on chain complexes/model structure on cosimplicial abelian groups
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structures in a cohesive (∞,1)-topos
See model structure on simplicial presheaves.
There are many model category structures on the category of simplicial presheaves derived from the model structure on simplicial sets.
The local such model structures are of interest in that they model infinity-stacks so that they are a presentation of the (infinity,1)-category of (infinity,1)-sheaves on the given site.
They can be thought of as being obtained from global model structures, of which there are two:
the global projective model structure has weak equivalences and fibrations being objectwise those of simplicial sets;
the global injective model structure has weak equivalences and cofibrations being objectwise those of simplicial sets;
These two model structures are Quillen equivalent (DHI04 p. 5 with the Quillen equivalence given by the identity functor). They can be defined on any domain category $S$, not necessarily a site. If we do have a structure of a site on $S$ then there is a notion of local weak equivalences of simplicial presheaves on $S$, defined below. One gets local projective and local injective model structures by applying left Bousfield localization of the above model structures at local weak equivalences (see p. 6 of DHI04)
the local projective model structure (weak equivalences are locally (usually stalkwise) and cofibrations are those that have the left lifting property against objectwise acyclic fibrations);
the local injective (weak equivalences are locally (usually stalkwise) and fibrations are those that have the right lifting property against the objectwise acyclic cofibrations).
Warning Since the (homotopy classes) of weak equivalences do not form a small set, the general existence theorem recalled at Bousfield localization of model categories does not apply. The existence of the Bousfield localization has to be shown by hand. For the injective structure this is what Joyal and Jardine accomplished.
Again, the injective and projective local model structures are Quillen equivalent by the identity functors between the underlying categories and hence provide projective and injective versions of the corresponding homotopy theory of infinity-stacks.
In the local injective structure all objects are cofibrant, so that the opposite category of simplicial presheaves with the local injective model structure is a category of fibrant objects.
Both local model structures are proper simplicially enriched categories (DHI04 p. 5).
The local injective model structure on simplicial presheaves is originally due to Jardine, following the construction of the Quillen equivalent local model structure on simplicial sheaves by Joyal. It was only later realized in DHI04 as a left Bousfield localization of the global injective model structure.
In between the injective and the projective model structures there are many other model structures obtained by varying the class of generating global cofibrations.
In the following let $S$ be a small site and denote by $SimpPr(S)$ be the category of simplicial presheaves on $S$.
One usually says that a local model structure on a category of presheaves is one whose weak equivalences are not defined objectwise but on covers and/or on stalks.
There are different equivalent ways to define local weak equivalences of simplicial presheaves on a site $S$.
(see section 2 of Jardine07)
We want to say that a local weak equivalence of simplicial presheaves is one which is “over each point” an isomorphism of homotopy groups.
we need the following terminology about sheaves of simplicial homtopy groups:
for $X$ a simplicial set, write $\pi_0(X)$ for its set of connected components and $\pi_n(X,x)$, $n \geq 1$, $x \in X_0$, for its $n$th simplicial homotopy group at $x$ (the homotopy group of its geometric realization), $\pi_n(X,x) = \pi_n(|X|, x)$. This yields functors $\pi_0 : SimpSet \to Set$ and $\pi_n : SimpSet \to Grps$.
By postcomposition these functors induce functors $\pi_0 : SimpSet^{S^{op}} \to Set^{S^{op}}$ and $\pi_n : SimpSet^{S^{op}} \to Grps^{S^{op}}$.
By postcomposition with the sheafification functor this yields functors $\tilde \pi_0 : SimpSet^{S^{op}} \to Sh(S)$ and $\tilde \pi_n : SimpSet^{S^{op}} \to Sh(S,Grps)$.
A local weak equivalence of simplicial presheaves is a morphism $f : X \to Y$ such that
the morphism $\tilde \pi_0(f) : \tilde \pi_0 X \to \tilde \pi_0 Y$ is an isomorphism in $Sh(S)$;
the diagrams
are pullback diagrams in $Sh(S,SimpSet)$, for all $n \geq 1$, where $\tilde X_0$ denotes the sheaf associated to $X_0$.
Equivalently a morphism $f : X \to Y$ of simplicial presheaves is, equivalently, a local weak equivalence if all induced morphisms of sheaves
are isomorphisms for all $U \in S$, for $X|_U, Y|_U$ the pullbacks to the over-category site $S/U$, for all $x \in X_0(U)$ and all $n \geq 0$.
