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.


From one perspective, sheaves, stacks, infinity-stacks on a given site SS with their descent conditions are nothing but a way of talking about the infinity-category of infinity-categories modeled on SS, in the sense of space and quantity: the \infty-category of \infty-category-valued presheaves/sheaves on SS.

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 XSX \in S a representable \infty-category valued presheaf, YXY \stackrel{\simeq}{\to} X a weakly equivalent replacement of XX, descent says that the usual statement of the Yoneda lemma for an \infty-category valued presheaf A\mathbf{A} – that [X,A]A(X)[X,\mathbf{A}] \simeq \mathbf{A}(X) – extends along the weak equivalence to yield also [Y,A]\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 SS are nothing but \infty-categories consistently modeled on SS. For instance a 0-stack=sheaf modeled on S=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 SS. By the yoga of enriched homotopy theory the second point means that these presheaves take values in a closed monoidal homotopical category.

So let


From this VV-enriched perspective it is natural to generalize to the case where the site SS is not just locally small, i.e. enriched over SetsSets, but is enriched over VV itself. If one does this one speaks of derived \infty-stacks.

The VV-enriched homotopical category CC is our generic model for an \infty-category of infinity-categories modeled on SS.


  • Let S=ptS = pt be the terminal category so that C=VC = V and take VV 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 SS 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 SS 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.

The enriched homotopy category

The Ho VHo_V-enriched category Ho CHo_C is now our model for the \infty-category if \infty-categories modeled on SS. The claim is:

  • \infty-stacks on SS are nothing but the objects in Ho CHo_C;

  • the descent condition on these morphisms is an extension of the statement of the Yoneda lemma – which says that for XSCX \in S \subset C a space and YCY \in C a cover YXY \stackrel{\simeq}{\to} X we have [X,A]A(X) [X,\mathbf{A}] \simeq \mathbf{A}(X) – extends to a statement which respects the weak equivalence YXY \stackrel{\simeq}{\to} X in that also

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

    RHom:C op×CV R Hom : C^op \times C \to V


    • for fixed XSX \in S the functor

      RHom(X,):Sh(S,V)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 ASh(S,V)\mathbf{A} \in Sh(S,V) the functor

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

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


  • for S=ptS = pt, V=V = CrossedComplexes: this is the context of results about cohomology in nonabelian algebraic topology;

  • for V=V = SimpSet these are pretty much the statements in ToenHDS:

    • CC is the category of simplicial sheaves on SS (middle of p. 11);

    • the right derived (V=SimpSet)(V =SimpSet)-enriched Hom is denoted there Map(F,G):=RHom(F,G)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)

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

Last revised on June 24, 2009 at 18:26:15. See the history of this page for a list of all contributions to it.