aa r X i v : . [ m a t h . AG ] J u l SHEAVES ON T -TOPOLOGIES MÁRIO J. EDMUNDO AND LUCA PRELLI
Abstract.
The aim of this paper is to give a unifying description of variousconstructions of sites (subanalytic, semialgebraic, o-minimal) and consider thecorresponding theory of sheaves. The method used applies to a more generalcontext and gives new results in semialgebraic and o-minimal sheaf theory.
Contents
Introduction 11. Sheaves on locally weakly quasi-compact spaces 31.1. Locally weakly quasi-compact spaces 31.2. Sheaves on locally weakly quasi-compact spaces 41.3. c-soft sheaves on locally weakly quasi-compact spaces 92. Sheaves on T -spaces. 102.1. T -sheaves 112.2. T -coherent sheaves 132.3. T -flabby sheaves 172.4. T -sheaves on locally weakly quasi-compact spaces 212.5. T loc -sheaves 242.6. T -spectrum 263. Examples 283.1. The semialgebraic site 283.2. The subanalytic site 283.3. The conic subanalytic site 293.4. The o-minimal site 29References 30 Introduction
Sheaf theory in some tame contexts such as semi-algebraic geometry ([10]), sub-analytic geometry ([28, 35]) and o-minimal geometry ([19]) has had recently differ-ent applications in various fields of mathematics such as model theory [4, 5, 20],analysis [28, 30, 31, 36] and representation theory [1, 2, 37]. Each one of the above
Mathematics Subject Classification.
Keywords and phrases:
Sheaf theory, Grothendieck topologies.
MÁRIO J. EDMUNDO AND LUCA PRELLI theories is very useful for the mentioned applications but has some elements whichare missing in the other ones: the aim of this paper is to give a unifying descriptionof all these various constructions (subanalytic, semialgebraic, o-minimal) using amodification of the notion of T -topology introduced by Kashiwara and Schapira in[28].The idea is the following: on a topological space X one chooses a subfamily T of open subsets of X satisfying some suitable hypothesis, and for each U ∈ T onedefines the category of coverings of U as the topological coverings { U i } i ∈ I ⊂ T of U admitting a finite subcover. In this way one defines a site X T and studies thecategory of sheaves on X T (called Mod( k T ) ). This idea was already present in [28].However in [28], the space X is assumed to be Hausdorff, locally compact and theelements of T are assumed to have finitely many connected components.The exigence to treat in a unifying way all the previous constructions, to treatalso some non Hausdorff cases (as conic subanalytic sheaves which are related to theextension of the Fourier-Sato transform [36]) and the non-standard setting whichappears naturally in the o-minimal context (where the elements of T are totallydisconnected and never locally compact), motivates a modification of the definitionof [28]. In particular, in our definition we replace “connectedness” by the notionof T -connectedness (which in the standard o-minimal context is connectedness).Remark that there are many important o-minimal expansions M = ( R , <, , , + , · , ( f ) f ∈F ) of the ordered field of real numbers. For example R an , R exp , R an , exp , R an ∗ , R an ∗ , exp see resp., [12, 40, 15, 17, 18]. For each such we have κ many non-isomorphic nonstandard o-minimal models for each κ > cardinality of the language. There ishowever a non-standard o-minimal structure M = [ n ∈ N R (( t n )) , <, , , + , · , ( f p ) p ∈ R [[ ζ ,...,ζ n ]] ! which does not came from a standard one ([32, 23]).With this more general notion of T -space X we study the category of sheaveson the site X T . The natural functor of sites ρ : X → X T induces relations betweenthe categories of sheaves on X and X T , given by the functors ρ ∗ and ρ − . Thefunctor ρ ∗ is fully faithful. Moreover when X is locally weakly quasi-compact thereis a right adjoint to the functor ρ − , denoted by ρ ! . The functor ρ ! is exact, com-mutes with lim −→ and ⊗ and is fully faithful. We introduce the category of T -flabbysheaves (known as sa -flabby in [10] and as quasi-injective in [35]): F ∈ Mod( k T ) is T -flabby if the restriction Γ( U ; F ) → Γ( V ; F ) is surjective for each U, V ∈ T with U ⊇ V . We prove that T -flabby sheaves are stable under lim −→ and ⊗ and are acyclicwith respect to the functor Γ( U ; • ) , for U ∈ T . More generally, if one introducesthe category Coh( T ) ⊂ Mod( k X ) of coherent sheaves (i.e. sheaves admitting afinite resolution consisting of finite sums of k U i , U i ∈ T ), then T -flabby sheavesare acyclic with respect to Hom k T ( ρ ∗ G, • ) , for G ∈ Coh( T ) . Coherent sheaves alsogive a description of sheaves on X T : for each F ∈ Mod( k T ) there exists a filtrantinductive family { F i } i ∈ I such that F ≃ lim −→ i ρ ∗ F i . In fact, we have an equivalence HEAVES ON T -TOPOLOGIES between the categories Mod( k T ) and Ind(Coh( T )) the indization of the category Coh( T ) .All of the above results and methods are new in the o-minimal context and mostof them are new even in the semialgebraic case as well. On the other hand, wealso introduce a method for studying the category Mod( k T ) of sheaves on T -spaceswhich is the fundamental tool in the semialgebraic and o-minimal case, namely, weprove that as in [19] the category of sheaves on X T is equivalent to the category ofsheaves on a locally quasi-compact space e X T , the T -spectrum of X , which general-izes the notion of o-minimal spectrum as well as the real spectrum of commutativerings from real algebraic geometry. In particular, sheaves on the subanalytic siteare sheaves on the T -spectrum associated to the family of relatively compact sub-analytic subsets. Such a result was not present in [28].This theory can then be specialized to each of the examples we mentioned above:when T is the category of semialgebraic open subsets of a locally semialgebraic space X we obtain the constructions (and the generalizations) of results of [10], in par-ticular, when X is a Nash manifold, we recover the setting of [37]. When T is thecategory of relatively compact subanalytic open subsets of a real analytic manifold X we obtain the constructions and results of [28, 35]. Moreover, when T is thecategory of conic subanalytic open subsets of a real analytic manifold X we obtaina suitable category of conic subanalytic sheaves considered in [36]. Finally, when T is the category of definable open subsets of a locally definable space X we obtainin the definable case the constructions of [19] and we obtain new results in theo-minimal context generalizing those of the two previous cases.The paper is organized in the following way. In Section 1 we introduce the locallyweakly quasi-compact spaces and study some properties of sheaves on such spaces.The results of this section will be used in two crucial ways on the theory of sheaveson T -spaces, they are required to show that: (i) when a T -space X is locally weaklyquasi-compact, then there is a right adjoint ρ ! to the functor ρ − induced by thenatural functor of sites ρ : X → X T ; (ii) for a T -space X , the category of sheaveson X T is equivalent to the category of sheaves on a locally quasi-compact space e X T , the T -spectrum of X . In Section 2 we introduce the T -spaces and develop thetheory of sheaves on such spaces as already described above.1. Sheaves on locally weakly quasi-compact spaces
Let X be a non necessarily Hausdorff topological space. One denotes by Op( X ) the category whose objects are the open subsets of X and the morphisms are theinclusions. In this section we generalize some classical results about sheaves onlocally compact spaces. For classical sheaf theory our basic reference is [26]. Werefer to [39] for an introduction to sheaves on Grothendieck topologies.1.1. Locally weakly quasi-compact spaces.Definition 1.1.1.
An open subset U of X is said to be relatively weakly quasi-compact in X if, for any covering { U i } i ∈ I of X , there exists J ⊂ I finite, such that U ⊂ S i ∈ J U i . MÁRIO J. EDMUNDO AND LUCA PRELLI
We will write for short U ⊂⊂ X to say that U is a relatively weakly quasi-compactopen set in X , and we will call Op c ( U ) the subcategory of Op( U ) consisting of opensets V ⊂⊂ U . Note that, given V, W ∈ Op c ( U ) , then V ∪ W ∈ Op c ( U ) . Definition 1.1.2.
A topological space X is locally weakly quasi-compact if satisfiesthe following hypothesis for every U, V ∈ Op( X ) LWC1.
Every x ∈ U has a fundamental neighborhood system { V i } with V i ∈ Op c ( U ) . LWC2.
For every U ′ ∈ Op c ( U ) and V ′ ∈ Op c ( V ) one has U ′ ∩ V ′ ∈ Op c ( U ∩ V ) . LWC3.
For every U ′ ∈ Op c ( U ) there exists W ∈ Op c ( U ) such that U ′ ⊂⊂ W . Of course an open subset U of a locally weakly quasi-compact space X is alsoa locally weakly quasi-compact space. Let us consider some examples of locallyweakly quasi-compact spaces: Example 1.1.3.
A locally compact topological space X is a locally weakly quasi-compact. In this case, for U, V ∈ Op( X ) we have V ⊂⊂ U if and only if V isrelatively compact subset of U . Example 1.1.4.
Let X be a topological space with a basis of quasi-compact (i.e.each open covering admits a finite subcover) open subsets closed under taking finiteintersections. Then X is locally weakly quasi-compact and, for U, V ∈ Op( X ) wehave V ⊂⊂ U if and only if V is contained in a quasi-compact subset of U . In thissituation we have the following particular cases:(i) X is a Noetherian topological space (each open subset of X is quasi-compact). This includes in particular: (a) algebraic varieties over alge-braically closed fields; (b) complex varieties (reduced, irreducible complexanalytic spaces) with the Zariski topology.(ii) X is a spectral topological space (in addition: (i) X is quasi-compact; (ii) T ; (iii) every irreducible closed subset is the closure of a unique point). Thisincludes in particular: (a) real algebraic varieties over real closed fields; (b)the o-minimal spectrum of a definable space in some o-minimal structure.(iii) X is an increasing union of open spectral topological spaces X i ’s, i.e. X isthe space S i ∈ I X i . This space X has a basis of quasi-compact open subsetsclosed under taking finite intersections and in addition is: (i) not quasi-compact in general unless I is finite; (ii) T . This includes in particular:(a) the semialgebraic spectrum of locally semialgebraic space; (b) moregenerally, the o-minimal spectrum of a locally definable space in some o-minimal structure. Example 1.1.5.
Let E be a real vector bundle over a locally compact space Z endowed with the natural action µ of R + (the multiplication on the fibers). Let ˙ E = E \ Z , and for U ∈ Op( E ) set U Z = U ∩ Z and ˙ U = U ∩ ˙ E . Let E R + denotethe space E endowed with the conic topology i.e. open sets of E R + are open sets of E which are µ -invariant. With this topology E R + is a locally weakly quasi-compactspace and, for U, V ∈ Op( E R + ) we have V ⊂⊂ U if and only if V Z ⊂⊂ U Z in Z and ˙ V ⊂⊂ ˙ U in ˙ E R + (the later is ˙ E with the induced conic topology).1.2. Sheaves on locally weakly quasi-compact spaces.
Recall that X is a nonnecessarily Hausdorff topological space. HEAVES ON T -TOPOLOGIES Definition 1.2.1.
