Simplicial presheaves equipped with the model structure on simplicial presheaves are one model/presentation for the (∞,1)-category of (∞,1)-sheaves on a given site.
The fibrant object $\bar X$ that a simplicial presheaf $X : S^{op} \to SSet$ is weakly equivalent to with respect to this model structure is the ∞-stackification of $X$. One expects that ∞-stacks/(∞,1)-sheaves are precisely those (∞,1)-presheaves which satisfy a kind of descent condition.
Precsisely what this condition is like for the particular model constituted by simplicial presheaves with the given Jardine model structure on simplicial presheaves was worked out in
and
recalled as corollary 6.5.3.13 in
The following is a summary of these results.
The main point is that the fibrant objects are essentially those simplicial presheaves, which satisfy descent with respect not just to covers, but to hypercovers.
Localizations of (∞,1)-presheaves at hypercovers are called hypercompletions in section 6.5.3 of Higher Topos Theory. Notice that in section 6.5.4 of Higher Topos Theory it is argued that it may be more natural not to localize at hypercovers, but just at covers after all.
A well-studied class of models/presentations for an (∞,1)-category of (∞,1)-sheaves is obtained using the model structure on simplicial presheaves on an ordinary (1-categorical) site $S$, as follows.
Let $[S^{op}, SSet]$ be the SSet-enriched category of simplicial presheaves on $S$.
Recall from model structure on simplicial presheaves that there is the global and the local injective simplicial model structure on $[S^{op}, SSet]$, and that the local model structure is a (Bousfield-)localization of the global model structure.
According to section 6.5.2 of HTT we have:
the full simplicial subcategory on fibrant-cofibrant objects of $[S^{op}, SSet]$ with respect to the global injective model structure is (the SSet-enriched category realization of) the $(\infty,1)$-category $PSh_{(\infty,1)}(S)$ of (∞,1)-presheaves on $S$.
the full simplicial subcategory on fibrant-cofibrant objects of $[S^{op}, SSet]$ with respect to the local injective model structure is (the SSet-enriched category realization of) the $(\infty,1)$-category $\bar{Sh}_{(\infty,1)}(S)$ which is the hypercompletion of the $(\infty,1)$-category $Sh_{(\infty,1)}(S)$ of (∞,1)-sheaves on $S$.
Since with respect to the local or global injective model structure all objects are automatically cofibrant, this means that $\bar Sh_{(\infty,1)}(S)$ is the full sub-$(\infty,1)$-category of $PSh_{(\infty,1)}(S)$ on simplicial presheaves which are fibrant with respect to the local injective model structure: these are the ∞-stacks in this model.
By the general properties of localization of an (∞,1)-category there should be a class of morphisms $f : Y \to X$ in $PSh_{(\infty,1)}(S)$ – hence between injective-fibrant objects in $[S^{op}, PSh(S)]$ – such that the simplicial presheaves representing $\infty$-stacks are precisely the local objects with respect to these morphisms.
This was worked out in
We now describe central results of that article.
For $X \in S$ an object in the site regarded as a simplicial presheaf and $Y \in [S^{op}, SSet]$ a simplicial presheaf on $S$, a morphism $Y \to X$ is a hypercover if it is a local acyclic fibration, i.e. of for all $V \in S$ and all diagrams
there exists a covering sieve $\{U_i \to V\}$ of $V$ with respect to the given Grothendieck topology on $S$ such that for every $U_i \to V$ in that sieve the pullback of the abve diagram to $U$ has a lift
If $S$ is a Verdier site then every such hypercover $Y \to X$ has a refinement by a hypercover which is cofibrant with respect to the projective global model structure on simplicial presheaves. We shall from now on make the assumption that the hypercovers $Y \to X$ we discuss are cofibrant in this sense. These are called split hypercovers. (This works in many cases that arise in practice, see the discussion after DHI, def. 9.1.)
The objects of $Sh_{(\infty,1)}(S)$ – i.e. the fibrant objects with respect to the projective model structure on $[S^{op}, SSet]$ – are precisely those objects $A$ of $PSh_{(\infty,1)}(S)$ – i.e. Kan complex-valued simplicial presheaves – which satisfy descent for all split hypercovers, i.e. those for which for all split hypercover $f : Y \to X$ in $[S^{op}, SSet]$ we have that
This is DHI, thm 1.3 formulated in the light of DHI, lemma 4.4 ii).
Notice that by the co-Yoneda lemma every simplicial presheaf $F : S^{op} \to SSet$, which we may regard as a presheaf $F : \Delta^{op}\times S^{op} \to Set$, is isomorphic to the weighted colimit
which is equivalently the coend
where $F_n$ is the Set-valued presheaf of $n$-cells of $F$ regarded as an $SSet$-valued presheaf under the inclusion $Set \hookrightarrow SSet$, and where the SSet-weight is the canonical cosimplicial simplicial set $\Delta$, i.e. for all $X \in S$
In particular therefore for $A$ a Kan complex-valued presheaf the descent condition reads
With the shorthand notation introduced above the descent condition finally reads, for all global-injective fibrant simplicial presheaves $A$ and hypercovers $U \to X$:
The right hand here is often denoted $Desc(Y_\bullet \to X, A)$, in which case this reads
(expanded version of remark 2.1 in DHI)
Using the Bousfield-Kan map every simplicial presheaf $F$ is also weakly equivalent to the weighted limit over $F_\bullet$ with weight given by $N(\Delta/(-)) : \Delta \to SSet$.
But by the discussion at weighted limit, the left hand computes the homotopy limit of $F_\bullet$ (since $F_\bullet$ is objectwise fibrant, since $F_n$ factors through $Set \hookrightarrow SSet$), hence we have a weak equivalence
Often the descent condition is therefore formulated with the cover $U$ replaced by its homotopy limit, whence it reads
With $A$ global-injective fibrant this is equivalent to
Using the notation introduced above this becomes finally