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infinity-stack homotopically

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.

Idea

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.

Setup in enriched homotopy theory

In the context of enriched homotopy theory we assume that our model for infinity-categories can be thought of as

1. generalized spaces modeled on the objects in a locally small category, and

2. 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.

Remark

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.

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The $V$-enriched homotopical category $C$ is our generic model for an $\infty$-category of infinity-categories modeled on $S$.

Examples

• Let $S = pt$ be the terminal category so that $C = V$ and take $V$ to be any of the examples listed at monoidal model category, such as Cat, 2Cat, probably omegaCat (but here the pushout-product axiom still needs to be checked), or SimpSet. Even though for such simple $S$ there is no nontrivial “topology” in the game, the notion of descent resulting from this setup is still interesting: it encodes for instance nonabelian cohomology of finite (really: discrete) groups, $\infty$-groups, $\infty$-groupoids.

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 enriched homotopy category

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 $C$

  $$R Hom : C^op \times C \to V$$

  and

  • for fixed $X \in S$ the functor

    $$R Hom(X, -) : Sh(S,V) \to V$$

    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

    $$R Hom(-, \mathbf{A})$$

    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 $X$).

Examples

• 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.

…

References

• 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)

• ToenSNAC – B. Toën, Stacks and non-abelian cohomology

• ToenHDS – B. Toën, Higher and derived stacks: a global overview (arXiv)