This entry is about models/presentations for an (infinity,1)-category of (infinity,1)-sheaves in terms of model categories of $\infty$-presheaves, in particular in terms of the Brown-Joyal-Jardine model structure on simplicial presheaves.
For other notions see infinity-stack and in general see Higher Topos Theory.
From one perspective, sheaves, stacks, infinity-stacks on a given site $S$ with their descent conditions are nothing but a way of talking about the infinity-category of infinity-categories modeled on $S$, in the sense of space and quantity: the $\infty$-category of $\infty$-category-valued presheaves/sheaves on $S$.
In particular, the all-important descent condition is from this perspective nothing but the condition that the Yoneda lemma extends to respect higher categorical equivalences:
for $X \in S$ a representable $\infty$-category valued presheaf, $Y \stackrel{\simeq}{\to} X$ a weakly equivalent replacement of $X$, descent says that the usual statement of the Yoneda lemma for an $\infty$-category valued presheaf $\mathbf{A}$ – that $[X,\mathbf{A}] \simeq \mathbf{A}(X)$ – extends along the weak equivalence to yield also $\cdots \simeq [Y,\mathbf{A}]$.
The $\infty$-category valued presheaves satisfying this condition represent the objects in the proper $\infty$-category of $\infty$-category valued presheaves/sheaves, which is usefully conceived as a suitable enriched homotopy category: these are the $\infty$-stacks.
Switching back perspective from presheaves to spaces, and reading the Yoneda lemma as the consistency condition on this interpretation (as indicated at Yoneda lemma), this says that $\infty$-stacks on a site $S$ are nothing but $\infty$-categories consistently modeled on $S$. For instance a 0-stack=sheaf modeled on $S =$Diff may be a generalized smooth space, while a 1-stack=stack modeled on Diff may be a differentiable stack representing a smooth groupoid.
Instead of committing the following discussion to a fixed model for infinity-categories or omega-categories I describe the situation in a setup which aims to come close to making the minimum number of necessary assumptions on the ambient context. After discussing the general idea I give concrete examples in concrete realizations of $\infty$-categorical contexts.
In the context of enriched homotopy theory we assume that our model for infinity-categories can be thought of as
generalized spaces modeled on the objects in a locally small category, and
such that there is a good notion of homotopy between maps into these spaces;
By the yoga of space and quantity the first point means that our infinity-categories are presheaves on a locally small category $S$. By the yoga of enriched homotopy theory the second point means that these presheaves take values in a closed monoidal homotopical category.
So let
$S$ be a site;
$V$ be a closed monoidal homotopical category;
$C := Sh(S,V)$ be the category of $V$-valued sheaves on $S$ such that it becomes a $V$-enriched homotopical category with some induced (usually local) notion of weak equivalences.
From this $V$-enriched perspective it is natural to generalize to the case where the site $S$ is not just locally small, i.e. enriched over $Sets$, but is enriched over $V$ itself. If one does this one speaks of derived $\infty$-stacks.
The $V$-enriched homotopical category $C$ is our generic model for an $\infty$-category of infinity-categories modeled on $S$.
In all of the following examples notice that if one wants to take the site $S$ to be something like Top or Diff, as one often does, then one needs to beware of the size issues of sheaves on large sites.
Simplicial presheaves, using the model structure on simplicial presheaves or the model structure on simplicial sheaves: If $S$ is any site and $V =$ SimpSet, then the natural notion of local weak equivalences in $Sh(S,V)$ are those morphisms $f : \mathbf{A} \to \mathbf{B}$ which induce isomorphisms of sheaves of simplicial homotopy groups under all functors $\pi_n : SimpSet \to Groups$. If $S$ has enough points (such as the site of open sets of a topological space), then this is equivalent to $f$ being a stalkwise weak equivalence of simplicial sets, using the model structure on simplicial sets (see for instance section 1 of JardStackSSh). Then I think that the $Ho_V$-enriched category $Ho_C$ is the $SimpSet$-enriched homotopy category of simplicial presheaves that is discussed in ToenSNAC and ToenHDS.
Let $S =$ Diff and with its standard Grothendieck topology and $V =$ omegaCat. The $C$ is the model for smooth $\infty$-categories used in Differential Nonabelian Cohomology.
Restrict $V$ either from simplicial sets to Kan complexes or from strict omega-categories to omega-groupoids hence equivalently to crossed complexes and then further to complexes of abelian groups to make contact, below, with ordinary notions of sheaf cohomology.
The $Ho_V$-enriched category $Ho_C$ is now our model for the $\infty$-category if $\infty$-categories modeled on $S$. The claim is:
$\infty$-stacks on $S$ are nothing but the objects in $Ho_C$;
the descent condition on these morphisms is an extension of the statement of the Yoneda lemma – which says that for $X \in S \subset C$ a space and $Y \in C$ a cover $Y \stackrel{\simeq}{\to} X$ we have $[X,\mathbf{A}] \simeq \mathbf{A}(X)$ – extends to a statement which respects the weak equivalence $Y \stackrel{\simeq}{\to} X$ in that also
\mathbf{A}(X) \stackrel{\simeq}{\to} Desc(Y,\mathbf{A}) := [Y,\mathbf{A}]$$;
the morphisms of $\mathbf{A}(X) \stackrel{\simeq}{\to} Desc(Y,\mathbf{A}) := [Y,\mathbf{A}]$ are (of course) computed by the right-derived Hom-functor in $C$
and
for fixed $X \in S$ the functor
is the functor which computes sheaf cohomology in the form of being the right-derived functor of the global section functor;
for fixed $\mathbf{A} \in Sh(S,V)$ the functor
is the functor which computes sheaf cohomology of the sheaf $\mathbf{A}$ in the form of Čech cohomology (by mapping out of $\infty$-categorical resolutions aka hypercovers of a space $X$).
for $S = pt$, $V =$ CrossedComplexes: this is the context of results about cohomology in nonabelian algebraic topology;
for $V =$ SimpSet these are pretty much the statements in ToenHDS:
$C$ is the category of simplicial sheaves on $S$ (middle of p. 11);
the right derived $(V =SimpSet)$-enriched Hom is denoted there $Map(F,G) := R Hom(F,G)$ (for instance middle of p. 14)
sheaf cohomology is reproduced as indicated, for instance p. 7 of ToenSNAC.
…
JardStackSSh – J. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology, homotopy and applications, vol. 3(2), 2001 p. 361-284 (pdf)
JardSimpSh – J. Jardine, Fields Lectures: Simplicial presheaves (pdf)
Last revised on June 24, 2009 at 18:26:15. See the history of this page for a list of all contributions to it.