+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Model category theory +--{: .hide} [[!include model category theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea The **Čech model structure on simplicial presheaves** on a [[site]] $C$ is a model for the [[topological localization]] of the [[(∞,1)-category of (∞,1)-presheaves]] on $C$ to the [[(∞,1)-category of (∞,1)-sheaves]]. It is obtained from the [[global model structure on simplicial presheaves]] on $C$ by the [[Bousfield localization of model categories|left Bousfield localization]] at [[Čech cover]]s: its fibrant objects are [[∞-stack]]s that satisfy [[descent]] over [[Čech cover]]s but not necessarily over [[hypercover]]s. Further [[Bousfield localization of model categories|left Bousfield localization]] at [[hypercover]]s 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. ## Definition Let $C$ be a small [[site]] and write $[C^{op}, sSet]_{proj}$ and $[C^{op}, sSet]_{inj}$ for the projective and injective [[global model structure on simplicial presheaves]], respectively. For $\{U_i \to V\}_i$ a covering family in the [[site]] $C$, let $$ C(\{U_i\}) := \left( \cdots\stackrel{\to}{\stackrel{\to}{\to}}\coprod_{i j} U_{i j}\stackrel{\to}{\to}\coprod_i U_i \right) $$ be the corresponding [[Cech nerve]], regarded as a [[simplicial presheaf]] on $C$. This comes canonically with a morphism $$ C(\{U_i\}) \to V $$ of simplicial presheaves, the corresponding _Čech cover morphism_ . Notice that by the discussion at [model structure on simplicial presheaves - fibrant and cofibrant objects](http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#FibAndCofibObjects) this is a morphism between cofibrant objects. +-- {: .un_defn} ###### Definition The injective (projective) **Čech model structure on simplicial presheaves** $[C^{op},sSet]_{Cech}$ on $C$ is the [[Bousfield localization of model categories|left Bousfield localization]] of $[C^{op}, sSet]_{inj}$ ($[C^{op}, sSet]_{proj}$) at the set of Čech cover morphisms. =-- By the general properties of [[Bousfield localization of model categories|Bousfield localization]] this means that the fibrant-cofibrant objects $A$ of $[C^{op},sSet]_{Cech}$ are precisely those that are fibrant-cofibrant in the global model structure and in addition satisfy the [[descent]] condition that for all covers $\{U_i \to V\}$ the morphism of simplicial sets $$ A(V) \cong [C^op,sSet](V,A) \to [C^{op},sSet](C(\{U_i\}), A) $$ 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 $(\infty,1)$-presheaf $(\infty,1)$-topos. ## Model structures on simplicial sheaves We may form the [[transferred model structure]] on simplicial _[[sheaf|sheaves]]_ by transferring along the degreewise [[sheafification]] [[adjunction]] $$ Sh(C) \stackrel{\overset{sh}{\leftarrow}}{\underset{}{\hookrightarrow}} PSh(C) \,. $$ This defines fibrations and weak equivalences in $sSh(C)$ to be those morphisms that are fibrations or weak equivalences, respectively, as morphism in $sPSh(C)_{Cech} = [C^{op},sSet]_{Cech}$. As discussed there, a necessary and sufficient condition for this to be a model structure is that * [[transfinite compositions]] of [[cobase changes]] of generating [[acyclic cofibrations]] are weak equivalences in $Sh(C)$. Here the generating (acyclic) cofibrations in $Sh(C)$ are obtained by applying the associated sheaf functor to generating (acyclic) cofibrations in $PSh(C)$. In the category $Sh(C)$, [[colimits]] like [[transfinite compositions]] and [[cobase changes]] are computed by applying the [[associated sheaf]] functor to the corresponding [[colimit]] in $PSh(C)$. The latter [[colimit]] in $PSh(C)$ does yield a [[weak equivalence]] in $PSh(C)$ because $PSh(C)$ admits a [[model structure]]. By the 2-out-of-3 property, applying the [[associated sheaf]] functor yields a weak equivalence again. ## References A detailed though unfinished account of the Čech model structure is given in * Daniel Dugger, _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf)) But beware of this document is unfinished. Some aspects of this appeared in * [[Daniel Dugger]], _[[DuggerUniv.pdf:file]]_ [[!redirects Cech model structure on simplicial presheaves]]