Let U = { U i } i ∈ I and U ′ = { U ′ j } j ∈ J be two families of opensubsets of X . One says that U ′ is a refinement of U if for each U i ∈ U there is U ′ j ∈ U ′ with U ′ j ⊆ U i . One denotes by
Cov( U ) the category whose objects are the coverings of U ∈ Op( X ) and the morphisms are the refinements, and by Cov f ( U ) its full subcate-gory consisting of finite coverings of U .Given V ∈ Op( U ) and S ∈ Cov( U ) , one sets S ∩ V = { U ∩ V } U ∈ S ∈ Cov( V ) . Definition 1.2.2.
The site X f on the topological space X is the category Op( X ) endowed with the following topology: S ⊂ Op( U ) is a covering of U if and only ifit has a refinement S f ∈ Cov f ( U ) . Definition 1.2.3.
Let
U, V ∈ Op( X ) with V ⊂ U . Given S = { U i } i ∈ I ∈ Cov( U ) and T = { V j } j ∈ J ∈ Cov( V ) , we write T ⊂⊂ S if T is a refinement of S ∩ V , and V j ⊂ U i if and only if V j ⊂⊂ U i . Let us recall the definitions of presheaf and sheaf on a site.
Definition 1.2.4.
A presheaf of k -modules on X is a functor Op( X ) op → Mod( k ) .A morphism of presheaves is a morphism of such functors. One denotes by Psh( k X ) the category of presheaves of k -modules on X . Let F ∈ Psh( k X ) , and let S ∈ Cov( U ) . One sets F ( S ) = ker (cid:16) Y W ∈ S F ( W ) ⇒ Y W ′ ,W ′′ ∈ S F ( W ′ ∩ W ′′ ) (cid:17) . Definition 1.2.5.
A presheaf F is separated (resp. is a sheaf ) if for any U ∈ Op( X ) and for any S ∈ Cov( U ) the natural morphism F ( U ) → F ( S ) is a monomor-phism (resp. an isomorphism). One denotes by Mod( k X ) the category of sheavesof k -modules on X . Let F ∈ Psh( k X ) , one defines the presheaf F + by setting F + ( U ) = lim −→ S ∈ Cov( U ) F ( S ) . One can show that F + is a separated presheaf and if F is a separated presheaf,then F + is a sheaf. Let F ∈ Psh( k X ) , the sheaf F ++ is called the sheaf associatedto the presheaf F . Lemma 1.2.6.
For F ∈ Psh( k X ) , and let U ∈ Op( X ) . If F is a sheaf on X f ,then for any V ∈ Op c ( U ) the morphism (1.1) F + ( U ) → F + ( V ) factors through F ( V ) . Proof.
Let S ∈ Cov( U ) , and set S ∩ V = { W ∩ V } W ∈ S . Since V ∈ Op c ( U ) ,there is a finite refinement T f ∈ Cov f ( V ) of S ∩ V . Then the morphism (1.1) is MÁRIO J. EDMUNDO AND LUCA PRELLI defined by F + ( U ) ≃ lim −→ S ∈ Cov( U ) F ( S ) → lim −→ S ∈ Cov( U ) F ( S ∩ V ) → lim −→ T f ∈ Cov f ( V ) F ( T f ) → lim −→ T ∈ Cov( V ) F ( T ) ≃ F + ( V ) . The result follows because F ( T f ) ≃ F ( V ) . (cid:3) Corollary 1.2.7.
With the hypothesis of Lemma 1.2.6, we consider two coverings S ∈ Cov( U ) and T ∈ Cov( V ) . If T ⊂⊂ S , then the morphism (1.2) F + ( S ) → F + ( T ) factors through F ( T ) . In particular, if T is finite, then the morphism (1.2) factorsthrough F ( V ) . From now on we will assume the following hypothesis:(1.3) the topological space X is locally weakly quasi-compact. Lemma 1.2.8.
Let U ∈ Op( X ) , and consider a subset V ⊂⊂ U . Then for any S f ∈ Cov f ( U ) there exists T f ∈ Cov f ( V ) with T f ⊂⊂ S f . Proof.
Let S f = { U i } . For each x ∈ U and U i ∋ x , consider a V x,i ∈ Op c ( U i ) containing x . Set V x = T i V x,i , the family { V x } forms a covering of U . Thenthere exists a finite subfamily { V j } containing V . By construction V j ∩ V ⊂⊂ U i whenever V j ⊂ U i . (cid:3) Lemma 1.2.9.
Let F ∈ Psh( k X ) , and let U ∈ Op( X ) . If F is a sheaf on X f , thenfor any V ∈ Op c ( U ) the morphism (1.4) F ++ ( U ) → F ++ ( V ) factors through F ( V ) . Proof.
Since X is locally weakly quasi-compact, there exists W ∈ Op c ( U ) with V ⊂⊂ W . As in Lemma 1.2.6 we obtain a diagram F ++ ( U ) / / (cid:15) (cid:15) F ++ ( W ) / / (cid:15) (cid:15) F ++ ( V )lim −→ S f ∈ Cov f ( W ) F + ( S f ) / / rrrrrrrrrrrr lim −→ T f ∈ Cov f ( V ) F + ( T f ) . rrrrrrrrrrrr Since X is locally weakly quasi-compact then by Lemma 1.2.8 for any S f ∈ Cov f ( W ) there exists T f ∈ Cov f ( V ) with T f ⊂⊂ S f . By Corollary 1.2.7 themorphism F + ( S f ) → F + ( T f ) HEAVES ON T -TOPOLOGIES factors through F ( T f ) ≃ F ( V ) . Then the morphism lim −→ S f ∈ Cov f ( W ) F + ( S f ) → lim −→ T f ∈ Cov f ( V ) F + ( T f ) factors through F ( V ) and the result follows. (cid:3) Corollary 1.2.10.
Let F ∈ Psh( k X ) . If F is a sheaf on X f , then: (i) for any V ∈ Op c ( X ) one has the isomorphism lim −→ U ⊃⊃ V F ( U ) ∼ → lim −→ U ⊃⊃ V F ++ ( U ) . (ii) for any U ∈ Op( X ) one has the isomorphism lim ←− V ⊂⊂ U F ( V ) ∼ → lim ←− V ⊂⊂ U F ++ ( V ) . Proof. (i) By Lemma 1.2.9 for each U ∈ Op( X ) with U ⊃⊃ V we have acommutative diagram F ++ ( U ) / / % % ▲▲▲▲▲▲▲▲▲▲ F ++ ( V ) F ( U ) O O / / F ( V ) O O This implies that the identity morphism of lim −→ U ⊃⊃ V F ( U ) factors through lim −→ U ⊃⊃ V F ++ ( U ) .On the other hand this also implies that the identity morphism of lim −→ U ⊃⊃ V F ++ ( U ) factors through lim −→ U ⊃⊃ V F ( U ) . Then lim −→ U ⊃⊃ V F ( U ) ∼ → lim −→ U ⊃⊃ V F ++ ( U ) .The proof of (ii) is similar. (cid:3) Corollary 1.2.11.
Let X be a quasi-compact and locally weakly quasi-compactspace, and let F ∈ Psh( k X ) . If F is a sheaf on X f , then the natural morphism (1.5) F ( X ) → F ++ ( X ) is an isomorphism. Proof.
It follows immediately from Corollary 1.2.10 (i) with V = X . (cid:3) Let { F i } i ∈ I be a filtrant inductive system in Mod( k X ) . One sets “lim −→ ” i F i = inductive limit in the category of presheaves, lim −→ i F i = inductive limit in the category of sheaves.Recall that lim −→ i F i = (“lim −→ ” i F i ) ++ . Proposition 1.2.12.
Let { F i } i ∈ I be a filtrant inductive system in Mod( k X ) andlet U ∈ Op( X ) . Then for any V ∈ Op c ( U ) the morphism Γ( U ; lim −→ i F i ) → Γ( V ; lim −→ i F i ) factors through lim −→ i Γ( V ; F i ) . MÁRIO J. EDMUNDO AND LUCA PRELLI
Proof.
By Lemma 1.2.9 it is enough to show that “lim −→ ” i F i is a sheaf on X f . Let U ∈ Op( X ) and S ∈ Cov f ( U ) . Since lim −→ i commutes with finite projective limitswe obtain the isomorphism (“lim −→ ” i F i )( S ) ≃ lim −→ i F i ( S ) . The result follows because F i ∈ Mod( k X ) for each i ∈ I . (cid:3) Corollary 1.2.13.
Let { F i } i ∈ I be a filtrant inductive system in Mod( k X ) . (i) For any V ∈ Op c ( X ) one has the isomorphism lim −→ U ⊃⊃ V,i Γ( U ; F i ) ∼ → lim −→ U ⊃⊃ V Γ( U ; lim −→ i F i ) . (ii) For any U ∈ Op( X ) one has the isomorphism lim ←− V ⊂⊂ U lim −→ i Γ( V ; F i ) ∼ → lim ←− V ⊂⊂ U Γ( V ; lim −→ i F i ) . Proof.
It follows from Corollary 1.2.10 with F = “lim −→ ” i F i . (cid:3) Corollary 1.2.14.
Let X be a quasi-compact and locally weakly quasi-compactspace. Then the natural morphism lim −→ i Γ( X ; F i ) → Γ( X ; lim −→ i F i ) is an isomorphism. Proof.
It follows from Corollary 1.2.11 with F = “lim −→ ” i F i . (cid:3) Example 1.2.15.
Let us consider the formula (1.6) lim −→ U ⊃⊃ V,i Γ( U ; F i ) ∼ → lim −→ U ⊃⊃ V Γ( U ; lim −→ i F i ) (i) Let X be a Noetherian space and let V ∈ Op( X ) . Then Γ( V ; F ) ≃ lim −→ U ⊃⊃ V Γ( U ; F ) ,since every open set is quasi-compact and (1.6) becomes lim −→ i Γ( V ; F i ) ≃ Γ( V ; lim −→ i F i ) . (ii) Assume that X has a basis of quasi-compact open subsets and let V ∈ Op c ( X ) . Then V is contained in a quasi-compact open subset of X and lim −→ U ⊃⊃ V Γ( U ; F ) ≃ lim −→ W ⊃ V Γ( W ; F ) , where W ranges through the family of quasi-compact subsets of X . (iii) Let X be a locally compact space and let V ∈ Op c ( X ) . Then Γ( V ; F ) ≃ lim −→ U ⊃⊃ V Γ( U ; F ) , and (1.6) becomes lim −→ i Γ( V ; F i ) ≃ Γ( V ; lim −→ i F i ) . (iv) Let E R + be a vector bundle endowed with the conic topology, and let V ∈ Op c ( E R + ) . Then lim −→ U ⊃⊃ V Γ( U ; F ) ≃ Γ( K ; F ) , where K is the union of theclosures of V Z in Z and ˙ V in ˙ E R + , and (1.6) becomes lim −→ i Γ( K ; F i ) ≃ Γ( K ; lim −→ i F i ) . Lemma 1.2.16.
Let F ∈ Psh( k X ) . Then we have the isomorphism lim ←− V ⊂⊂ X lim −→ V ⊂⊂ W F ( W ) ∼ → lim ←− V ⊂⊂ X F ( V ) . HEAVES ON T -TOPOLOGIES Proof.
