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Adopting the perspective on cohomology as described at cohomology, one understands cohomology as
$\mathrm{H}^0(X,A) = \pi_0 \mathrm{Hom}(X,A),$in the appropriate context of some $(\infty,1)$-topos. Then, perspectives which are perhaps more familiar are described as ways of computing this using presentations of the relevant $(\infty,1)$-topos.
Here’s my interpretation of two such perspectives, Čech cohomology and abelian sheaf cohomology, at a very naive level.
For Čech cohomology, I start with a topological space $X$, and the Čech nerve provides a cofibrant replacement for $X$ in an $(\infty,1)$-category of simplicial topological spaces (I insist on simplicial topological spaces and not simplicial sets), ignoring potential oversights which are corrected by using hypercovers. For abelian sheaf cohomology, starting with a sheaf of abelian groups $A$, a fibrant replacement is given, in an $(\infty,1)$-category of simplicial abelian sheaves, by an injective resolution (through the lens of the Dold–Kan correspondence).
In my mind, in both cases, one has gone from objects to simplicial objects, and I am left wondering how to fully justify this leap. For instance, in the Čech case, we have gone from the Quillen model structure for the $(\infty,1)$-topos $\mathrm{Top}$ to some (?) model structure on simplicial topological spaces. Why did we have to leave our original $(\infty,1)$-category, and what guarantees that this process allows the computation to give the correct answer? Can we always use this tool of “taking simplicial objects” to help in the computation of the cohomology, for any $(\infty,1)$-topos we start with?
One is not “leaving” the $\infty$-category (or rather a model category presenting it), or at least one does not have to.
The Cech nerve of a good enough cover or more generally a choice of split hypercover provides a cofibrant resolution of the given domain in the given projective model category. Similarly the injective resolution provides a fibrant resolution of the codomain, just as you say.
Sorry, I don’t follow. If I start with the Quillen model structure on $\mathrm{Top}$, in what way am I not leaving it when I start dealing with simplicial objects? As you mention, it changes the focus from cofibrant objects in $\mathrm{Top}$ to cofibrant objects in the functor category with projective model structure. Is that not something you can do more generally to aid in computing cohomology?
I was referring to the model structure on simplicial presheaves over some site, the one that presents the $\infty$-topos.
That’s the context in which Cech cohomology and sheaf has its place. Of course under mild conditions $L_{whe} Top$ sits inside there as the locally constant sheaves, and using this embedding one can compute ordinary cohomology using Cech methods. But in either case the reply is: if you are doing Cech/sheaf cohomology from the $\infty$-categorical perspective you may go to one model category presenting your $\infty$-topos and then the traditional technology for computing cohomology groups turns out to be just about cofibrant and fibrant resolutions in that model category.
The hom-space (or at least its homotopy type) is part of the data of an $(\infty, 1)$-category; but if you start with a 1-category, then you don’t have that data. Even if you have a model category, you will first have to consider simplicial and/or cosimplicial resolutions of your original objects in order to compute the hom-spaces. (Unless you have a simplicial model category – in which case you can just take fibrant and/or cofibrant replacements.)
The story with abelian sheaves is a bit more complicated. First of all, the only reasonable model structure on the category of abelian sheaves is the “discrete” model structure where the weak equivalences are the isomorphisms. So we have to embed it somewhere – but the category of simplicial abelian sheaves is not the answer, because simplicial abelian sheaves correspond to chain complexes concentrated in non-negative degrees, whereas injective resolutions are cochain complexes. Nonetheless, the classical theory tells us that injective resolutions calculate what we want. So we can consider the category of cochain complexes of abelian sheaves whose cohomology is concentrated in degree 0. Inverting the quasi-isomorphisms in this category recovers the category of abelian sheaves; however, magically this category has enough “flexibility” to make the general notion of right derived functor agree with the classical one.
I understand now that Čech cohomology and sheaf cohomology, by virtue of involving sheaves, fundamentally exist in a $(\infty,1)$-topos of $\infty$-stacks (as opposed to something like $\mathrm{Top}$ with Quillen model structure)… and we are not leaving this context in any way. Rather, the computational tools reflect the understanding that one can consider $(\infty,1)$-sheaves as simplicial presheaves. So it seems to me talking about “looking at simplicial objects” was a red herring, when the leap was in fact “looking at sheaves”. Thanks for the answers.
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