# Cech model structure and the homotopy descent condition

Let $$\text{Cart}$$ be the category of cartesian spaces which has as its objects the collection of sets $$U$$ for which there exists $$n \in \mathbb{N}$$ so that $$U \subset \mathbb{R}^n$$ and $$U$$ is diffeomorphic to $$\mathbb{R}^n$$. The category $$\text{Cart}$$ turns into a site by endowing it with the coverage of good open covers, i.e., open covers $$\mathcal{U} = \{ U_j \}_{j \in \mathcal{J}}$$ such that $$U_j \cap U_k$$ is either empty or diffeomorphic to $$\mathbb{R}^n$$. For any good cover $$\mathcal{U}$$ of an object $$V \in \text{Cart}$$ as above we may then define the Cech nerve $$c^\mathcal{U}$$ as the simplicial presheaf on $$\text{Cart}$$ with $$m$$-simplices $$c^\mathcal{U}_m = \coprod\limits_{\zeta \colon m} j_\text{Cart}(U_\zeta)$$ where, first of all, $$\zeta \colon m$$ should mean that $$\zeta$$ runs over all those $$(m+1)$$-tuples $$(\zeta_0, \ldots, \zeta_m) \in \mathcal{J}^{m+1}$$ for which we have $$U_\zeta \equiv \bigcap\limits_{i = 0}^m U_{\zeta_i} \neq \emptyset$$ Moreover, the functor $$j_\text{Cart} \colon \text{Cart} \to \text{Psh}_\Delta(\text{Cart})$$, which has as its codomain the category of simplicial presheaves on $$\text{Cart}$$, takes a cartesian space $$U \in \text{Cart}$$ and maps it onto the Yoneda embedding of the object $$U$$ but viewed as a constant simplicial object in the category of presheaves on $$\text{Cart}$$. There is then a unique map induced by the universal property of the respective coproducts: $$\psi^\mathcal{U} \colon c^\mathcal{U} \to j_\text{Cart}(V)$$ The Cech model structure on the category of presheaves is then defined to be the the left Bousfield localization of the injective model structure $$\text{Psh}_\Delta(\text{Cart})_\text{inj}$$ with respect to the family of morphisms $$\psi^\mathcal{U} \colon c^\mathcal{U} \to j_\text{Cart}(V)$$ Call this new model structure $$\text{Psh}_\Delta(\text{Cart})_\text{loc}$$. Fibrant objects are precisely those objects $$X \in \text{Psh}_\Delta(\text{Cart})$$ which are fibrant with respect to the injective model structure and for which $$\mathbb{R}\text{Map}(\psi^\mathcal{U}, X) \colon \mathbb{R}\text{Map}(j_\text{Cart}(V), X) \to \mathbb{R}\text{Map}(c^\mathcal{U},X)$$ is a weak equivalence of simplicial sets, where $$\text{Map}$$ is the simplicially enriched hom-functor and $$\mathbb{R}\text{Map}$$ is the right derived version of the former.

How does one deduce that the above weak equivalence boils down to the statement that we have a weak equivalence $$X(V) \overset{\sim}{\to} \text{holim}_\zeta X(U_\zeta)$$

My attempt is this: Since $$X$$ is fibrant in the injective model structure and any object is cofibrant with respect to the injective model structure, it suffices to consider $$\text{Map}(\psi^\mathcal{U}, X) \colon \text{Map}(j_\text{Cart}(V), X) \to \text{Map}(c^\mathcal{U},X)$$ The domain of this map certainly boils down to $$X(V)$$ by a simple application of the Yoneda Lemma. The right hand-side is seen to be naturally isomorphic to $$\prod\limits_{\zeta \colon -} \text{Map}(U_\zeta, X) \cong \prod_{\zeta\colon -} X(U_\zeta)$$ This looks already pretty good. It seems as if this thing should boil down to a homotopy limit, but i don't really see how to make this rigorous. Any help is welcome :)

• It would be a good exercise for you to prove the analogous statement for ordinary sheaves of sets first. Also the title of your question is nonsense (but I appreciate it is difficult to summarise the question in the space provided). Commented Nov 7, 2022 at 22:15
• @ZhenLin I already did that for ordinary sheaves of sets. However, I feel like my problem is more about me not having enough knowledge of homotopy limits and how to work with right derived mapping spaces. Commented Nov 8, 2022 at 12:56
• It is completely analogous. The proof takes the same form. (I am assuming you didn't prove it it an overly hands-on way with elements...) Commented Nov 8, 2022 at 14:10

Any simplicial presheaf is a homotopy colimit (over $$\def\op{{\sf op}}Δ^\op$$) of its individual layers, which are presheaves of sets.
Thus, we have $$\def\R{{\bf R}}\def\Map{\mathop{\rm Map}}\def\holim{\mathop{\rm holim}}\def\hocolim{\mathop{\rm hocolim}}\R\Map(c^U,X)≃\R\Map(\hocolim_{n∈Δ^\op}c^U_n,X)≃\holim_{n∈Δ}\Map(c^U_n,X).$$
Next, since every $$c^U_n$$ is a coproduct (hence homotopy coproduct) of representables, we have $$\def\ho{{\sf h}}\holim_{n∈Δ}\Map(c^U_n,X)≃\holim_{n∈Δ}\R\Map(\coprod^\ho_{ζ:n}j(U_ζ),X)≃\holim_{n∈Δ}\prod^\ho_{ζ:n}\R\Map(j(U_ζ),X)≃\holim_{n∈Δ,ζ:n}X(U_ζ),$$ as desired.