The result follows since for each V ∈ Op c ( X ) there exists W ∈ Op c ( X ) such that V ⊂⊂ W since X is locally weakly compact. Let U, V ⊂⊂ X such that U ⊃⊃ V . The restriction morphism F ( U ) → F ( V ) factors through lim −→ W ⊃⊃ V F ( W ) .Taking the projective limit we obtain the result. (cid:3) Remark 1.2.17.
The notion of locally weakly quasi-compact can be extended tothe case of a site, by generalizing the hypothesis LWC1-LWC3. For our purpose weare interested in the topological setting and we refer to [34] for this approach. c-soft sheaves on locally weakly quasi-compact spaces.
Let X be alocally weakly quasi-compact space, and consider the category Mod( k X ) . Definition 1.3.1.
We say that a sheaf F on X is c-soft if the restriction morphism Γ( W ; F ) → lim −→ U ⊃⊃ V Γ( U ; F ) is surjective for each V, W ∈ Op c ( X ) with V ⊂⊂ W . It follows from the definition that injective sheaves and flabby sheaves are c-soft.Moreover, it follows from Corollary 1.2.13 that filtrant inductive limits of c-softsheaves are c-soft.
Proposition 1.3.2.
Let → F ′ → F → F ′′ → be an exact sequence in Mod( k X ) ,and assume that F ′ is c-soft. Then the sequence → lim −→ U ⊃⊃ V Γ( U ; F ′ ) → lim −→ U ⊃⊃ V Γ( U ; F ) → lim −→ U ⊃⊃ V Γ( U ; F ′′ ) → is exact for any V ∈ Op c ( X ) . Proof.
Let s ′′ ∈ lim −→ U ⊃⊃ V Γ( U ; F ′′ ) . Then there exists U ⊃⊃ V such that s ′′ is represented by s ′′ U ∈ Γ( U ; F ′′ ) . Let { U i } i ∈ I ∈ Cov( U ) such that there exists s i ∈ Γ( U i ; F ) whose image is s ′′ U | U i for each i . There exists W ∈ Op c ( U ) with W ⊃⊃ V , a finite covering { W j } nj =1 of W and a map ε : J → I of the index setssuch that W j ⊂⊂ U ε ( j ) . We may argue by induction on n . If n = 2 , set U i = U ε ( i ) , i = 1 , . Then ( s − s ) | U ∩ U belongs to Γ( U ∩ U ; F ′ ) , and its restriction defines anelement of lim −→ W ′ ⊃⊃ W ∩ W Γ( W ′ ; F ′ ) , hence it extends to s ′ ∈ Γ( U ; F ′ ) . By replacing s with s − s ′ on W we may assume that s = s on W ∩ W . Then there exists s ∈ Γ( W ∪ W ; F ) with s | W i = s i . Thus the induction proceeds. (cid:3) Proposition 1.3.3.
Let → F ′ → F → F ′′ → be an exact sequence in Mod( k X ) ,and assume F ′ , F c-soft. Then F ′′ is c-soft. Proof.
Let
V, W ∈ Op c ( X ) with V ⊂⊂ W and let us consider the diagrambelow Γ( W ; F ) α (cid:15) (cid:15) / / Γ( W ; F ′′ ) γ (cid:15) (cid:15) lim −→ U ⊃⊃ V Γ( U ; F ) β / / lim −→ U ⊃⊃ V Γ( U ; F ′′ ) . The morphism α is surjective since F is c-soft and β is surjective by Proposition1.3.2. Then γ is surjective. (cid:3) Proposition 1.3.4.
The family of c-soft sheaves is injective respect to the functor lim −→ U ⊃⊃ V Γ( U ; • ) for each V ∈ Op c ( X ) . Proof.
The family of c-soft sheaves contains injective sheaves, hence it is co-generating. Then the result follows from Propositions 1.3.2 and 1.3.3. (cid:3)
Assume the following hypothesis(1.7) X has a countable cover { U n } n ∈ N with U n ∈ Op c ( X ) , ∀ n ∈ N . Lemma 1.3.5.
Assume (1.7) . Then there exists a covering { V n } n ∈ N of X suchthat V n ⊂⊂ V n +1 and V n ∈ Op c ( X ) for each n ∈ N . Proof.
Let { U n } n ∈ N be a countable cover of X with U n ∈ Op c ( X ) for each n ∈ N . Set V = U . Given { V i } ni =1 with V i +1 ⊃⊃ V i , i = 1 , . . . , n − , letus construct V n +1 ⊃⊃ V n . Consider x / ∈ V n . Up to take a permutation of N we may assume x ∈ U n +1 . Since X is locally weakly quasi-compact there exists V n +1 ∈ Op c ( X ) such that V n ∪ U n +1 ⊂⊂ V n +1 . (cid:3) Proposition 1.3.6.
Assume (1.7) . Then the category of c-soft sheaves is injectiverespect to the functor Γ( X ; • ) . Proof.
Take an exact sequence → F ′ → F → F ′′ → , and suppose F ′ c-soft.By Lemma 1.3.5 there exists a covering { V n } n ∈ N of X such that V n ⊂⊂ V n +1 (and V n ∈ Op c ( X ) ) for each n ∈ N . All the sequences → lim −→ U n ⊃⊃ V n Γ( U n ; F ′ ) → lim −→ U n ⊃⊃ V n Γ( U n ; F ) → lim −→ U n ⊃⊃ V n Γ( U n ; F ′′ ) → are exact by Proposition 1.3.2, and the morphism lim −→ U n +1 ⊃⊃ V n +1 Γ( U n +1 ; F ′ ) → lim −→ U n ⊃⊃ V n Γ( U n ; F ′ ) is surjective for all n . Then by Proposition 1.12.3 of [26] the sequence → lim ←− n lim −→ U n ⊃⊃ V n Γ( U n ; F ′ ) → lim ←− n lim −→ U n ⊃⊃ V n Γ( U n ; F ) → lim ←− n lim −→ U n ⊃⊃ V n Γ( U n ; F ′′ ) → is exact. By Lemma 1.2.16 lim ←− n lim −→ U n ⊃⊃ V n Γ( U n ; G ) ≃ Γ( X ; G ) for any G ∈ Mod( k X ) and the result follows. (cid:3) Example 1.3.7.
Let us consider some particular cases(i) When X is Noetherian c-soft sheaves are flabby sheaves.(ii) When X has a basis of quasi-compact open subsets, then F ∈ Mod( k X ) isc-soft if the restriction morphism Γ( U ; F ) → Γ( V ; F ) is surjective, for anyquasi-compact open subsets U, V of X with U ⊇ V .(iii) When X is a locally compact space countable at infinity, then we recoverc-soft sheaves as in chapter II of [26].(iv) When E R + is a vector bundle endowed with the conic topology, then F ∈ Mod( k E R + ) is c-soft if the restriction morphism Γ( E R + ; F ) → Γ( K ; F ) issurjective, where K is defined as in Example 1.2.15.2. Sheaves on T -spaces. In the following we shall assume that k is a field and X is a topological space.Below we give the definition of T -space, adapting the construction of Kashiwara and HEAVES ON T -TOPOLOGIES Schapira [28]. We study the category of sheaves on X T generalizing results alreadyknown in the case of subanalytic sheaves. Then we prove that as in [19] the categoryof sheaves on X T is equivalent to the category of sheaves on a locally weakly-compact topological space e X T , the T -spectrum, which generalizes the notion ofo-minimal spectrum.2.1. T -sheaves. Let X be a topological space and let us consider a family T ofopen subsets of X . Definition 2.1.1.
The topological space X is a T -space if the family T satisfiesthe hypotheses below (2.1) (i) T is a basis for the topology of X , and ∅ ∈ T , (ii) T is closed under finite unions and intersections , (iii) every U ∈ T has finitely many T -connected components,where we define: • a T -subset is a finite Boolean combination of elements of T ; • a closed (resp. open) T -subset is a T -subset which is closed (resp. open)in X ; • a T -connected subset is a T -subset which is not the disjoint union of twoproper T -subsets which are closed and open. Example 2.1.2.
Let R = ( R, <, , , + , · ) be a real closed field. Let X be alocally semialgebraic space ([10, 11]) and consider the subfamily of Op( X ) definedby T = { U ∈ Op( X ) : U is semialgebraic } . The family T satisfy (2.1). Note alsothat the T -subsets of X are exactly the semialgebraic subsets of X ([7]). Example 2.1.3.
Let X be a real analytic manifold and consider the subfamily of Op( X ) defined by T = Op c ( X sa ) = { U ∈ Op( X sa ) : U is subanalytic relativelycompact } . The family T satisfies (2.1). Example 2.1.4.
Let X be a real analytic manifold endowed with a subanalyticaction µ of R + . In other words we have a subanalytic map µ : X × R + → X, which satisfies, for each t , t ∈ R + : ( µ ( x, t t ) = µ ( µ ( x, t ) , t ) ,µ ( x,
1) = x. Denote by X R + the topological space X endowed with the conic topology, i.e. U ∈ Op( X R + ) if it is open for the topology of X and invariant by the action of R + .We will denote by Op c ( X R + ) the subcategory of Op( X R + ) consisting of relativelyweakly quasi-compact open subsets. Consider the subfamily of Op( X R + ) definedby T = Op c ( X sa, R + ) = { U ∈ Op c ( X R + ) : U is subanalytic } . The family T satisfies(2.1). Example 2.1.5.
Let M = ( M, <, ( c ) ∈C , ( f ) f ∈F , ( R ) R ∈R ) be an arbitrary o-minimalstructure. Let X be a locally definable space ([3]) and consider the subfamily of Op( X ) defined by T = Op( X def ) = { U ∈ Op( X ) : U is definable } . The family T satisfies (2.1). Note also that (i) the T -subsets of X are exactly the definablesubsets of X (by the cell decomposition theorem in [13], see [19] Proposition 2.1). Let X be a T -space. One can endow the category T with a Grothendieck topol-ogy, called the T -topology, in the following way: a family { U i } i in T is a coveringof U ∈ T if it admits a finite subcover. We denote by X T the associated site, writefor short k T instead of k X T , and let ρ : X → X T be the natural morphism of sites.We have functors(2.2) Mod( k X ) ρ ∗ / / Mod( k T ) . ρ − o o Proposition 2.1.6.
We have ρ − ◦ ρ ∗ ≃ id . Equivalently, the functor ρ ∗ is fullyfaithful. Proof.
Let V ∈ Op( X ) and let G ∈ Mod( k T ) . Then ρ − G = ( ρ ← F ) ++ , where ρ ← G ∈ Psh( k X ) is defined by Op( X ) ∋ V lim −→ U ⊇ V,U ∈T G ( U ) . In particular, when U ∈ T , ρ ← G ( U ) = G ( U ) .Let F ∈ Mod( k X ) and denote by ι : Mod( k X ) → Psh( k X ) the forgetful functor.The adjunction morphism ρ ← ◦ ρ ∗ → id in Psh( k X ) defines ρ ← ρ ∗ F → ιF . Thismorphism is an isomorphism on T , since ρ ← ρ ∗ F ( U ) ≃ ρ ∗ F ( U ) ≃ F ( U ) ≃ ιF ( U ) when U ∈ T . By (2.1) (i) T forms a basis for the topology of X , hence we get anisomorphism ρ − ρ ∗ F ≃ ( ρ ← ρ ∗ F ) ++ ≃ ( ιF ) ++ ≃ F and the result follows. (cid:3) Proposition 2.1.7.
