nLab
model structure on simplicial presheaves

Idea

Model structures on simplicial presheaves present (∞,1)-categories of (∞,1)-presheaves and localizations of these, such as notably the left exact localizations that are (∞,1)-categories of (∞,1)-sheaves: these model structures are models for ∞-stack (∞,1)-toposes.

Recall that

• a model category is a way to present an (∞,1)-category;

• in the context of (∞,1)-categories presheaves on an $(\mathrm{\infty }\mathrm{,1})$-category $C$ are given by (∞,1)-functors $C^{\mathrm{op}}\to$ SSet.

This suggests that the (∞,1)-category of (∞,1)-sheaves on some site $C$ can be presented by a model category structure on the ordinary functor category

$$[C^{\mathrm{op}},\mathrm{SSet}]\simeq [{\Delta }^{\mathrm{op}},\mathrm{PSh}(C)]$$[C^{op},SSet] \simeq [\Delta^{op}, PSh(C)]

– the category of simplicial presheaves .

Various interrelated flavors of model structures on the category of simplicial presheaves on $C$ have been introduced and studied since the 1970s, originally by K. Brown and Andre Joyal and then developed in detail by Jardine.

Notice that when regarded as a presentation of an (∞,1)-sheaf, i.e. of an ∞-stack, a simplicial presheaf – being an ordinary functor instead of a pseudofunctor – corresponds to a rectified ∞-stack. It might therefore seem that a model given by simplicial presheaves is too restrictive to capture the full expected flexibility of a notion of ∞-stack. But this is not so.

In

• Jacob Lurie, Higher Topos Theory

a fully general definition of a (∞,1)-category of ∞-stacks is given it is shown – proposition 6.5.2.1 – that, indeed, the Brown–Joyal–Jardine model is a presentation of that.

More precisely

• the global model structure on simplicial presheaves on a category is a presentation of the (∞,1)-category of (∞,1)-presheaves;

• the Čech model structure on simplicial presheaves on a site is a presentation of the (∞,1)-category of (∞,1)-sheaves;

• the local model structure on simplicial presheaves on a site is a presentation of the hypercompletion of the (∞,1)-category of (∞,1)-sheaves (see the discussion at hypercover).

• the Bousfield localization of the global model category structure to the local one presents the corresponding localization of an (∞,1)-category from presheaves to sheaves, mimicking the corresponding statement for a category of sheaves.

Originally K. Brown had considered in BrownAHT not a model structure on simplicial presheaves but

• a category of fibrant objects structure on locally Kan simplicial sheaves (see there for details)

and Joyal had originally considered a

• local model structure on simplicial sheaves.

Joyal’s local model structure on simplicial sheaves is Quillen equivalent to the injective local model structure on simplicial presheaves.

By repackaging Kan complexes as simplicial groupoids one obtains a model structure on presheaves of simplicial groupoids which is also Quillen equivalent to the above.

If K. Brown’s category of fibrant objects on locally Kan simplicial sheaves is restricted to globally Kan simplicial sheaves on a topos with enough point then it is the full subcategory on the fibrant objects in the projective local model structure on simplicial sheaves.

But since in all cases the weak equivalences are the same (where they apply, for Brown’s model if the topos has enough points), all these local homotopical categories define equivalent homotopy categories.

By Lurie’s result these are in each case in turn equivalent to the homtopy category of the (∞,1)-topos of ∞-stacks. So in particular they serve as a home for general cohomology.

Various old results appear in a new light this way. For instance using the old result of BrownAHT on the way ordinary abelian sheaf cohomology is embedded in the homotopy theory of simplicial sheaves, one sees that the old right derived functor definition of abelian sheaf cohomology really computes the ∞-stackification of a sheaf of chain complexes regarded under the Dold–Kan correspondence as a simplicial sheaf.

The different model structures and their interrelation

It is the very point of model category structures on a given homotopical category that there may be several of them: each presenting the same (∞,1)-category but also each suited for different computational purposes.

So it is good that there are many model structures on simplicial (pre)sheaves, as there are.