If the site $S$ has enough points then this condition is equivalent to saying that $f$ is a weak equivalence in the model structure on simplicial sets over every stalk (see p. 363 of Jardine01).
(see DI02, i.e. Dugger and Isaksen, Weak equivalences of simplicial presheaves )
If $X$ and $Y$ are local fibrations there is a characterisation in terms of local homotopy liftings. Write $P$ for the pushout of the diagram $\partial \Delta^n \leftarrow \partial \Delta^n\times \Delta^1 \rightarrow \Delta^n\times \Delta^1$. Then there are two maps $\Delta^n\rightarrow P$ by restriction of $\Delta^n\times \Delta^1\rightarrow P$ along the cofaces.
Then a local weak equivalence is a morphism $f : X \to Y$ such that for all commuting diagrams
with $U$ simplicially constantly representable there exists a covering sieve $R$ of $U$ such that for every $V\in R$ there are morphisms $g:\Delta^n \otimes V\rightarrow X$ and $h:P\otimes V\rightarrow Y$ for which $g\circ i_V=\partial\Delta^n \otimes V\rightarrow \partial\Delta^n \otimes U \rightarrow X$ and $\Delta^n\otimes V \rightarrow \Delta^n \otimes U\rightarrow Y= \Delta^n\otimes V\rightarrow P\otimes V\rightarrow Y$ and in addition the square
commutes.
The local injective model structure on simplicial presheaves on a site $C$ is the left Bousfield localization $SPr(C)_{loc inj}$ of the injective global model structure $SPr(C)_{inj}$ at the class of local weak equivalences described above.
So
cofibrations are precisely the objectwise cofibrations of simplicial sets, i.e. the monomorphisms in $SPr(S)$;
weak equivalences are the local weak equivalences from above.
The inclusion of sheaves into simplicial presheaves $SimpSh(S) \hookrightarrow SimpPr(S)$ and the sheafification functor $SimpPr(S) \to SimpSh(S)$ constitute a Quillen equivalence with respect to the above local injective model structure on $SimpPr(S)$ ans the local model structure on simplicial sheaves.
See Jardine07, theorem 5.
The fibrant objects in the local injective model structure $SPr(C)_{loc inj}$ are those simplicial presheaves that
are fibrant in the global injective model structure;
satisfy descent for all hypercovers.
DHI04, theorem 1.1
The local projective model structure on simplicial presheaves on a site $C$ is the left Bousfield localization $SPr(C)_{loc proj}$ of the projective global model structure $SPr(C)_{proj}$ at the class of local weak equivalences described above.
So
cofibrations are precisely the cofibrations in the global projective structure (defined by left lifting property with respect to global Kan fibrations)
weak equivalences are the local weak equivalences from above.
Remark. Notice that this is still using left Bousfield localization. If we used right Bousfield localization the local projective fibrations would simply be the global Kan fibrations. Instead we have the following.
The local injective model structure $SPr(C)_{loc inj}$ is Quillen equivalent to the “universal homtopy thepory” $U C/S$ constructed by
formally adding homotopy colimits to the category $C$ to create the category $U C$.
imposing relations requiring that for every hypercover $U \to X$, the morphism $hocolim_n U_n \to X$ is a weak equivalence.
DHI04, theorem 1.2
In $U C/S$ the fibrant objects have a simpler description than in $SPr(C)_{loc inj}$: they still need to satisfy descent but the implicit fibrancy condition with respect to the global injective structure is replaced by the fibrancy condition with respect to the global projective structure
The fibrant objects in $U C/S$ are those simplicial presheaves $A$ that
are objectwise fibrant (i.e. take values in Kan complexes)
satisfy descent for all hypercovers.
DHI04, theorem 1.3
One can regard the projective and the injective model structure as two extrema of a poset of model structures on simplicial presheaves.
See Jardine07 page 12
See model structure on simplicial presheaves.
The local projective model structure on simplicial presheaves appears as theorem 1.6 in
Its analog for sheaves, theorem 2.1 there, is due to
That the local projective model structure (directly defined) is indeed the left Bousfield localization of the global projective model structure is lemma 4.3 there.