Let { F i } i ∈ I be a filtrant inductive system in Mod( k T ) and let U ∈ T . Then lim −→ i Γ( U ; F i ) ∼ → Γ( U ; lim −→ i F i ) . Proof.
Denote by “lim −→ ” i F i the presheaf V lim −→ i Γ( V ; F i ) on X T . Let U ∈ T and let S be a finite covering of U . Since lim −→ i commutes with finite projective limitswe obtain the isomorphism (“lim −→ ” i F i )( S ) ∼ → lim −→ i F i ( S ) and F i ( U ) ∼ → F i ( S ) since F i ∈ Mod( k T ) for each i . Moreover the family of finite coverings of U is cofinal in Cov( U ) . Hence “lim −→ ” i F i ∼ → (“lim −→ ” i F i ) + . Applying once again the functor ( · ) + weget “lim −→ ” i F i ≃ (“lim −→ ” i F i ) + ≃ (“lim −→ ” i F i ) ++ ≃ lim −→ i F i . Hence applying the functor Γ( U ; · ) we obtain the isomorphism lim −→ i Γ( U ; F i ) ∼ → Γ( U ; lim −→ i F i ) for each U ∈ T . (cid:3) Proposition 2.1.8.
Let F be a presheaf on X T and assume that (i) F ( ∅ ) = 0 , (ii) For any
U, V ∈ T the sequence → F ( U ∪ V ) → F ( U ) ⊕ F ( V ) → F ( U ∩ V ) isexact.Then F ∈ Mod( k T ) . HEAVES ON T -TOPOLOGIES Proof.
Let U ∈ T and let { U j } nj =1 be a finite covering of U . Set for short U ij = U i ∩ U j . We have to show the exactness of the sequence → F ( U ) → ⊕ ≤ k ≤ n F ( U k ) → ⊕ ≤ i Example 2.1.9. Let us see some examples of sites associated to T -topologies:(i) When T is the family of Example 2.1.2 we obtain the semi-algebraic site of[10, 11].(ii) When T is the family of Example 2.1.3 we obtain the subanalytic site X sa of [28, 35].(iii) When T is the family of Example 2.1.4 we obtain the conic subanalytic siteof [36].(iv) When T is the family of Example 2.1.5 we obtain the o-minimal site X def .It is the one considered in [19] when X is a definable space.2.2. T -coherent sheaves. Let us consider the category Mod( k X ) of sheaves of k X -modules on X , and denote by K the subcategory whose objects are the sheaves F = ⊕ i ∈ I k U i with I finite and U i ∈ T for each i . The following definition isextracted from [28]. Definition 2.2.1. Let T be a subfamily of Op( X ) satisfying (2 . , and let F ∈ Mod( k X ) . (i) F is T -finite if there exists an epimorphism G ։ F with G ∈ K . (ii) F is T -pseudo-coherent if for any morphism ψ : G → F with G ∈ K , ker ψ is T -finite. (iii) F is T -coherent if it is both T -finite and T -pseudo-coherent. Remark that (ii) is equivalent to the same condition with “ G is T -finite" insteadof “ G ∈ K ". One denotes by Coh( T ) the full subcategory of Mod( k X ) consistingof T -coherent sheaves. It is easy (see [29], Exercise 8.23) to prove that Coh( T ) isadditive and stable by kernels. Lemma 2.2.2. Let F, G ∈ K . Then, given ϕ : F → G , we have ker ϕ ∈ K . Proof. We have F = ⊕ li =1 k W i , G = ⊕ mj =1 k W ′ j . Composing with the projection p j , j = 1 , ..., m on each factor of G , ker ϕ will be the intersection of the ker p j ◦ ϕ sothat, if each one has the desired form, the same will happen to their intersection.Therefore it is sufficient to assume m = 1 , let us say, G = k W . A morphism ϕ : F → G is then defined by a sequence v = ( v , . . . , v l ) , where v i is the image by ϕ of the section of k W i defined by on W i , so v i = 0 if W i W . More precisely,if s = ( s , ..., s l ) is a germ of F in y , we have ϕ ( s , ..., s l ) = P li =1 v iy s i . So, given s = ( s , ..., s l ) ∈ ker ϕ , if, for a given i , we have v iy s i = 0 , then s defines a germ of H i =: ⊕ i ′ = i k W i ′ ∩ W i in y .Accordingly, ker ϕ ≃ ⊕ li =1 H i . (cid:3) Therefore, according to the definition of Coh( T ) and to Lemma 2.2.2, any F ∈ Coh( T ) admits a finite resolution K • := 0 → K → · · · → K n → F → consisting of objects belonging to K . Proposition 2.2.3. Let U ∈ T and consider the constant sheaf k U X T ∈ Mod( k T ) .We have k U X T ≃ ρ ∗ k U . Proof. Let F be the presheaf on X T defined by F ( V ) = k if V ⊂ U , F ( V ) = 0 otherwise. This is a separated presheaf and k U X T = F ++ . Moreover there is aninjective arrow F ( V ) ֒ → ρ ∗ k U ( V ) for each V ∈ Op( X T ) . Hence F ++ ֒ → ρ ∗ k U sincethe functor ( · ) ++ is exact. Let S ⊆ T be the sub-family of T -connected elements.Then S forms a basis for the Grothendieck topology of X T . For each W ∈ S we have F ( W ) ≃ ρ ∗ k U ( W ) ≃ k if W ⊂ U and F ( W ) = 0 otherwise. Then F ++ ≃ ρ ∗ k U . (cid:3) Proposition 2.2.4. The restriction of ρ ∗ to Coh( T ) is exact. Proof. Let us consider an epimorphism G ։ F in Coh( T ) , we have to provethat ψ : ρ ∗ G → ρ ∗ F is an epimorphism. Let U ∈ T and let = s ∈ Γ( U ; ρ ∗ F ) ≃ Hom k X ( k U , F ) (by adjunction). Set G ′ = G × F k U = ker( G ⊕ k U ⇒ F ) . Then G ′ ∈ Coh( T ) and moreover G ′ ։ k U . There exists a finite { U i } i ∈ I ⊂ T of T -connected elements such that ⊕ i k U i ։ G ′ . The composition k U i → G ′ → k U isgiven by the multiplication by a i ∈ k . Set I = { k U i ; a i = 0 } , we may assume a i = 1 . We get a diagram ⊕ i ∈ I k U i $ $ $ $ ❍❍❍❍❍❍❍❍❍ / / G ′ (cid:15) (cid:15) (cid:15) (cid:15) / / G (cid:15) (cid:15) (cid:15) (cid:15) k U s / / F. The composition k U i → G ′ → G defines t i ∈ Hom k X ( k U i , G ) ≃ Γ( U i ; ρ ∗ G ) . Hencefor each s ∈ Γ( U ; ρ ∗ F ) there exists a finite covering { U i } of U and t i ∈ Γ( U i ; ρ ∗ G ) such that ψ ( t i ) = s | U i . This means that ψ is surjective. (cid:3) Notation 2.2.5. Since the functor ρ ∗ is fully faithfull and exact on Coh( T ) , wewill often identify Coh( T ) with its image in Mod( k T ) and write F instead of ρ ∗ F for F ∈ Coh( T ) . HEAVES ON T -TOPOLOGIES Theorem 2.2.6. The following hold: (i) The category Coh( T ) is stable by finite sums, kernels, cokernels and exten-sions in Mod( k T ) . (ii) The category Coh( T ) is stable by • ⊗ k T • in Mod( k T ) . Proof. (i) The result follows from a general result of homological algebra of[27], Appendix A.1. With the notations of [27] let P be the set of finite families ofelements of T , for U = { U i } i ∈ I ∈ P set L ( U ) = ⊕ i k U i , for V = { V j } j ∈ J ∈ P set Hom P ( U , V ) = Hom k T ( L ( U ) , L ( V )) = ⊕ i ⊕ j Hom k T ( k U i , k V j ) and for F ∈ Mod( k T ) set H ( U , F ) = Hom k T ( L ( U ) , F ) = ⊕ i Hom k T ( k U i , F ) . By Proposition A.1 of [27] in order to prove (i) it is enough to prove the properties(A.1)-(A.4) below:(A.1) For any U = { U i } ∈ P the functor H ( U , • ) is left exact in Mod( k T ) .(A.2) For any morphism g : V → W in P , there exists a morphism f : U → V in P such that U f → V g → W is exact.(A.3) For any epimorphism f : F → G in Mod( k T ) , U ∈ P and ψ ∈ H ( U , G ) ,there exists V ∈ P and an epimorphism g ∈ Hom P ( V , U ) and ϕ ∈ H ( V , F ) such that ψ ◦ g = f ◦ ϕ .(A.4) For any U , V ∈ P and ψ ∈ H ( U , L ( V )) there exists W ∈ P and an epi-morphism f ∈ Hom P ( W , U ) and a morphism g ∈ Hom P ( W , U ) such that L ( g ) = ψ ◦ f in Hom k T ( L ( W ) , L ( V )) .It is easy to check that the axioms (A.1)-(A.4) are satisfied.(ii) Let F ∈ Coh( T ) . Then F has a resolution ⊕ j ∈ J k U j → ⊕ i ∈ I k U i → F → with I and J finite. Let V ∈ T . The sequence ⊕ j ∈ J k V ∩ U j → ⊕ i ∈ T k V ∩ U i → F V → is exact. Then it follows from (i) that F V is coherent. Let G ∈ Coh( T ) . Thesequence ⊕ j ∈ J G U j → ⊕ i ∈ I G U i → G ⊗ k T F → is exact. The sheaves G U i and G U j are coherent for each i ∈ I and each j ∈ J .Hence it follows by (i) that G ⊗ k T F is coherent as required. (cid:3) Corollary 2.2.7. The following hold: (i) The category Coh( T ) is stable by finite sums, kernels, cokernels in Mod( k X ) . (ii) The category Coh( T ) is stable by • ⊗ k X • in Mod( k X ) . Proof. (i) The stability under finite sums and kernels is easy, see [29], Exercise8.23. Let F, G ∈ Coh( T ) and let ϕ : F → G be a morphism in Mod( k X ) . Then ρ ∗ ( ϕ ) is a morphism in Mod( k T ) and coker( ρ ∗ ϕ ) ∈ Coh( T ) by Theorem 2.2.6.We have coker( ρ ∗ ϕ ) ≃ ρ ∗ coker ϕ since ρ ∗ is exact on Coh( T ) by Proposition 2.2.4.Composing with ρ − and applying Proposition 2.1.6 we obtain coker ϕ ∈ Coh( T ) .(ii) The proof of the stability by • ⊗ k X • is similar to that of Theorem 2.2.6. (cid:3) Theorem 2.2.8. (i) Let G ∈ Coh( T ) and let { F i } be a filtrant inductive systemin Mod( k T ) . Then we have the isomorphism lim −→ i Hom k T ( ρ ∗ G, F i ) ∼ → Hom k T ( ρ ∗ G, lim −→ i F i ) . (ii) Let F ∈ Mod( k T ) . There exists a small filtrant inductive system { F i } i ∈ I in Coh( T ) such that F ≃ lim −→ i ρ ∗ F i . Proof. (i) There exists an exact sequence G → G → G → with G , G finite direct sums of constant sheaves k U with U ∈ T . Since ρ ∗ is exact on Coh( T ) and commutes with finite sums, by Proposition 2.2.3 we are reduced to prove theisomorphism lim −→ i Γ( U ; F i ) ∼ → Γ( U ; lim −→ i F i ) . Then the result follows from Proposition2.1.7.