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.)