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Consider the following category $\mathcal C$:

  • An object of $\mathcal C$ is a pair $(X,\mathcal F)$ where $X$ is a space and $\mathcal F$ is a sheaf on $X$.
  • A morphism $(X,\mathcal F)\to(Y,\mathcal G)$ is a map of spaces $f:X\to Y$ together with a map of sheaves $f^*\mathcal G\to\mathcal F$ (equivalently $\mathcal G\to f_*\mathcal F$).

The words 'space' and 'sheaf' above can be taken in a number of senses: topological spaces and all sheaves, schemes and coherent sheaves, complex analytic spaces and coherent sheaves, etc. My question is somewhat general, and I will leave it up to the reader which context they wish to work in.

Cohomology $H^*$ is a functor from $\mathcal C$ to the category of graded abelian groups, however we can say more. Namely, let $W$ denote the class of morphisms $(X,\mathcal F)\to(Y,\mathcal G)$ in $\mathcal C$ for which $\mathcal G\xrightarrow\sim f_*\mathcal F\xrightarrow\sim Rf_*\mathcal F$ are both isomorphisms. It follows by the Leray spectral sequence that $H^*$ sends morphisms in $W$ to isomorphisms. The functor $H^*:\mathcal C\to\operatorname{AbGrp}$ thus factors through the localization $\mathcal C\to\mathcal C[W^{-1}]$ (note though that we have not argued that this localization exists).

The above suggests that cohomology makes sense not only for objects of $\mathcal C$ but also for what we might get by gluing together various objects of $\mathcal C$ along morphisms in $W$. For instance, take two objects $(X_1,\mathcal F_1),(X_2,\mathcal F_2)\in\mathcal C$. We could glue these together along a common open subset $(U_1,\mathcal F_1|_{U_1})\xrightarrow\sim(U_2,\mathcal F_2|_{U_2})$ to obtain another object of $\mathcal C$. On the other hand, we should also be able to, at least formally speaking, glue together along any morphism $(U_1,\mathcal F_1|_{U_1})\xrightarrow\sim(U_2,\mathcal F_2|_{U_2})$ in $W$ and obtain some sort of generalized object (though not an object of $\mathcal C$) to which it still makes sense to apply the functor $H^*$.

I can imagine various ways of making the above discussion precise (i.e. making sense out of "objects of $\mathcal C$ glued together along morphisms in $W$"), however they all suffer from the following deficiency: there is no simple way to describe morphisms between two such objects. This is similar to how a topological groupoid presents a stack, but given two topological groupoids it is somewhat cumbersome to describe, purely in terms of topological groupoids, the space of morphisms between the associated stacks. The notion of a stack solves this issue (of describing morphism spaces) beautifully: a morphism of stacks (on a site $\mathcal D$) is just a natural transformation of functors $\mathcal D\to\operatorname{Groupoids}$. I can finally form my question:

How can we form a category whose objects are "generalized objects" of $\mathcal C$ (i.e. gluings of objects of $\mathcal C$ along morphisms in $W$) and whose morphisms admit some sort of simple description?

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  • $\begingroup$ Just for confirmation: when you talk about "sheaves", do you mean sheaves of abelian groups or similar objects (i.e. not sheaves of sets), right? (it looks like it because you're talking about "cohomology", but it might be helpful to say it explicitly since it threw me off for a moment) $\endgroup$ Dec 2, 2019 at 8:52
  • $\begingroup$ Can you describe more explicitly what do you mean by "gluing together"? Are you perhaps using the isomorphism $(U_1,\mathcal F_1|_{U_1})\xrightarrow\sim(U_2,\mathcal F_2|_{U_2})$ to identify a subobject of $(X_1, \mathcal F_1)$ and $(X_2,\mathcal F_2)$ along which you do a pushout? $\endgroup$
    – fosco
    Dec 2, 2019 at 11:24

1 Answer 1

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Here is a construction which I think is at least close to what you're driving at.