(ii) Let F ∈ Mod( k T ) , and define I := { ( U, s ) : U ∈ T , s ∈ Γ( U ; F ) } G := ⊕ ( U,s ) ∈ I ρ ∗ k U The morphism ρ ∗ k U → F , where the section ∈ Γ( U ; k U ) is sent to s ∈ Γ( U ; F ) defines un epimorphism ϕ : G → F . Replacing F by ker ϕ we construct a sheaf G = ⊕ ( V,t ) ∈ I ρ ∗ k V and an epimorphism G ։ ker ϕ . Hence we get an exactsequence G → G → F → . For J ⊂ I set for short G J = ⊕ ( U,s ) ∈ J ρ ∗ k U anddefine similarly G J . Set J = { ( J , J ); J k ⊂ I k , J k is finite and im ϕ | G J ⊂ G J } . The category J is filtrant and F ≃ lim −→ ( J ,J ) ∈ J coker( G J → G J ) . (cid:3) Corollary 2.2.9. Let G ∈ Coh( T ) and let { F i } be a filtrant inductive system in Mod( k T ) . Then we have an isomorphism lim −→ i H om k T ( G, F i ) ∼ → H om k T ( G, lim −→ i F i ) . Proof. Let U ∈ T . We have the chain of isomorphisms Γ( U ; lim −→ i H om k T ( G, F i )) ≃ lim −→ i Γ( U ; H om k T ( G, F i )) ≃ lim −→ i Hom k T ( G U , F i )) ≃ Hom k T ( G U , lim −→ i F i )) ≃ Γ( U ; H om k T ( G, lim −→ i F i )) , where the first and the third isomorphism follow from Theorem 2.2.8 (i). the factthat G U ∈ Coh( T ) follows from Theorem 2.2.6 (ii). (cid:3) As in [28], we can define the indization of the category Coh( T ) . Recall that thecategory Ind(Coh( T )) , of ind- T -coherent sheaves is the category whose objects arefiltrant inductive limits of functors lim −→ i Hom Coh( T ) ( • , F i ) ( “lim −→ ” i F i for short) , HEAVES ON T -TOPOLOGIES where F i ∈ Coh( T ) , and the morphisms are the natural transformations of suchfunctors. Note that since Coh( T ) is a small category, Ind(Coh( T )) is equivalent tothe category of k -additive left exact contravariant functors from Coh( T ) to Mod( k ) . See [29] for a complete exposition on indizations of categories. We can extend thefunctor ρ ∗ : Coh( T ) → Mod( k T ) to λ : Ind(Coh( T )) → Mod( k T ) by setting λ (“lim −→ ” i F i ) := lim −→ i ρ ∗ F i . Corollary 2.2.10. The functor λ : Ind(Coh( T )) → Mod( k T ) is an equivalence ofcategories. Proof. Let F = “lim −→ ” j F j , G = “lim −→ ” i G i ∈ I(Coh( T )) . By Theorem 2.2.8 (i) andthe fact that the functor ρ ∗ is fully faithfull on Coh( T ) we have Hom k T ( λ ( F ) , λ ( G )) ≃ Hom k T (lim −→ j ρ ∗ F j , lim −→ i ρ ∗ G i ) ≃ lim ←− j lim −→ i Hom k T ( ρ ∗ F j , ρ ∗ G i ) ≃ lim ←− j lim −→ i Hom Coh( T ) ( F j , G i ) ≃ Hom Ind(Coh( T )) ( F, G ) , hence λ is fully faithful. By Theorem 2.2.8 (ii) for each F ∈ Mod( k T ) there exists G = “lim −→ ” i F i ∈ Ind(Coh( T )) such that λ ( G ) = lim −→ i ρ ∗ F i ≃ F , hence λ is essentiallysurjective. (cid:3) T -flabby sheaves.Definition 2.3.1. We say that an object F ∈ Mod( k T ) is T -flabby if for each U, V ∈ T with V ⊇ U the restriction morphism Γ( V ; F ) → Γ( U ; F ) is surjective. Remark 2.3.2. Remark that the category Mod( k T ) is a Grothendieck category,hence it has enough injectives. It follows from the definition that injective sheavesare T -flabby. This implies that the family of T -flabby objects is cogenerating in Mod( k T ) . Example 2.3.3. Let us see some examples of T -flabby sheaves:(i) When T is the family of Example 2.1.2 we obtain the family of sa -flabbyobjects of [10].(ii) When T is the family of Example 2.1.3 we obtain the family of quasi-injective objects of [35]. Proposition 2.3.4. The following hold: (i) Let F i be a filtrant inductive system of T -flabby sheaves. Then lim −→ i F i is T -flabby. (ii) Products of T -flabby objects are T -flabby. Proof. We will only prove (i) since the proof of (ii) is similar since takingproducts is exact and commutes with taking sections. Let U ∈ T . Then for each i the restriction morphism Γ( V ; F i ) → Γ( U ; F i ) is surjective. Applying the exact lim −→ i and using Proposition 2.1.7, the morphism Γ( V ; lim −→ i F i ) ≃ lim −→ i Γ( V ; F i ) → lim −→ i Γ( U ; F i ) ≃ Γ( U ; lim −→ i F i ) is surjective. (cid:3) Proposition 2.3.5. The full additive subcategory of Mod( k T ) of T -flabby objectis Γ( U ; • ) -injective for every U ∈ T , i.e.: (i) For every F ∈ Mod( k T ) there exists a T -flabby object F ′ ∈ Mod( k T ) andan exact sequence → F → F ′ . (ii) Let → F ′ → F → F ′′ → be an exact sequence in Mod( k T ) and assumethat F ′ is T -flabby. Then the sequence → Γ( U ; F ′ ) → Γ( U ; F ) → Γ( U ; F ′′ ) → is exact. (iii) Let F ′ , F, F ′′ ∈ Mod( k T ) , and consider the exact sequence → F ′ → F → F ′′ → . Suppose that F ′ is T -flabby. Then F is T -flabby if and only if F ′′ is T -flabby. Proof. (i) It follows from the definition that injective sheaves are T -flabby. So(i) holds since it is true for injective sheaves. Indeed, as a Grothendieck category, Mod( k T ) admits enough injectives.(ii) Let s ′′ ∈ Γ( U ; F ′′ ) , and let { V i } ni =1 ∈ Cov( U ) be such that there exists s i ∈ Γ( V i ; F ) whose image is s ′′ | V i . For n ≥ on V ∩ V s − s defines a sectionof Γ( V ∩ V ; F ′ ) which extends to s ′ ∈ Γ( U ; F ′ ) since F ′ is T -flabby. Replace s with s − s ′ (identifying s ′ with it’s image in F ). We may suppose that s = s on V ∩ V . Then there exists t ∈ Γ( V ∪ V , F ) such that t | V i = s i , i = 1 , . Thus theinduction proceeds.(iii) Let U, V ∈ T with V ⊇ U and let us consider the diagram below / / Γ( V ; F ′ ) α (cid:15) (cid:15) / / Γ( V ; F ) β (cid:15) (cid:15) / / Γ( V ; F ′′ ) γ (cid:15) (cid:15) / / / / Γ( U ; F ′ ) / / Γ( U ; F ) / / Γ( U ; F ′′ ) / / where the row are exact by (ii) and the morphism α is surjective since F ′ is T -flabby.It follows from the five lemma that β is surjective if and only if γ is surjective. (cid:3) Theorem 2.3.6. Let F ∈ Mod( k T ) . Then the following hold: (i) F is T -flabby if and only if the functor Hom k T ( • , F ) is exact on Coh( T ) . (ii) If F is T -flabby then the functor H om k T ( • , F ) is exact on Coh( T ) . Proof. (i) is a consequence of a general result of homological algebra (see Theo-rem 8.7.2 of [29]). For (ii), let F ∈ Mod( k T ) be T -flabby. There is an isomorphismof functors Γ( U ; H om k T ( • , F )) ≃ Hom k T (( • ) U , F ) for each U ∈ T . By Theorem 2.2.6 and (i) the functor Hom k T (( • ) U , F ) is exact on Coh( T ) and so the functor H om k T ( • , F ) is also exact on Coh( T ) . (cid:3) HEAVES ON T -TOPOLOGIES Theorem 2.3.7. Let G ∈ Coh( T ) . Then the following hold: (i) The family of T -flabby sheaves is injective with respect to the functor Hom k T ( G, • ) . (ii) The family of T -flabby sheaves is injective with respect to the functor H om k T ( G, • ) . Proof. (i) Let G ∈ Coh( T ) . Let → F ′ → F → F ′′ → be an exact sequencein Mod( k T ) and assume that F ′ is T -flabby. We have to show that the sequence → Hom k T ( G, F ′ ) → Hom k T ( G, F ) → Hom k T ( G, F ′′ ) → is exact.There is an epimorphism ϕ : ⊕ i ∈ I k U i → G where I is finite and U i ∈ T foreach i ∈ I . The sequence → ker ϕ → ⊕ i ∈ I k U i → G → is exact. We set forshort G = ker ϕ and G = ⊕ i ∈ I k U i . We get the following diagram where the firstcolumn is exact by Theorem 2.3.6 (i) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / Hom k T ( G, F ′ ) (cid:15) (cid:15) / / Hom k T ( G, F ) (cid:15) (cid:15) / / Hom k T ( G, F ′′ ) (cid:15) (cid:15) / / / / Hom k T ( G , F ′ ) (cid:15) (cid:15) / / Hom k T ( G , F ) (cid:15) (cid:15) / / Hom k T ( G , F ′′ ) (cid:15) (cid:15) / / / / Hom k T ( G , F ′ ) (cid:15) (cid:15) / / Hom k T ( G , F ) (cid:15) (cid:15) / / Hom k T ( G , F ′′ ) (cid:15) (cid:15) / / 00 0 0 The second row is exact by Proposition 2.3.5 (ii), hence the top row is exact bythe snake lemma.