Injective/projective - local/global - presheaves/sheaves

The following diagram is a map for part of the territory:

$$\begin{array}{ccccccccc}& & (\mathrm{\infty }\mathrm{,1})\mathrm{Sh}(C)& & & (\mathrm{\infty }\mathrm{,1})\mathrm{PSh}(C)& & & (\mathrm{\infty }\mathrm{,1})\mathrm{Sh}(C)\\ & & {\uparrow }^{\mathrm{presentation}}& & & {\uparrow }^{\mathrm{presentation}}& & & {\uparrow }^{\mathrm{presentation}}\\ \mathrm{SSh}(C{)}_{\mathrm{inj}}^{l\mathrm{loc}}& \stackrel{\stackrel{\mathrm{sheafification}}{\leftarrow }}{\stackrel{\mathrm{embedding}}{\to }}& \mathrm{SPSh}(C{)}_{\mathrm{inj}}^{l\mathrm{loc}}& \stackrel{}{\leftarrow }\mid & \mathrm{SPSh}(C{)}_{\mathrm{inj}}& \stackrel{\stackrel{\mathrm{Id}}{\leftarrow }}{\stackrel{\mathrm{Id}}{\to }}& \mathrm{SPSh}(C{)}_{\mathrm{proj}}& \stackrel{}{\mapsto }& \mathrm{SPSh}(C{)}_{\mathrm{proj}}^{l\mathrm{loc}}& \stackrel{\stackrel{\mathrm{sheafification}}{\to }}{\stackrel{\mathrm{embedding}}{\leftarrow }}& \mathrm{SSh}(C{)}_{\mathrm{proj}}^{l\mathrm{loc}}\\ \mathrm{Joyal}& \stackrel{\mathrm{Quillen}\mathrm{equivalence}}{\leftrightarrow }& \mathrm{Jardine}& \stackrel{\mathrm{left}\mathrm{Bousf}.\mathrm{localization}}{\leftarrow \mid }& \mathrm{Heller}& \stackrel{\mathrm{Quillen}\mathrm{equivalence}}{\leftrightarrow }& \mathrm{Bousfield}-\mathrm{Kan}& \stackrel{\mathrm{left}\mathrm{Bousf}.\mathrm{localization}}{\mapsto }& \mathrm{Blander}& \stackrel{\mathrm{Quillen}\mathrm{equivalence}}{\leftrightarrow }& \mathrm{Brown}-\mathrm{Gersten}\\ \\ & \mathrm{everything}\mathrm{cofibrant};\\ & \mathrm{fibrant}=\mathrm{global}\mathrm{injective}\mathrm{fib}...\\ & ...\mathrm{satisfying}\mathrm{descent}& & & & & & & & \mathrm{cofibrant}=\mathrm{global}\mathrm{projective}\mathrm{cofib};\\ & & & & & & & & & \mathrm{fibrant}=\mathrm{Kan}\mathrm{valued}\mathrm{and}...\\ & & & & & & & & & ...\mathrm{satisfying}\mathrm{descent}\end{array}$$\array{ && (\infty,1)Sh(C) &&& (\infty,1)PSh(C) &&& (\infty,1)Sh(C) \\ && \uparrow^{presentation} &&& \uparrow^{presentation} &&& \uparrow^{presentation} \\ SSh(C)^{l loc}_{inj} & \stackrel{\stackrel{sheafification}{\leftarrow}} {\stackrel{embedding}{\to}}& SPSh(C)^{l loc}_{inj} &\stackrel{}{\leftarrow}|& SPSh(C)_{inj} &\stackrel{\stackrel{Id}{\leftarrow}} {\stackrel{Id}{\rightarrow}}& SPSh(C)_{proj} &\stackrel{}{\mapsto}& SPSh(C)_{proj}^{l loc} & \stackrel{\stackrel{sheafification}{\to}} {\stackrel{embedding}{\leftarrow}}& SSh(C)_{proj}^{l loc} \\ Joyal &\stackrel{Quillen equivalence}{\leftrightarrow}& Jardine &\stackrel{left Bousf. localization}{\leftarrow|}& Heller &\stackrel{Quillen equivalence}{\leftrightarrow}& Bousfield-Kan &\stackrel{left Bousf. localization}{\mapsto}& Blander &\stackrel{Quillen equivalence}{\leftrightarrow}& Brown-Gersten \\ \\ & everything cofibrant; \\ & fibrant = global injective fib... \\ \;\;\; & ...satisfying descent &&&&&&&& cofibrant = global projective cofib; \\ &&&&&&&&& fibrant = Kan valued and... \\ &&&&&&&&& \;\;\; ...satisfying descent }

Here

• “inj” denotes the injective model structure: fibrations are objectwise fibrations

• “proj” denotes the projective model structure: fibrations are objectwise fibrations

• no “loc” subscript means global model structure: weak equivalences are the objectwise weak equivalences:

• “l loc” denotes left Bousfield localization at hypercovers (at stalkwise acyclic fibrations if the topos has enough points)

The identity functor on the category $\mathrm{SPSh}(C)$ of simplicial presheaves is a Quillen adjunction for the projective and injective global model structure and this is a Quillen equivalence.

