Homotopy descent and cohomology

Question

I've been reading "Structured Brown representability via concordance" by D.Pavlov (https://dmitripavlov.org/concordance.pdf) and I'm strugeling with a point and was wondering if someone could help me with my confusion. In the text there is a criterion which says that if a simplicial presheaf $$F:Man^{op}\rightarrow \text{sSet}$$ satisfies homotopy descent, where $Man$ is the category of smooth manifolds, then there exists a K such that $$F\cong [-,K].$$ As a sanity check, or more to see if I could actually use this criterion, I wanted to show that singular cohomology satisfies this condition. Let $(U_i\rightarrow M)$ be an open cover and denote by $$U_{\underline{i}}=U_{i_0}\cap \ldots \cap U_{i_m}.$$By section 5.8 in https://pages.uoregon.edu/ddugger/hocolim.pdf, we know that $$[\text{hocolim } U_{\underline{i}},K(\mathbb{Z},n)]\cong \text{holim }[U_{\underline{i}},K(\mathbb{Z},n)]$$ so singular cohomology should satisfy homotopy descent. I would like to prove this without using the existence of Eilenberg-Maclane spaces. As an example, choose some manifold $M$ and an open cover $U,V$. Then we need to show that $$H^n(\text{hocolim }\left( U\leftarrow U\cap V\rightarrow V\right),\mathbb{Z})\cong \text{holim }\left[H^n(U,\mathbb{Z})\rightarrow H^n(U\cap V,\mathbb{Z})\leftarrow H^n(V,\mathbb{Z})\right].$$ In section 18 of https://pages.uoregon.edu/ddugger/hocolim.pdf, we are given spectal sequences computing the cohomology of a homotopy colimit. In the case of a homotopy pushout, this is just the information we get from the Mayer-Vietoris sequence, i.e. $$H^n(\text{hocolim }\left( U\leftarrow U\cap V\rightarrow V\right),\mathbb{Z})\cong \text{coker}\left( H^{n-1}(U)\oplus H^{n-1}(V)\rightarrow H^{n-1}(U\cap V) \right)\oplus \text{ker}\left(H^n(U)\oplus H^n(V)\rightarrow H^n(U\cap V)\right). $$ So now I'm left with showing that this Mayer-Vietoris data is weakly equivalent to $\text{holim }\left[H^n(U,\mathbb{Z})\rightarrow H^n(U\cap V,\mathbb{Z})\leftarrow H^n(V,\mathbb{Z})\right]$, but that is where I get stuck. I would very much appreciate any guidence, thanks!

Answer 1

if a simplicial presheaf F:Man^op→sSet satisfies homotopy descent, where Man is the category of smooth manifolds, then there exists a K such that F≅[−,K].

Here one must also mention that F is required to be concordance-invariant (alias R-local), i.e., the map F(X)→F(R⨯X) must be a weak equivalence.

By section 5.8 in https://pages.uoregon.edu/ddugger/hocolim.pdf, we know that [hocolim U_i,K(Z,n)]≅holim [U_i,K(Z,n)]

Here [-,-] must be the whole mapping space, not just the set of homotopy classes of maps, just like in the cited text by Dugger. So in particular, one must use the whole mapping space into K(Z,n) (equivalently, the whole singular cochain complex) instead of H^n (the homology group).

Concerning the homotopy descent property for singular cochains, note that the cited paper provides an independent proof in Proposition 2.8, by applying the main theorem in reverse.

First, it is a classical result (proved using subdivisions) that the cochain complex C of sheaves of local singular cochains is locally weakly equivalent to the cochain complex of presheaves of singular cochains.

The remainder of the proof uses a simple observation that the simplicial object k↦C^n_closed(Δ^k⨯X), once converted to a chain complex using the Dold–Kan correspondence, becomes quasi-isomorphic to the n-truncated singular cochain complex via an explicit map that is constructed using subdivisions.

Theorem 0.2 then provides the desired homotopy descent property by showing that the n-truncated singular cochain complex is homotopy representable, hence satisfies homotopy descent.