3
$\begingroup$

Let $\mathcal{U}$ be an open cover of a topological space $B$. As we see, for example in this question, the associated Cech diagram $B_{\mathcal{U}}$ constitutes a simplicial space with the weak equivalence $$ \text{hocolim } B_{\mathcal{U}} \to B.$$

Now, let $p : E \to B$ be a (Serre? Hurewicz?) fibration, the cover $\mathcal{U}$ pulls back via $p$ to a cover of $E$, which in turn gives us another simplicial space and weak equivalence $$ \text{hocolim } E_{\mathcal{U}} \to E.$$

We then get a cosimplicial space on the level of sections $\Gamma(E)_\mathcal{U}$ (considered as objects in $\mathrm{Top}$ with the compact-open topology), since if $U \subset V$, we can restrict a section of $\Gamma(E|_V)$ to a section of $\Gamma(E|_U)$, and we may form a homotopy limit, which is equipped with a map $$ \text{holim } \Gamma(E)_{\mathcal{U}} \leftarrow \Gamma(E).$$

1. Is this a weak equivalence?

Lastly, regardless whether the answer to the previous question is false or not: We can then hit $\Gamma(E)_{\mathcal{U}}$ with the contravariant singular cochain functor $\mathrm{Sing}^\bullet$ into the category of differential graded algebras, to get another simplicial space, and a map

$$ \text{coholim} \left(\mathrm{Sing}^\bullet \Gamma(E) \right)_{\mathcal{U}} \to \mathrm{Sing}^\bullet \Gamma(E). \tag{$*$}$$

2. Is this a weak equivalence?


Disclaimer: I cannot tell whether these questions are incredibly easy for people versed in this language, or very hard -- or perhaps even nonsensical? The actual subject I am interested in are factorization algebras and homotopy cosheaves, see here. I would like it if the assignment $U \mapsto \left(\mathrm{Sing}^\bullet \Gamma(E |_U)\right)$ would constitute such a homotopy cosheaf (of DGAs) (which would basically amount to $(*)$ being a weak equivalence), so I could use techniques from that theory, but unfortunately, the generalities of homotopy limits and colimits are quite mysterious to me, and getting into the game looks like it will take a lot of time and effort.

$\endgroup$
0

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.