Čech model structure on simplicial presheaves

**model category** ## Definitions ## * category with weak equivalences * weak factorization system * homotopy * homotopy category * small object argument * resolution ## Morphisms ## * Quillen adjunction * Quillen equivalence * Quillen bifunctor * derived functor ## Universal constructions ## * homotopy Kan extension * homotopy limit/homotopy colimit * Bousfield-Kan map ## Refinements ## * monoidal model category * monoidal Quillen adjunction * enriched model category * enriched Quillen adjunction * simplicial model category * simplicial Quillen adjunction * cofibrantly generated model category * combinatorial model category * cellular model category * algebraic model category * compactly generated model category * proper model category * cartesian closed model category, locally cartesian closed model category * stable model category ## Producing new model structures * on functor categories (global) * Reedy model structure * on overcategories * Bousfield localization * transferred model structure * model structure on algebraic fibrant objects ## Presentation of $(\infty,1)$-categories ## * (∞,1)-category * simplicial localization * (∞,1)-categorical hom-space * presentable (∞,1)-category ## Model structures ## * Cisinski model structure ### for $\infty$-groupoids for ∞-groupoids * on topological spaces * Strom model structure * Thomason model structure * model structure on presheaves over a test category * on simplicial sets, on semi-simplicial sets * for right/left fibrations * model structure on simplicial groupoids * on cubical sets * on strict ω-groupoids, on groupoids * on chain complexes/model structure on cosimplicial abelian groups related by the Dold-Kan correspondence * model structure on cosimplicial simplicial sets ### for $n$-groupoids * for n-groupoids/for n-types * for 1-groupoids ### for $\infty$-groups * model structure on simplicial groups * model structure on reduced simplicial sets ### for $\infty$-algebras #### general * on monoids * on simplicial T-algebras, on homotopy T-algebras * on algebas over a monad * on algebras over an operad, on modules over an algebra over an operad #### specific * on dg-algebras over an operad * on dg-algebras and on on simplicial rings/on cosimplicial rings related by the monoidal Dold-Kan correspondence * for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras * model structure on dg-modules ### for stable/spectrum objects * model structure on spectra * model structure on presheaves of spectra ### for $(\infty,1)$-categories * on categories with weak equivalences * Joyal model for quasi-categories * on sSet-categories * for complete Segal spaces * for Cartesian fibrations ### for stable $(\infty,1)$-categories * on dg-categories ### for $(\infty,1)$-operads * on operads, for Segal operads on algebras over an operad, on modules over an algebra over an operad * on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations ### for $(n,r)$-categories * for (n,r)-categories as Θ-spaces * for weak ω-categories as weak complicial sets * on cellular sets * on higher categories in general * on strict ω-categories ### for $(\infty,1)$-sheaves / $\infty$-stacks * on homotopical presheaves * on simplicial presheaves global model structure/Cech model structure/local model structure on simplicial sheaves on presheaves of simplicial groupoids on sSet-enriched presheaves * model structure for (2,1)-sheaves/for stacks

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The Čech model structure on simplicial presheaves on a site CC is a model for the topological localization of an (∞,1)-category of (∞,1)-presheaves on CC to the (∞,1)-category of (∞,1)-sheaves.

It is obtained from the the global model structure on simplicial presheaves on CC by left Bousfield localizations at Cech covers: its fibrant objects are ∞-stacks that satisfy descent over Čech covers but not necessarily over hypercovers.

Accordingly, the (∞,1)-topos presented by the Čech model structure has as its cohomology theory Čech cohomology.

Further left Bousfield localization at hypercovers leads from the Čech model structure to the Joyal-Jardine local model structure on simplicial presheaves that presents the hypercomplete (∞,1)-topos which is the hypercompletion of that presented by the Čech model structure.


Let CC be a small site and write [C op,sSet] proj[C^{op}, sSet]_{proj} and [C op,sSet] inj[C^{op}, sSet]_{inj} for the projective and injective global model structure on simplicial presheaves, respectively.

For {U iV} i\{U_i \to V\}_i a covering family in the site CC, let

C({U i}):=( ijU ij iU i) C(\{U_i\}) := \left( \cdots\stackrel{\to}{\stackrel{\to}{\to}}\coprod_{i j} U_{i j}\stackrel{\to}{\to}\coprod_i U_i \right)

be the corresponding Cech nerve, regarded as a simplicial presheaf on CC. This comes canonically with a morphism

C({U i})V C(\{U_i\}) \to V

of simplicial presheaves, the corresponding Čech cover morphism .