(ii) Let G ∈ Coh( T ) . It is enough to check that for each U ∈ T and each exactsequence → F ′ → F → F ′′ → with F ′ T -flabby, the sequence → Γ( U ; H om k T ( G, F ′ )) → Γ( U ; H om k T ( G, F )) → Γ( U ; H om k T ( G, F ′′ )) → is exact. We have Γ( U, H om k T ( G, • )) ≃ Hom k T ( G U , • ) , and, by (i) and the fact that G U ∈ Coh( T ) (Theorem 2.2.6 (ii)), T -flabby objectsare injective with respect to the functor Hom k T ( G U , • ) for each G ∈ Coh( T ) , andfor each U ∈ T . (cid:3) Proposition 2.3.8. Let F ∈ Mod( k T ) . Then F is T -flabby if and only if H om k T ( G, F ) is T -flabby for each G ∈ Coh( T ) . Proof. Suppose that F is T -flabby, and let G ∈ Coh( T ) . We have Hom k T ( • , H om k T ( G, F )) ≃ Hom k T ( • ⊗ k T G, F ) and Hom k T ( • ⊗ k T G, F ) is exact on Coh( T ) by Theorems 2.2.6 (ii) and 2.3.6 (i). Suppose that H om k T ( G, F ) is T -flabby for each G ∈ Coh( T ) . Let U, V ∈ T with V ⊇ U . For each W ∈ T the morphism Γ( V ; Γ W F ) → Γ( U ; Γ W F ) is surjective.Hence the morphism Γ( V ; F ) ≃ Γ( V ; Γ V F ) → Γ( U ; Γ V F ) ≃ Γ( U ; F ) is surjective. (cid:3) Let us consider the following subcategory of Mod( k T ) : P X T := { G ∈ Mod( k T ); G is Hom k T ( • , F ) -acyclic for each F ∈ F X T } , where F X T is the family of T -flabby objects of Mod( k T ) .This category is generating. In fact if { U j } j ∈ J ∈ T , then ⊕ j ∈ J k U j ∈ P X T byTheorem 2.3.7 (and the fact that ΠHom k T ( • , • ) ≃ Hom k T ( ⊕• , • ) and products are exact). Moreover P X T is stable by • ⊗ k T K , where K ∈ Coh( T ) .In fact if G ∈ P X T and F ∈ F X T we have Hom k T ( G ⊗ k T K, F ) ≃ Hom k T ( G, H om k T ( K, F )) and H om k T ( K, F ) is T -flabby by Proposition 2.3.8. In particular, if G ∈ P X T then G U ∈ P X T for every U ∈ Op( X T ) . Theorem 2.3.9. The category ( P opX T , F X T ) is injective with respect to the functors Hom k T ( • , • ) and H om k T ( • , • ) . Proof. (i) Let G ∈ P X T and consider an exact sequence → F ′ → F → F ′′ → with F ′ T -flabby. We have to prove that the sequence → Hom k T ( G, F ′ ) → Hom k T ( G, F ) → Hom k T ( G, F ′′ ) → is exact. Since the functor Hom k T ( G, • ) is acyclic on T -flabby sheaves we obtainthe result.Let F be T -flabby, and let → G ′ → G → G ′′ → be an exact sequence on P X T . Since the objects of P X T are Hom k T ( • , F ) -acyclic the sequence → Hom k T ( G ′′ , F ) → Hom k T ( G, F ) → Hom k T ( G ′ , F ) → is exact.(ii) Let G ∈ P X T , and let → F ′ → F → F ′′ → be an exact sequence with F ′ T -flabby. We shall show that for each U ∈ T the sequence → Γ( U ; H om k T ( G, F ′ )) → Γ( U ; H om k T ( G, F )) → Γ( U ; H om k T ( G, F ′′ )) → is exact. This is equivalent to show that for each U ∈ T the sequence → Hom k T ( G U , F ′ ) → Hom k T ( G U , F ) → Hom k T ( G U , F ′′ ) → is exact. This follows since G U ∈ P X T as we saw above. The proof of the exactnessin P opX T is similar. (cid:3) Proposition 2.3.10. Let F ∈ Mod( k T ) . The following assumptions are equivalent (i) F is T -flabby, (ii) F is Hom k T ( G, • ) -acyclic for each G ∈ Coh( T ) , (iii) R Hom k T ( k V \ U , F ) = 0 for each U, V ∈ T . HEAVES ON T -TOPOLOGIES Proof. (i) ⇒ (ii) follows from Theorem 2.3.7, (ii) ⇒ (iii) setting G = k V \ U with U, V ∈ T , (iii) ⇒ (i) since if R Hom k T ( k V \ U , F ) = 0 for each U, V ∈ T with V ⊇ U , then the restriction Γ( V ; F ) → Γ( U ; F ) is surjective. (cid:3) Let X, Y be two topological spaces and let T ⊂ Op( X ) , T ′ ⊂ Op( Y ) satisfy(2.1). Let f : X → Y be a continuous map. If f − ( T ′ ) ⊂ T then f defines amorphism of sites f : X T → Y T ′ . Proposition 2.3.11. Let f : X T → Y T ′ be a morphism of sites. T -flabby sheavesare injective with respect to the functor f ∗ . The functor f ∗ sends T -flabby sheavesto T ′ -flabby sheaves. Proof. Let us consider V ∈ T ′ . There is an isomorphism of functors Γ( V ; f ∗ • ) ≃ Γ( f − ( V ); • ) . It follows from Proposition 2.3.5 that T -flabby are injective withrespect to the functor Γ( f − ( V ); • ) for any V ∈ T ′ .Let F be T -flabby and let U, V ∈ T ′ with V ⊃ U . Then the morphism Γ( V ; f ∗ F ) = Γ( f − ( V ); F ) → Γ( f − ( U ); F ) = Γ( U ; f ∗ F ) is surjective. (cid:3) T -sheaves on locally weakly quasi-compact spaces. Assume that X isa locally weakly quasi-compact space. Lemma 2.4.1. For each U ∈ Op c ( X ) there exists V ∈ T such that U ⊂⊂ V ⊂⊂ X . Proof. Since X is locally weakly quasi-compact we may find W ∈ Op c ( X ) suchthat U ⊂⊂ W . By (2.1) (i) we may find a covering { W i } i ∈ I of X with W i ∈ T and W i ⊂⊂ X for each i ∈ I . Then there exists a finite family { W j } ℓj =1 whose union V = S ℓj =1 W j contains W . Then V ∈ T and U ⊂⊂ V ⊂⊂ X . (cid:3) When X is locally weakly quasi-compact we can construct a left adjoint to thefunctor ρ − . Proposition 2.4.2. Let F ∈ Mod( k T ) , and let U ∈ Op( X ) . Then Γ( U ; ρ − F ) ≃ lim ←− V ⊂⊂ U,V ∈T Γ( V ; F ) Proof. By Theorem 2.2.8 we may assume F = lim −→ i ρ ∗ F i , with F i ∈ Coh( T ) .Then ρ − F ≃ lim −→ i ρ − ρ ∗ F i ≃ lim −→ i F i . We have the chain of isomorphisms Γ( U ; ρ − F ) ≃ lim ←− V ⊂⊂ U,V ∈T lim −→ V ⊂⊂ W Γ( W ; ρ − F ) ≃ lim ←− V ⊂⊂ U,V ∈T lim −→ V ⊂⊂ W Γ( W ; lim −→ i ρ − ρ ∗ F i ) ≃ lim ←− V ⊂⊂ U,V ∈T lim −→ V ⊂⊂ W,i Γ( W ; ρ − ρ ∗ F i ) ≃ lim ←− V ⊂⊂ U,V ∈T lim −→ i Γ( V ; ρ − ρ ∗ F i ) ≃ lim ←− V ⊂⊂ U,V ∈T lim −→ i Γ( V ; ρ ∗ F i ) ≃ lim ←− V ⊂⊂ U,V ∈T Γ( V ; F ) , where the first and the fourth isomorphisms follow from Lemma 1.2.16, the thirdisomorphism is a consequence of Corollary 1.2.13, and the last isomorphism followsfrom Proposition 2.1.7. (cid:3) Proposition 2.4.3. The functor ρ − admits a left adjoint, denoted by ρ ! . It sat-isfies (i) for F ∈ Mod( k X ) and U ∈ T , ρ ! F is the sheaf associated to the presheaf U lim −→ U ⊂⊂ V Γ( V ; F ) , (ii) For U ∈ Op( X ) one has ρ ! k U ≃ lim −→ V ⊂⊂ U,V ∈T k V . Proof. Let e F ∈ Psh( k T ) be the presheaf U lim −→ U ⊂⊂ V Γ( V ; F ) , and let G ∈ Mod( k T ) . We will construct morphisms Hom Psh( k T ) ( e F , G ) ξ / / Hom k X ( F, ρ − G ) ϑ o o . To define ξ , let ϕ : e F → G and U ∈ Op( X ) . Then the morphism ξ ( ϕ )( U ) : F ( U ) → ρ − G ( U ) is defined as follows F ( U ) ≃ lim ←− V ⊂⊂ U,V ∈T lim −→ V ⊂⊂ W F ( W ) ϕ −→ lim ←− V ⊂⊂ U,V ∈T G ( V ) ≃ ρ − G ( U ) . On the other hand, let ψ : F → ρ − G and U ∈ T . Then the morphism ϑ ( ψ )( U ) : e F ( U ) → G ( U ) is defined as follows e F ( U ) ≃ lim −→ U ⊂⊂ V ∈T F ( V ) ψ −→ lim −→ U ⊂⊂ V ∈T ρ − G ( V ) → G ( U ) . By construction one can check that the morphism ξ and ϑ are inverse to eachothers. Then (i) follows from the chain of isomorphisms Hom Psh( k T ) ( e F , G ) ≃ Hom k T ( e F ++ , G ) ≃ Hom k T ( e F ++ , G ) . To show (ii), consider the following sequence of isomorphisms Hom k T ( ρ ! k U , F ) ≃ Hom k X ( k U , ρ − F ) ≃ lim ←− V ⊂⊂ U,V ∈T Hom k T ( k V , F ) ≃ Hom k T ( lim −→ V ⊂⊂ U,V ∈T k V , F ) , where the second isomorphism follows from Proposition 2.4.2. (cid:3) Proposition 2.4.4. The functor ρ ! is exact and commutes with lim −→ and ⊗ . Proof. It follows by adjunction that ρ ! is right exact and commutes with lim −→ ,so let us show that it is also left exact. With the notations of Proposition 2.4.3,let F ∈ Mod( k X ) , and let e F ∈ Psh( k T ) be the presheaf U lim −→ U ⊂⊂ V Γ( V ; F ) . Then ρ ! F ≃ e F ++ , and the functors F e F and G G ++ are left exact.Let us show that ρ ! commutes with ⊗ . Let F, G ∈ Mod( k X ) , the morphism lim −→ U ⊂⊂ V F ( V ) ⊗ k lim −→ U ⊂⊂ V G ( V ) → lim −→ U ⊂⊂ V ( F ( V ) ⊗ k G ( V )) defines a morphism in Mod( k T ) ρ ! F ⊗ k T ρ ! G → ρ ! ( F ⊗ k X G ) by Proposition 2.4.3 (i). Since ρ ! commutes with lim −→ we may suppose that F = k U and G = k V and the result follows from Proposition 2.4.3 (ii). (cid:3) HEAVES ON T -TOPOLOGIES Proposition 2.4.5. The functor ρ ! is fully faithful. In particular one has ρ − ◦ ρ ! ≃ id . Moreover, for F ∈ Mod( k X ) and G ∈ Mod( k T ) one has ρ − H om k T ( ρ ! F, G ) ≃ H om k X ( F, ρ − G ) . Proof. For F, G ∈ Mod( k X ) by adjunction we have Hom k X ( ρ − ρ ! F, G ) ≃ Hom k X ( F, ρ − ρ ∗ G ) ≃ Hom k X ( F, G ) . This also implies that ρ ! is fully faithful, in fact Hom k T ( ρ ! F, ρ ! G ) ≃ Hom k X ( F, ρ − ρ ! G ) ≃ Hom k X ( F, G ) . Now let K, F ∈ Mod( k X ) and G ∈ Mod( k T ) , we have Hom k X ( K, ρ − H om k T ( ρ ! F, G )) ≃ Hom k T ( ρ ! K, H om k T ( ρ ! F, G )) ≃ Hom k T ( ρ ! K ⊗ k T ρ ! F, G ) ≃ Hom k T ( ρ ! ( K ⊗ k X F ) , G ) ≃ Hom k X ( K ⊗ k X F, ρ − G ) ≃ Hom k X ( K, H om k X ( F, ρ − G )) . (cid:3) Finally let us consider sheaves of rings in Mod( k T ) . If A is a sheaf of rings in Mod( k X ) , then ρ ∗ A and ρ ! A are sheaves of rings in Mod( k T ) .Let A be a sheaf of unitary k -algebras on X , and let e A ∈ Psh( k T ) be thepresheaf defined by the correspondence T ∋ U lim −→ U ⊂⊂ V Γ( V ; A ) . Let F ∈ Psh( k T ) ,and assume that, for V ⊂ U , with U, V ∈ T , the following diagram is