A first course in topos quantum theory. Lecture Notes in Physics.Springer (2013)[18] Freyd, P.J.: Abelian categories. Harper’s Series in Modern Mathematics.Harper & Row, New York (1964)[19] Girard, J.Y.: Linear logic. Theoretical computer science (1), 1–101(1987)[20] Godement, R.: Topologie algébrique et théorie des faisceaux, nouveautirage edn. Actualites scientifiques et industrielles. Hermann, Paris (1958)[21] Gray, J.W.: Fragments of the history of sheaf theory. In: Applications ofsheaves, pp. 1–79. Springer (1979)[22] Grothendieck, A.: Sur quelques points d’algèbre homologique. TohokuMathematical Journal (2), 119–221 (1957)[23] Grothendieck, A.: Espaces vectoriels topologiques: Curso de extensão uni-versitária da Faculdade de Filosofia, Ciências e Letras da Universidade deSão Paulo. Publicação da Sociedade de Matemática de São Paulo (1958)[24] Grothendieck, A.: Éléments de géométrie algébrique : I. le langage desschémas. Publications Mathématiques de l’IHÉS , 5–228 (1960)[25] Grothendieck, A.: Revêtement étales et groupe fondamental (SGA1). Lec-ture Note in Mathematics (1961)[26] Grothendieck, A.: Éléments de géométrie algébrique : II. Étude globale élé-mentaire de quelques classes de morphismes. Publications Mathématiquesde l’IHÉS , 5–222 (1961)[27] Grothendieck, A.: Éléments de géométrie algébrique : III. Étude coho-mologique des faisceaux cohérents, Première partie. Publications Mathé-matiques de l’IHÉS , 5–167 (1961)[28] Grothendieck, A.: Fondements de la géométrie algébrique: extraits duSéminaire Bourbaki, 1957-1962. Secrétariat mathématique, Paris (1962)[29] Grothendieck, A.: Théorie des topos et cohomologie Étale des schemas(SGA4). Lecture Notes in Mathematics , 299–519 (1963)[30] Grothendieck, A.: Éléments de géométrie algébrique : III. étude coho-mologique des faisceaux cohérents, Seconde partie. Publications Mathéma-tiques de l’IHÉS , 5–91 (1963)[31] Grothendieck, A.: Éléments de géométrie algébrique : IV. Étude localedes schémas et des morphismes de schémas, Première partie. PublicationsMathématiques de l’IHÉS , 5–259 (1964)3632] Grothendieck, A.: Éléments de géométrie algébrique : IV. Étude localedes schémas et des morphismes de schémas, Seconde partie. PublicationsMathématiques de l’IHÉS , 5–231 (1965)[33] Grothendieck, A.: Éléments de géométrie algébrique : IV. Étude localedes schémas et des morphismes de schémas, troisièeme partie. PublicationsMathématiques de l’IHÉS , 5–255 (1966)[34] Grothendieck, A.: Éléments de géométrie algébrique : IV. Étude localedes schémas et des morphismes de schémas, quatrième partie. PublicationsMathématiques de l’IHÉS , 5–361 (1967)[35] Hartshorne, R.: Algebraic geometry. Graduate texts in mathematics.Springer, New York (1977)[36] Hyland, M., de Paiva, V.: Full intuitionistic linear logic. Annals of Pureand Applied Logic (3), 273–291 (1993)[37] Iversen, B.: Cohomology of Sheaves. Universitext. Springer (1986)[38] Johnstone, P.: Topos Theory. London Mathematical Society Monographs.Academic Press, London (1977)[39] Johnstone, P.: Sketches of an Elephant: A Topos Theory Compendium -Volume 1. Oxford Logic Guides. Oxford University Press, New York (2002)[40] Johnstone, P.: Sketches of an Elephant: A Topos Theory Compendium -Volume 2. Oxford Logic Guides. Oxford University Press, New York (2002)[41] Joyal, A., Tierney, M.: An extension of the galois theory of grothendieck.Memoirs of the American Mathematical Society (309) (1984)[42] Koszul, J.: Faisceaux et cohomologie. Curso de extensão universitária daFaculdade de Filosofia, Ciências e Letras da Universidade de São Paulo.Instituto de Matemática Pura e Aplicada do CNPq (1957)[43] Lambek, J., Scott, P.J.: Introduction to higher order categorical logic.Studies in Advanced Mathematics. Cambridge University Press (1986)[44] Lawvere, F.W.: Metric spaces, generalized logic, and closed categories.Rendiconti del seminario matématico e fisico di Milano (1), 135–166(1973)[45] Leinster, T., Shulman, M.: Magnitude homology of enriched categories andmetric spaces. arXiv preprint arXiv:1711.00802 (2017)[46] Leray, J.: Sur la forme des espces topologiques et sur les points fixes desrepresentations. J. Math. Pures Appl. , 95–248 (1945)[47] Leray, J.: L’anneau spectral et l’anneau filtré d’homologie d’un espace lo-calement compact et d’une application continue. Journal de MathématiquesPures et Appliquées (9), 1–139 (1950)3748] MacLane, S.: Categories for the working mathematician, 2nd edn. Gradu-ate Texts in Mathematics. Springer, New York (1998)[49] MacLane, S., Moerdijk, I.: Sheaves in geometry and logic: A first intro-duction to topos theory. Universitext. Springer, New York (1992)[50] McLarty, C.: Elementary categories, elementary toposes. Oxford LogicGuides. Oxford University Press, Oxford (1992)[51] Meckes, M.W.: Magnitude, diversity, capacities, and dimensions of metricspaces. Potential Analysis (2), 549–572 (2015)[52] Mendes, C.A., Mariano, H.L.: Sheaf-like categories and applications tocontinuous logic. in preparation (2020)[53] Moerdijk, I.: Classifying spaces and classifying topoi. Lecture Notes inMathematics. Springer, Berlin (1995)[54] Moerdijk, I.: Classifying spaces for toposes with enough points. Rendicontidel seminario matématico e fisico di Milano , 377–389 (1996)[55] Moerdijk, I., Gonzalo, E.R.: Models for Smooth Infinitesimal Analysis.Springer, Berlin (1991)[56] Mulvey, C.: Intuitionistic algebra and representations of rings. Memoirs ofthe American Mathematical Society (2013) (1974)[57] Petersen, P.: Riemannian geometry. Graduate Texts in Mathematics.Springer, New York (2006)[58] Posur, S.: Constructive category theory and applications to equiv-ariant sheaves. Ph.D. thesis, Universität Siegen (2017). URL https://dspace.ub.uni-siegen.de/handle/ubsi/1179 [59] Seely, R.A.: Linear logic,*-autonomous categories and cofree coalgebras.In: In Categories in Computer Science and Logic, pp. 371–382. AmericanMathematical Society (1987)[60] Serre, J.P.: Faisceaux algebriques coherents. Annals of Mathematics (2),197–278 (1955)[61] Serre, J.P.: Géométrie algébrique et géométrie analytique. In: Annales del’institut Fourier, vol. 6, pp. 1–42 (1956)[62] Shulman, M.: Linear logic for constructive mathematics. arXiv preprintarXiv:1805.07518 (2018)[63] Tennison, B.R.: Sheaf Theory. London Mathematical Society Lecture NoteSeries. Cambridge University Press, Cambridge (1975)[64] Weibel, C.A.: An introduction to homological algebra. Studies in AdvancedMathematics. Cambridge University Press (1994)3865] Wildeshaus, J.: On derived functors on categories without enough injec-tives. Journal of Pure and Applied Algebra (2), 207 – 213 (2000)[66] Willerton, S.: On the magnitude of spheres, surfaces and other homoge-neous spaces. Geometriae Dedicata (1), 291–310 (2014)[67] Yetter, D.N.: Quantales and (noncommutative) linear logic. Journal ofSymbolic Logic55