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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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-category of $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-category valued presheaf $A$ – that $[X,A]\simeq A(X)$ – extends along the weak equivalence to yield also $\cdots \simeq [Y,A]$.

The $\mathrm{\infty }$-category valued presheaves satisfying this condition represent the objects in the proper $\mathrm{\infty }$-category of $\mathrm{\infty }$-category valued presheaves/sheaves, which is usefully conceived as a suitable enriched homotopy category: these are the $\mathrm{\infty }$-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 $\mathrm{\infty }$-stacks on a site $S$ are nothing but $\mathrm{\infty }$-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 $\mathrm{\infty }$-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;

• $C:=\mathrm{Sh}(S,V)$ 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 $\mathrm{Sets}$, but is enriched over $V$ itself. If one does this one speaks of derived $\mathrm{\infty }$-stacks.

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

Examples

• Let $S=\mathrm{pt}$ 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, $\mathrm{\infty }$-groups, $\mathrm{\infty }$-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 $\mathrm{Sh}(S,V)$ are those morphisms $f:A\to B$ which induce isomorphisms of sheaves of simplicial homotopy groups under all functors ${\pi }_n:\mathrm{SimpSet}\to \mathrm{Groups}$. 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 ${\mathrm{Ho}}_V$-enriched category ${\mathrm{Ho}}_C$ is the $\mathrm{SimpSet}$-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 $\mathrm{\infty }$-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 ${\mathrm{Ho}}_V$-enriched category ${\mathrm{Ho}}_C$ is now our model for the $\mathrm{\infty }$-category if $\mathrm{\infty }$-categories modeled on $S$. The claim is:

• $\mathrm{\infty }$-stacks on $S$ are nothing but the objects in ${\mathrm{Ho}}_C$;

• 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 $[X,A]\simeq A(X)$ – 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}]$$; \mathbf{A}(X) \stackrel{\simeq}{\to} Desc(Y,\mathbf{A}) := [Y,\mathbf{A}]$$;

• the morphisms of $$A(X)\stackrel{\simeq }{\to }\mathrm{Desc}(Y,A):=[Y,A]$$ are (of course) computed by the right-derived Hom-functor in $C$

  $$R\mathrm{Hom}:C^{\mathrm{op}}\times C\to V$$R Hom : C^op \times C \to V

  and

  • for fixed $X\in S$ the functor

    $$R\mathrm{Hom}(X,-):\mathrm{Sh}(S,V)\to V$$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 $A\in \mathrm{Sh}(S,V)$ the functor

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

    is the functor which computes sheaf cohomology of the sheaf $A$ in the form of Čech cohomology (by mapping out of $\mathrm{\infty }$-categorical resolutions aka hypercovers of a space $X$).

Examples

• for $S=\mathrm{pt}$, $V=$ 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 $(V=\mathrm{SimpSet})$-enriched Hom is denoted there $\mathrm{Map}(F,G):=R\mathrm{Hom}(F,G)$ (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)