commutative: Γ( U ; e A ) ⊗ k Γ( U ; F ) (cid:15) (cid:15) / / Γ( U ; F ) (cid:15) (cid:15) Γ( V ; e A ) ⊗ k Γ( V ; F ) / / Γ( V ; F ) . In this case one says that F is a presheaf of e A -modules on T . Proposition 2.4.6. Let A be a sheaf of k -algebras on X , and let F be a presheafof e A -modules on X T . Then F ++ ∈ Mod( ρ ! A ) . Proof. Let U ∈ T , and let r ∈ lim −→ U ⊂⊂ V Γ( V ; A ) . Then r defines a morphism lim −→ U ⊂⊂ V Γ( V ; A ) ⊗ k Γ( W ; F ) → Γ( W ; F ) for each W ⊆ U , W ∈ T , hence an en-domorphism of ( F ++ ) | U X T ≃ ( F | U X T ) ++ . This morphism defines a morphism ofpresheaves e A → E nd ( F ++ ) and e A ++ ≃ ρ ! A by Proposition 2.4.3. Then F ++ ∈ Mod( ρ ! A ) . (cid:3) Proposition 2.4.7. Assume that X is locally weakly quasi-compact. Let F ∈ Mod( k T ) be T -flabby. Then ρ − F is c-soft. Proof. Recall that if U ∈ Op( X ) then Γ( U ; ρ − F ) ≃ lim ←− V ⊂⊂ U Γ( V ; F ) , where V ∈ T . Let W ∈ Op( X ) , W ⊂⊂ X . It follows from Lemma 2.4.1 that every U ′ ⊃⊃ W , U ′ ∈ Op( X ) contains U ∈ T such that U ⊃⊃ W . Hence lim −→ U ′ Γ( U ′ ; F ) ≃ lim −→ U Γ( U ; F ) , where U ′ ⊃⊃ W , U ′ ∈ Op( X ) and U ∈ T such that U ⊃⊃ W . We have the chainof isomorphisms lim −→ U Γ( U ; ρ − F ) ≃ lim −→ U lim ←− V ⊂⊂ U Γ( V ; F ) ≃ lim −→ U Γ( U ; F ) where U ∈ T , U ⊃⊃ W and V ∈ T . The first isomorphism follows from Proposition2.4.2 and second one follows since for each U ⊃⊃ W , U ∈ T , there exists V ∈ T such that U ⊃⊃ V ⊃⊃ W .Let V, W ∈ Op c ( X ) with V ⊂⊂ W . Since F is T -flabby and filtrant inductivelimits are exact, the morphism lim −→ W ′ Γ( W ′ ; ρ − F ) ≃ lim −→ W ′ Γ( W ′ ; F ) → lim −→ U Γ( U ; F ) ≃ lim −→ U Γ( U ; ρ − F ) , where W ′ , U ∈ T , W ′ ⊃⊃ W , U ⊃⊃ V , is surjective. Hence Γ( W ; ρ − F ) → lim −→ U ⊃⊃ V Γ( U ; ρ − F ) is surjective. (cid:3) T loc -sheaves. Let X be a T -space and let(2.3) T loc = { U ∈ Op( X ) : U ∩ W ∈ T for every W ∈ T } . Clearly, ∅ , X ∈ T loc , T ⊆ T loc and T loc is closed under finite intersections. Definition 2.5.1. We make the following definitions: • a subset S of X is a T loc -subset if and only if S ∩ V is a T -subset for every V ∈ T ; • a closed (resp. open) T loc -subset is a T loc -subset which is closed (resp. open)in X ; • a T loc -connected subset is a T loc -subset which is not the disjoint union oftwo proper clopen T loc -subsets. Observe that if { S i } i is a family of T loc -subsets such that { i : S i ∩ W = ∅} isfinite for every W ∈ T , then the union and the intersection of the family { S i } i isa T loc -subset. Also the complement of a T loc -subset is a T loc -subset. Therefore the T loc -subsets form a Boolean algebra. Example 2.5.2. Let us see some examples of T loc subsets:(i) Let T be the family of Example 2.1.2. Then the T loc subsets are the locallysemi-algebraic subsets of X .(ii) Let T be the family of Example 2.1.3. Then the T loc subsets are the sub-analytic subsets of X .(iii) Let T be the family of Example 2.1.4. Then the T loc subsets are the conicsubanalytic subsets of X .(iv) Let T be the family of Example 2.1.5. Then the T loc subsets are the locallydefinable subsets of X . HEAVES ON T -TOPOLOGIES One can endow T loc with a Grothendieck topology in the following way: a family { U i } i in T loc is a covering of U ∈ T loc if for any V ∈ T , there exists a finite subfamilycovering U ∩ V . We denote by X T loc the associated site, write for short k T loc insteadof k X T loc , and let X ρ loc | | ③③③③③③③③ ρ ! ! ❇❇❇❇❇❇❇❇ X T loc / / X T be the natural morphisms of sites. Remark 2.5.3. The forgetful functor, induced by the natural morphism of sites X T loc → X T , gives an equivalence of categories Mod( k T loc ) ∼ → Mod( k T ) . The quasi-inverse to the forgetful functor sends F ∈ Mod( k T ) to F loc ∈ Mod( k T loc ) given by F loc ( U ) = lim ←− V ∈T F ( U ∩ V ) for every U ∈ T loc .Therefore, we can and will identify Mod( k T loc ) with Mod( k T ) and apply the pre-vious results for Mod( k T ) to obtain analogues results for Mod( k T loc ) . Recall that F ∈ Mod( k T ) is T -flabby if the restriction Γ( V ; F ) → Γ( U ; F ) issurjective for any U, V ∈ T with V ⊇ U . Assume that(2.4) X T loc has a countable cover { V n } n ∈ N with V n ∈ T , ∀ n ∈ N . Proposition 2.5.4. Let F ∈ Mod( k T ) . Then F is T -flabby if and only if therestriction Γ( X ; F ) → Γ( U ; F ) is surjective for any U ∈ T loc . Proof. Suppose that F is T -flabby. Consider a covering { V n } n ∈ N of X T loc satisfying (2.4). Set U n = U ∩ V n and S n = V n \ U n . All the sequences → k U n → k V n → k S n → are exact. Since F is T -flabby the sequence → Hom k T ( k S n , F ) → Hom k T ( k V n , F ) → Hom k T ( k U n , F ) → is exact. Moreover the morphism Hom k T ( k S n +1 , F ) → Hom k T ( k S n , F ) is surjectivefor all n since S n = S n +1 ∩ V n is open in S n +1 . Then by Proposition 1.12.3 of [26]the sequence → lim ←− n Hom k T ( k S n , F ) → lim ←− n Hom k T ( k V n , F ) → lim ←− n Hom k T ( k U n , F ) → is exact. The result follows since lim ←− n Γ( U n ; G ) ≃ Γ( U ; G ) for any G ∈ Mod( k T ) and U ∈ T loc . The converse is obvious. (cid:3) Proposition 2.5.5. The full additive subcategory of Mod( k T ) of T -flabby objectis Γ( U ; • ) -injective for every U ∈ T loc . Proof. Take an exact sequence → F ′ → F → F ′′ → , and suppose that F ′ is T -flabby. Consider a covering { V n } n ∈ N of X T loc satisfying (2.4). Set U n = U ∩ V n .All the sequences → Γ( U n ; F ′ ) → Γ( U n ; F ) → Γ( U n ; F ′′ ) → are exact by Proposition 2.3.5, and the morphism Γ( U n +1 ; F ′ ) → Γ( U n ; F ′ ) issurjective for all n . Then by Proposition 1.12.3 of [26] the sequence → lim ←− n Γ( U n ; F ′ ) → lim ←− n Γ( U n ; F ) → lim ←− n Γ( U n ; F ′′ ) → is exact. Since lim ←− n Γ( U n ; G ) ≃ Γ( U ; G ) for any G ∈ Mod( k T ) the result follows. (cid:3) Let X, Y be two topological spaces and let T ⊂ Op( X ) , T ′ ⊂ Op( Y ) satisfy(2.1). Let f : X → Y be a continuous map. If f − ( T ′ loc ) ⊆ T loc then f defines amorphism of sites f : X T loc → Y T ′ loc . Corollary 2.5.6. Let f : X T loc → Y T ′ loc be a morphism of sites. T -flabby sheavesare injective with respect to the functor f ∗ . The functor f ∗ sends T -flabby sheavesto T ′ -flabby sheaves. Proof. Let us consider V ∈ T ′ loc . There is an isomorphism of functors Γ( V ; f ∗ • ) ≃ Γ( f − ( V ); • ) . It follows from Proposition 2.5.5 that T -flabby are injective with re-spect to the functor Γ( f − ( V ); • ) for any V ∈ T ′ loc .Let F be T -flabby and let U, V ∈ T ′ with V ⊃ U . Then the morphism Γ( V ; f ∗ F ) = Γ( f − ( V ); F ) → Γ( f − ( U ); F ) = Γ( U ; f ∗ F ) is surjective by Proposition 2.5.4. (cid:3) Remark 2.5.7. An interesting case is when X is a locally weakly quasi-compactspace and there exists S ⊆ Op( X ) with T = { U ∈ S : U ⊂⊂ X } satisfying (2.1) .Assume that X satisfies (1.7) . Then X has a covering { V n } n ∈ N of X such that V n ∈ T and V n ⊂⊂ V n +1 for each n ∈ N . By Lemma 1.3.5 we may find a covering { U n } n ∈ N of X such that U n ∈ Op c ( X ) and U n ⊂⊂ U n +1 for each n ∈ N . ByLemma 2.4.1 for each n ∈ N there exists V n ∈ T such that U n ⊂⊂ V n ⊂⊂ U n +1 .In this situation Proposition 2.5.4 and 2.5.5 are satisfied. T -spectrum. Let X be a topological space and let P ( X ) be the power set of X . Consider a subalgebra F of the power set Boolean algebra hP ( X ) , ⊆i . Then F is closed under finite unions, intersections and complements. We refer to [25] foran introduction to this subject.The Boolean algebra F has an associated topological space, that we denote by S ( F ) , called its Stone space. The points in S ( F ) are the ultrafilters α on F . Thetopology on S ( F ) is generated by a basis of open and closed sets consisting of allsets of the form e A = { α ∈ S ( F ) : A ∈ α } , where A ∈ F . The space S ( F ) is a compact totally disconnected Hausdorff space.Moreover, for each A ∈ F , the subspace e A is Hausdorff and compact. Definition 2.6.1. Let X be a T -space and let F be the Boolean algebra of T loc -subsets of X (i.e. Boolean combinations of elements of T loc ). The topological space e X T is the data of: • the points of S ( F ) such that U ∈ α for some U ∈ T , • a basis for the topology is given by the family of subsets { e U : U ∈ T } .We call e X T the T -spectrum of X . HEAVES ON T -TOPOLOGIES With this topology, for U ∈ T , the set e U is quasi-compact in