Notice that by the discussion at model structure on simplicial presheaves - fibrant and cofibrant objects this is a morphism between cofibrant objects.


The injective (projective) Čech model structure on simplicial presheaves [C op,sSet] Cech[C^{op},sSet]_{Cech} on CC is the left Bousfield localization of [C op,sSet] inj[C^{op}, sSet]_{inj} ([C op,sSet] proj[C^{op}, sSet]_{proj}) at the set of Čech cover morphisms.

By the general poperties of Bousfield localization this means that the fibrant-cofibrant objects AA of [C op,sSet] Cech[C^{op},sSet]_{Cech} are precisely those that are fibrant-cofibrant in the global model structure and in addition satisfy the descent condition that for all covers {U iV}\{U_i \to V\} the morphism of simplicial sets

A(U)[C op,sSet](V,U)[C op,sSet](C({U i}),A) A(U) \simeq [C^op,sSet](V,U) \to [C^{op},sSet](C(\{U_i\}), A)

is a weak equivalence in the standard model structure on simplicial sets.

This is the model for the \infty-analog of the sheaf condition, modelling the topological localization of an (,1)(\infty,1)-presheaf (,1)(\infty,1)-topos.

Mike Shulman: Two questions, one (hopefully) easy and one (perhaps) hard:

  1. Is there a Quillen equivalent Čech model structure on simplicial sheaves? Can we just lift the model structure for simplicial presheaves along the sheafification adjunction?

  2. Is there a characterization of the weak equivalences in either Čech model structure?

I am particularly interested in this for the following reason. According to Beke in Sheafifiable homotopy model categories, the weak equivalences in the local model structure on simplicial sheaves are precisely those maps f:XYf\colon X\to Y of simplicial objects in the corresponding 1-topos of sheaves of sets such that the statement ”ff is a weak equivalence of simplicial sets” is true in the internal logic of the topos (at least, interperiting ”ff is a weak equivalence of simplicial sets” by one particular set of geometric sentences whose interpretation in SetSet is equivalent to saying that a simplicial map is a weak equivalence). But if, as HTT teaches us, Čech descent is often to be preferred to hyperdescent, then we should be interested in Čech weak equivalences instead. So I would really like to know what it means for a map of simplicial sheaves to be a Čech weak equivalence, in the internal logic of the 1-topos of sheaves of sets. If nothing else, I think such a characterization would help me understand the real meaning of hypercompletion. But any sort of characterization of them would be better than none.

Urs Schreiber: below is a reply to the first question.

Mike Shulman: Thanks for attacking this. I thought I should also mention, for anyone listening in, that this question is evidently also relevant to what the correct notion of internal ∞-groupoid may be.


We may form the transferred model structure on simplicial sheaves by transferring along the degreewise sheafification adjunction

Sh(C)shPSh(C). Sh(C) \stackrel{\overset{sh}{\leftarrow}}{\underset{}{\hookrightarrow}} PSh(C) \,.

This defines fibrations and weak equivalences in sSh(C)sSh(C) to be those morphisms that are fibrations or weak equivalences, respectively, as morphism in sPSh(C) Cech=[C op,sSet] CechsPSh(C)_{Cech} = [C^{op},sSet]_{Cech}.

As discussed there, sufficient conditions for this to be a model structure is that

  • the inclusion Sh(C)PSh(C)Sh(C) \hookrightarrow PSh(C) preserves filtered colimits;

  • sSh(C)sSh(C) has functorial fibrant replacement and functorial path objects for fibrant objects.

Since sheafification does preserve filtered colimits the first condition is satisfied degreewise and hence is satisfied.

Mike Shulman: I believe that sheafification preserves κ\kappa-filtered colimits for some sufficiently large κ\kappa, but if the site has covers of infinite cardinality, I don’t see why sheafification would preserve ω\omega-filtered colimits. But I think this is enough for the proof to work.

Since the small object argument holds in sSh(C)sSh(C) for generating acyclic cofibrations we have functorial fibrant replacement. And a path object is obtained just by forming objectwise the standard path object in sSet, as in [C op,sSet][C^{op}, sSet].

Mike Shulman: The small object argument doesn’t automatically produce functorial fibrant replacements in this context… isn’t the whole question whether the map to the “fibrant replacement” is still a weak equivalence (in the underlying category)? I.e. whether F(J)F(J)-cell complexes are still weak equivalences.


A detailed though unfinished account of the Čech model structure is given in

  • Daniel Dugger, Sheaves and homotopy theory (web, pdf)

But beware of this document is unfinished. Some aspects of this appeared in

Revised on March 17, 2010 01:28:48 by Mike Shulman (