Topology on cohomology of a sheaf of topological groups

Question

Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question:

• Is there a natural way to introduce topology on $H^i(X, \mathcal{F})$?

My guess is that for each open covering $\mathcal{U}$ the space of Čech cochains $\check{C}(\mathcal{U}, \mathcal{F})$ can be endowed with the compact-open topology. This topology can be restricted to the space of closed cochains, descends to $H^i(\mathcal{U}, \mathcal{F})$ and induce direct limit topology on $H^i(X, \mathcal{F})$. But does this construction make sense? Say, is it functorial? Is it true that $H^i(X, \mathcal{F})$ are commutative topological groups (with respect to the natural group operation)? Are the other reasonable choices of topology on Čech complex? Can they lead to other topologies?

And what is the best reference on this topic?

I don't think that this is important, but in the situation I am interested in, the space $X$ is a (finite-dimensional) manifold and $\mathcal{F}$ is a sheaf of Lie groups.

Answer 1

Both cases ($F$ is a sheaf of abelian topological groups or abelian Lie groups) can be treated using the same machinery.

The Yoneda embedding embeds abelian Lie groups as a fully faithful subcategory of the category of sheaves of abelian groups on the site of smooth manifolds, and the embedding functor preserves small limits. I refer to the latter category as the category of abelian smooth groups. This category is complete and cocomplete. It is, in fact, better behaved than the category of abelian topological groups, since the latter category is not an abelian category: a morphism of abelian topological groups can have a trivial kernel and cokernel, without being an isomorphism. On the other hand, the category of smooth groups is abelian.

From now on, we work either with presheaves of chain complexes of abelian topological groups or presheaves of chain complexes of smooth groups.

The category of such chain complexes can be equipped with the projective model structure transferred from simplicial topological spaces respectively simplicial sheaves on the site of smooth manifolds.

The category of presheaves of such chain complexes can itself be equipped with the projective model structure, which can then be further localized with respect to Čech nerves of open covers. Fibrant objects in the resulting model category are presheaves of chain complexes that satisfy the homotopy descent condition.

The fibrant replacement functor computes the (hyper)cohomology of sheaves. More precisely, if $F→RF$ is a fibrant replacement of the sheaf $F$, then $H^i(X,F)=H^i(Γ(X,RF))$.

The resulting cohomology theory is essentially (a reformulation of) the Segal–Mitchison cohomology in the topological case, or its smooth version by Brylinski.

In particular, the cohomology group $H^i(X,F)$ is by definition a smooth (respectively topological) group, since $H^i$ is computed in the category of smooth (respectively topological) groups.