Let $\mathcal S$ be our category of spaces, and let $Shv: \mathcal S \to Cat$ be the pseudofunctor taking a space to its category of sheaves $Shv(X)$ and a taking a map $f$ to its pushforward $f_\ast$. Then by the Grothendieck construction there is a corresponding fibration $\mathcal C \to \mathcal S$. Moreover, this is the same $\mathcal C$ as in the question. That is,

The category $\mathcal C$ is the result of applying the Grothendieck construction to the functor $Shv: \mathcal S \to Cat$.

Let us assume that

  1. $\mathcal S$ is locally presentable;

  2. For each $X \in \mathcal S$, the category $Shv(X)$ is locally presentable;

  3. For each $f: X \to Y$, the functor $f_\ast: Shv(X) \to Shv(Y)$ has a left adjoint $f^\ast$;

  4. The functor $Shv : \mathcal S \to Cat$ preserves $\kappa$-filtered colimits for some $\kappa$.

Then the fibration $\mathcal C \to \mathcal S$ is a presentable fibration, and by Thm 10.3 here, $\mathcal C$ is a locally presentable category.

Now for locally presentable categories, there is a very nice theory of localization. Assuming that $W$ is closed under colimits in the arrow category and also under pushouts along arbitrary maps (satisfies a mild set-theoretic hypothesis), the localization $\mathcal C[W^{-1}]$ is a reflective subcategory of $\mathcal C$, consisting of the $W$-local objects.


To connect to your setting, what I would do is take the $\infty$-categorical version of all of this (as in the paper of Gepner and Haugseng referenced above). So I would take $Shv(X)$ to be the $\infty$-category of sheaves on $X$ localized at the quasi-isomorphisms (i.e. the $\infty$-categorical enhancement of the derived category). From your examples, it sounds like neither $\mathcal S$ nor $Shv(X)$ is necessarily presentable -- you might have some finiteness conditions on them which prevent them from being cocomplete (every locally presentable category is cocomplete). But that's fine -- assuming that $\mathcal S$ and $Shv(X)$ have finite colimits, ($\infty$-categorical colimits in the latter case), you can just take the Ind-category of each to get something presentable.

Having done this, we have a presentable $\infty$-category $\mathcal C$. I would take $\mathcal W$ to be the class of morphisms inverted by the cohomology functor. In order to apply the theory of localizations of presentable $\infty$-categories, $\mathcal W$ needs to be closed under colimits in the arrow category and pushout along arbitrary morphisms. I believe the colimits in $\mathcal C$ are going to be related to colimits in $\mathcal S$, so in order for this to work, you will probably need it to be the case that cohomology behaves well (i.e. satisfies Mayer-Vietoris) with respect to all pushouts in $\mathcal S$, which is probably not the case -- it probably only behaves well with respect to some kind of "homotopy pushouts". So probably $\mathcal S$ also needs to be modified by localizing it at some kind of "homotopy equivalences" to obtain an $\infty$-category $\mathcal S_L$ such that the $\infty$-categorical pushouts of $\mathcal S_L$ are computed via the "homotopy pushouts" of $\mathcal S_L$. So we'll end up with an $\infty$-category $\mathcal C_L$ by applying the $\infty$-categorical Grothendieck construction to the functor $Shv: \mathcal S_L \to Cat_\infty$.

But once all of this is done, $\mathcal C_L[\mathcal W^{-1}]$ will be the full subcategory of $\mathcal C_L$ consisting of the $\mathcal W$-local objects. This may seems surprising, but really it's analogous to the fact that sheaves are a full subcategory of presheaves -- allowing more types of gluing makes your objects more local.

Probably the trickiest part in general is constructing $\mathcal S_L$. When $\mathcal S$ is spaces, for example, $\mathcal S_L$ would be the $\infty$-category of spaces; when $\mathcal S$ is schemes, $\mathcal S_L$ might be motivic spaces, etc. Probably $\mathcal S_L$ can itself be constructed as some localization of sheaves of spaces on $\mathcal S$, as in motivic homotopy theory.

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