It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ordinary sheaf $A \in \operatorname{Sh}(X, \operatorname{Ch}(\mathbb{Z}))$ of chain complexes on a topological space $X$,
- The functor that sends an open subset $U$ to the derived sections $\operatorname{R \Gamma}(U,A) \in D(\mathbb{Z})$ in the derived $\infty$-category is an $\infty$-sheaf.
- If $A = F[n]$ is concentrated in degree $n >0$, for $F$ an ordinary sheaf of abelian groups, then (under the Dold-Kan correspondence) $A$ corresponds to the Eilenberg-Maclane object $B^n F$ defined in HTT, as suggested in this n-lab article.
From the first statement (if it is true), one can show Theorem 3.8 on the correspondence between Cech and sheaf cohomology in the n-lab article on Cech cohomology, by using the fact that the Cech complex calculates the homotopy limit that describes descent for $\infty$-sheaves. I also want to prove Theorem 3.7:
- On a paracompact Hausdorff space $X$, Cech cohomology and sheaf cohomology agree: $$ \operatorname{\check{H}}^*(X,A) \cong \operatorname{H}^*(X,A) $$ Here, the Cech complex is defined by taking the colimit over all covers of $X$.
The mentioned nlab articles of course give references on the classical proofs of these statements (as well as, in the first case, a proof via simplicial cosheaves); but I am interested in whether, and how, one can abstractly deduce them from results in Lurie's books. In particular, Chapter 7 in HTT develops a lot of theory for paracompact Hausdorff spaces, and this would be an interesting application. I would be very greatful for some hints or references on how this works.
