TITLE: Modern versions of Verdier's hypercovering theorem?
QUESTION [14 upvotes]: 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.

$\operatorname{Ho} \mathbf{Hc}^\mathrm{op}$ is a filtered category and admits a small cofinal subcategory.
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 $U \to X$.
(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.)

REPLY [6 votes]: Here are some remarks on Charles Rezk's answer:

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. 
Proposition 3.5 in Calculating simplicial localizations says that locally fibrant presheaves have the same hammock spaces as the whole model category.
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.