e X T since it isquasi-compact in S ( F ) . Hence e X T is locally weakly quasi-compact with a basisof quasi-compact open subsets given by { e U : U ∈ T } . Note that if X ∈ T , then e X T = e X which is a spectral topological space. Remark 2.6.2. We may also define e X T by means of prime filters of elements of T . This is because T -subsets can be written as finite unions and intersections of T -open and T -closed subsets. In this situation an ultrafilter is determined by theprime filter contained in it. Proposition 2.6.3. Let X be a T -space. Then there is an equivalence of categories Mod( k T ) ≃ Mod( k e X T ) . Proof. Let us consider the functor ζ t : T → Op( e X T ) U e U . This defines a morphism of sites ζ : e X T → X T . Indeed, if V ∈ T , S ∈ Cov( V ) ,then e S = { e V i : V i ∈ S } ∈ Cov( e V ) . Let F ∈ Mod( k T ) and consider the presheaf ζ ← F ∈ Psh( k e X T ) defined by ζ ← F ( U ) = lim −→ U ⊆ e V F ( V ) . In particular, if U = e V , V ∈ T , ζ ← F ( U ) ≃ F ( V ) . In this case, by Corollary 1.2.11 we have the isomorphisms ζ − F ( e V ) = ( ζ ← F ) ++ ( e V ) ≃ ζ ← F ( e V ) ≃ F ( V ) . Then for V ∈ T we have ζ ∗ ζ − F ( V ) ≃ ζ − F ( e V ) ≃ F ( V ) . This implies ζ ∗ ◦ ζ − ≃ id . On the other hand, given α ∈ e X T and G ∈ Mod( k e X T ) , ( ζ − ζ ∗ G ) α ≃ lim −→ e U ∋ α,U ∈T ζ − ζ ∗ G ( e U ) ≃ lim −→ e U ∋ α,U ∈T ζ ∗ G ( U ) ≃ lim −→ e U ∋ α,U ∈T G ( e U ) ≃ G α since { e U : U ∈ T } forms a basis for the topology of e X T . This implies ζ − ◦ ζ ∗ ≃ id . (cid:3) Example 2.6.4. Let us see some examples of T -spectra.(i) When T is the family of Example 2.1.2 the T -spectrum e X T of X is thesemilagebraic spectrum of X ([10]). When X is semialgebraic, then e X T = e X , the semialgebraic spectrum of X from [9].(ii) When T is the family of Example 2.1.3 the T -spectrum e X T of X is thesubanalytic spectrum of X . The equivalence Mod( k e X sa ) ≃ Mod( k X sa ) wasused in [38] to bound the homological dimension of subanalytic sheaves.(iii) When T is the family of Example 2.1.5 the T -spectrum e X T of X is theo-minimal spectrum of X . When X is a definable space, then e X T = e X , theo-minimal spectrum of X from [33, 19]. Examples In this section we recall our main examples of T -sheaves. Good references ono-minimality are, for example, the book [13] by van den Dries and the notes [8] byCoste. For semialgebraic geometry relevant to this paper the reader should consultthe work by Delfs [10], Delfs and Knebusch [11] and the book [7] by Bochnak,Coste and Roy. For subanalytic geometry we refer to the work [6] by Bierstone andMilmann.3.1. The semialgebraic site. Let R = ( R, <, , , + , · ) be a real closed field.Let X be a locally semialgebraic space and consider the subfamily of Op( X ) de-fined by T = { U ∈ Op( X ) : U is semialgebraic } . The family T satisfies (2.1)and the associated site X T is the semialgebraic site on X of [10, 11]. Note alsothat: (i) the T -subsets of X are exactly the semialgebraic subsets of X ([7]); (ii) T loc = { U ∈ Op( X ) : U is locally semialgebraic } and (iii) the T loc -subsets of X areexactly the locally semialgebraic subsets of X ([11]).One can show (using triangulation of semialgebraic sets, as in [26]) that the fam-ily Coh( T ) corresponds to the family of sheaves which are locally constant on alocally semi-algebraic stratification of X . For each F ∈ Mod( k T ) there exists afiltrant inductive system { F i } i ∈ I in Coh( T ) such that F ≃ lim −→ i ρ ∗ F i .The subcategory of T -flabby sheaves corresponds to the subcategory of sa -flabby sheaves of [10] and it is injective with respect to Γ( U ; • ) , U ∈ Op( X T ) and Hom k T ( G, • ) , G ∈ Coh( T ) . Our results on T -flabby sheaves generalize thosefor sa -flabby sheaves from [10].We call in this case the T -spectrum e X T of X the semialgebraic spectrum of X .The points of e X T are the ultrafilters α of locally semialgebraic subsets of X suchthat U ∈ α for some U ∈ Op( X T ) . This is a locally weakly quasi-compact spacewith basis of quasi-compact open subsets given by { e U : U ∈ Op( X T ) } and there isan equivalence of categories Mod( k T ) ≃ Mod( k e X T ) . When X is semialgebraic, then e X T = e X , the semialgebraic spectrum of X from [9], and there is an equivalence ofcategories Mod( k T ) ≃ Mod( k e X ) ([10]).3.2. The subanalytic site. Let X be a real analytic manifold and consider thesubfamily of Op( X ) defined by T = Op c ( X sa ) = { U ∈ Op( X sa ) : U is subanalyticrelatively compact } . The family T satisfies (2.1) and the associated site X T is thesubanalytic site X sa of [28, 35]. In this case the T loc -subsets are the subanalyticsubsets of X .The family Coh( T ) corresponds to the family Mod c R - c ( k X ) of R -constructiblesheaves with compact support, and for each F ∈ Mod( k X sa ) there exists a filtrantinductive system { F i } i ∈ I in Mod c R - c ( k X ) such that F ≃ lim −→ i ρ ∗ F i .The subcategory of T -flabby sheaves corresponds to quasi-injective sheaves andit is injective with respect to Γ( U ; • ) , U ∈ Op( X sa ) and Hom k Xsa ( G, • ) , G ∈ Mod R - c ( k X ) . HEAVES ON T -TOPOLOGIES We call in this case the T -spectrum e X T of X the subanalytic spectrum of X and denote it by e X sa . The points of e X sa are the ultrafilters of subanalytic subsetsof X such that U ∈ α for some U ∈ Op c ( X sa ) . Then there is an equivalence ofcategories Mod( k X sa ) ≃ Mod( k e X sa ) .Let U ∈ Op( X sa ) and denote by U X sa the site with the topology induced by X sa . This corresponds to the site X T , where T = Op c ( X sa ) ∩ U . In this situation(2.1) is satisfied.3.3. The conic subanalytic site. Let X be a real analytic manifold endowedwith a subanalytic action µ of R + . In other words we have a subanalytic map µ : X × R + → X, which satisfies, for each t , t ∈ R + : ( µ ( x, t t ) = µ ( µ ( x, t ) , t ) ,µ ( x, 1) = x. Denote by X R + the topological space X endowed with the conic topology, i.e. U ∈ Op( X R + ) if it is open for the topology of X and invariant by the action of R + .We will denote by Op c ( X R + ) the subcategory of Op( X R + ) consisting of relativelyweakly quasi-compact open subsets.Consider the subfamily of Op( X R + ) defined by T = Op c ( X sa, R + ) = { U ∈ Op c ( X R + ) : U is subanalytic } . The family T satisfies (2.1) and the associatedsite X T is the conic subanalytic site X sa, R + . In this case the T loc -subsets are theconic subanalytic subsets.Set Coh( X sa, R + ) = Coh( T ) . For each F ∈ Mod( k X sa, R + ) there exists a filtrantinductive system { F i } i ∈ I in Coh( X sa, R + ) such that F ≃ lim −→ i ρ ∗ F i .The subcategory of T -flabby sheaves is injective with respect to Γ( U ; • ) , U ∈ Op( X sa, R + ) and Hom k Xsa, R + ( G, • ) , G ∈ Coh( X sa, R + ) .We call in this case the T -spectrum e X T of X the conic subanalytic spectrum of X and denote it by e X sa, R + . The points of e X sa, R + are the ultrafilters α of conicsubanalytic subsets of X such that U ∈ α for some U ∈ Op c ( X sa, R + ) . Then thereis an equivalence of categories Mod( k X sa, R + ) ≃ Mod( k e X sa, R + ) .3.4. The o-minimal site. Let M = ( M, <, ( c ) ∈C , ( f ) f ∈F , ( R ) R ∈R ) be an arbi-trary o-minimal structure. Let X be a locally definable space and consider thesubfamily of Op( X ) defined by T = Op( X def ) = { U ∈ Op( X ) : U is definable } .The family T satisfies (2.1) and the associated site X T is the o-minimal site X def of [19]. Note also that: (i) the T -subsets of X are exactly the definable subsetsof X (by the cell decomposition theorem in [13], see [19] Proposition 2.1); (ii) T loc = { U ∈ Op( X ) : U is locally definable } and (iii) the T loc -subsets of X areexactly the locally definable subsets of X . Set Coh( X def ) = Coh( T ) . For each F ∈ Mod( k X def ) there exists a filtrant in-ductive system { F i } i ∈ I in Coh( X def ) such that F ≃ lim −→ i ρ ∗ F i .The subcategory of T -flabby sheaves (or definably flabby sheaves) is injectivewith respect to Γ( U ; • ) , U ∈ Op( X def ) and Hom k X def ( G, • ) , G ∈ Coh( X def ) .We call in this case the T -spectrum e X T of X the definable or o-minimal spec-trum of X and denote it by e X def . The points of e X def are the ultrafilters α ofthe Boolean algebra of locally definable subsets of X such that U ∈ α for some U ∈ Op( X def ) . 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