Modern versions of Verdier's hypercovering theorem?

Question

Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a simplicial version of Verdier's hypercovering theorem:

Let $X$ be a locally fibrant simplicial presheaf on $\mathcal{C}$, let $\hat{X}$ be an associated hypersheaf (i.e. a fibrant replacement in the local Jardine model structure), let $\mathbf{Hc}$ be the category of hypercovers (of $1$), and let $\operatorname{Ho} \mathbf{Hc}$ be the same category modulo simplicial homotopy.

1. $\operatorname{Ho} \mathbf{Hc}^\mathrm{op}$ is a filtered category and admits a small cofinal subcategory.
2. There is a canonical bijection $$\mathop{\varinjlim_{\operatorname{Ho} \mathbf{Hc}^\mathrm{op}}} \pi_0 \underline{\mathrm{Hom}} (U, X) \cong \pi_0 \Gamma (\hat{X})$$ where $U$ is the functor sending a hypercover to the corresponding simplicial presheaf and $\underline{\mathrm{Hom}} (U, X)$ is the simplicial set of morphisms $U \to X$.
3. (An analogous statement for higher homotopy groups, where the colimit is indexed over a more complicated category if the basepoint is not a global section of $X$.)

Here is another version:

Let $X$ be a presheaf of Kan complexes on $\mathcal{C}$ and let $\hat{X}$ be an associated hypersheaf. Then we have a natural bijection $$\mathop{\varinjlim_{\operatorname{Ho} \mathbf{Hc}^\mathrm{op}}} \pi_0 \underline{\mathrm{Hom}} (Z \odot U, X) \cong \operatorname{Ho} \mathbf{sSet} (Z, \Gamma (\hat{X}))$$ for all simplicial sets $Z$.

Thus, we can compute the (weak) homotopy type of $\Gamma (\hat{X})$ in terms of $X$ and hypercovers.

Question. Is there a formula of the same kind that gives an actual "model" for $\Gamma (\hat{X})$?

For instance, it would be nice if we had $$\mathop{\varinjlim_{\mathbf{Hc}^\mathrm{op}}} \underline{\mathrm{Hom}} (U, X) \simeq \Gamma (\hat{X})$$ but I do not see how to prove this. (Is it even true? It is easy enough to show that we get a bijection in $\pi_0$, and I think we also get an equivalence of fundamental groupoids.)

Answer 1

Jardine has recent a paper called "The Verdier hypercovering theorem", based on an earlier paper called "Cocycle categories", which you should look at if you haven't.

He doesn't exactly say it this way, but it looks like the following is true: given presheaf such that $X$ is locally fibrant, then you can define the "hypercover cocycle category" $H_{hyp}(U,X)$ to be the category whose objects are diagrams $U\xleftarrow{p} V\rightarrow X$, where $p$ is a hypercover (i.e., local trivial fibration). Then the nerve of $H_{hyp}(1,X)$ is a simplicial set with the homotopy type of $\Gamma(\hat{X})$, and more generally $NH_{hyp}(U,X)\approx \Gamma(U,\hat{X})$.

This is in the spirit of what you are asking for. Perhaps it can be modified to involve simplical mapping spaces, if that's what you really need.

Added:

In any model category $M$, for objects $X$ and $Y$, Jardine defines the "cocycle category" $H(X,Y)$ to be the category whose objects are spans $X\xleftarrow{f} A\rightarrow Y$, where $f$ is a weak equivalence, and whose maps are $A\to A'$ compatible with the projections to $X$ and $Y$.

If the model category $M$ has functorial factorizations, is right proper, and has the property that weak equivalences are preserved under all finite products, then you can show that if $Y\to Y'$ is a weak equivalence, then the evident map $H(X,Y)\to H(X,Y')$ is a weak equivalence on nerves; in fact, it is a simplicial homotopy equivalence. This is basically Lemma 3 of Jardine's Cocycle categories paper; he describes it as a proof that $\pi_0H(X,Y)\to \pi_0H(X,Y')$, but in fact his proof gives (assuming factorizations are functorial) an explicit map $H(X,Y')\to H(X,Y)$ and simplicial homotopies from the composites to identity, making it a simplicial homotopy equivalence.

(If $M$ is simplicial presheaves on a site, and if $Y$ is locally fibrant then the evident inclusion $H_{hyp}(X,Y)\to H(X,Y)$ is also a simplicial homotopy equivalence, also by using a factorization argument; this is exactly Lemma 5 of Jardine's Verdier hypercovering paper.)

To understand the homotopy type of $H(X,Y)$, it suffices by the above remarks to consider the case that $Y$ is fibrant (in the model category structure). That $H(X,Y)$ computes the homotopy type of the derived mapping space follows from results in the papers of Dwyer and Kan on hammock localizations (Function complexes in homotopical algebra and, especially, Calculating simplicial localizations; Prop. 6.2 of the latter is what you want.).

Answer 2

Here are some remarks on Charles Rezk's answer:

1. Everything is happening in the category of locally fibrant presheaves, which has a homotopy calculus of right fractions in the sense of Dwyer and Kan (because it has suitable functorial factorisations), so the hammock spaces have the same homotopy type has Jardine's cocycle categories.
2. Proposition 3.5 in Calculating simplicial localizations says that locally fibrant presheaves have the same hammock spaces as the whole model category.
3. There is indeed a formula in terms of the simplicial hom spaces: writing $\mathcal{R}$ for the full subcategory of locally fibrant simplicial presheaves that are locally weakly equivalent to $1$, we have $$\Gamma (\hat{X}) \simeq \mathbf{R} \underline{\mathrm{Hom}} (1, X) \simeq {\mathrm{ho}{\varinjlim}}_{\mathcal{R}^\mathrm{op}} \underline{\mathrm{Hom}}(U, X)$$ where $U : \mathcal{R} \to \mathbf{sPsh} (\mathcal{C})$ is the inclusion, essentially because $$N(H_\mathrm{hyp} (1, \Delta^{\bullet} \pitchfork X)) \simeq {\mathrm{ho}{\varinjlim}}_{\mathcal{R}^\mathrm{op}} \operatorname{disc} \underline{\mathrm{Hom}} (U, X)_n$$ by e.g. Thomason's homotopy colimit theorem.