The local model structures on simplicial sheaves are just the restrictions of those on simplicial presheaves.

These are related by a Quillen adjunction given by the usual geometric embedding of the category of sheaves as a full subcategory of that of presheaves – with sheafification the left adjoint – and this is also Quillen equivalence.

The characteristic of the left Bousfield localizations is that for them the fibrant objects are those that satisfy descent: see descent for simplicial presheaves.

In either case

• the global model structures presents the (∞,1)-category of (∞,1)-presheaves

while

• the local model structures presents the (∞,1)-category of (∞,1)-sheaves (i.e. ∞-stacks).

The following diagram collection model categories that are presentations for the (∞,1)-category of (∞,1)-sheaves. All indicated morphism pairs are Quillen equivalences.

$$\begin{array}{ccccccccccc}\mathrm{PSh}(X,\mathrm{SGrpd})& \stackrel{\stackrel{\mathrm{embedding}}{\leftarrow }}{\stackrel{\mathrm{sheafification}}{\to }}& \mathrm{Sh}(X,\mathrm{SGrpd})& \stackrel{}{\leftrightarrow }& \mathrm{Sh}(X,\mathrm{SSet}{)}_{\mathrm{inj}}^{l\mathrm{loc}}& \stackrel{\stackrel{\mathrm{sheafification}}{\leftarrow }}{\stackrel{\mathrm{embedding}}{\to }}& \mathrm{PSh}(X,\mathrm{SSet}{)}_{\mathrm{inj}}^{l\mathrm{loc}}& \stackrel{\stackrel{\mathrm{Id}}{\leftarrow }}{\stackrel{\mathrm{Id}}{\to }}& \mathrm{PSh}(X,\mathrm{SSet}{)}_{\mathrm{proj}}^{l\mathrm{loc}}& \stackrel{\stackrel{\mathrm{embedding}}{\leftarrow }}{\stackrel{\mathrm{sheafification}}{\to }}& \mathrm{Sh}(X,\mathrm{SSet}{)}_{\mathrm{proj}}^{l\mathrm{loc}}\\ \\ \mathrm{Luo}-\mathrm{Bubenik}-\mathrm{Tim}& & \mathrm{Joyal}-\mathrm{Tierney}& & \mathrm{Joyal}& & \mathrm{Jardine}& & \mathrm{Blander}& & \mathrm{Brown}-\mathrm{Gersten}\end{array}$$\array{ PSh(X, SGrpd) &\stackrel{\stackrel{embedding}{\leftarrow}} {\stackrel{sheafification}{\to}}& Sh(X,SGrpd) &\stackrel{}{\leftrightarrow}& Sh(X, SSet)^{l loc}_{inj} &\stackrel{\stackrel{sheafification}{\leftarrow}} {\stackrel{embedding}{\to}}& PSh(X, SSet)^{l loc}_{inj} &\stackrel{\stackrel{Id}{\leftarrow}} {\stackrel{Id}{\to}}& PSh(X, SSet)^{l loc}_{proj} &\stackrel{\stackrel{embedding}{\leftarrow}} {\stackrel{sheafification}{\to}}& Sh(X, SSet)^{l loc}_{proj} \\ \\ Luo-Bubenik-Tim && Joyal-Tierney && Joyal && Jardine && Blander && Brown-Gersten }

On the right this lists the model structures on simplicial (pre)sheaves, here displayed as (pre)sheaves with values in simplicial sets, using $\mathrm{SPSh}(C)\simeq \mathrm{PSh}(C,\mathrm{SSet})$.

On the left we have the Joyal–Tierney and Luo–Bubenik–Tim model structures on presheaves of simplicial groupoids.

(…have to check here the relation $\mathrm{Sh}(X,\mathrm{SGrpd})\leftrightarrow \mathrm{PSh}(X,\mathrm{SGrpd})$)

Dependence on the underlying site

Proposition

Let $C,D$ be sites and let $f:C\to D$ be a functor that induces a morphism of sites in that $f_{*}:\mathrm{PSh}(D)\to \mathrm{PSh}(C)$ preserves sheaves and its left adjoint $f^{*}:\mathrm{PSh}(C)\to \mathrm{PSh}(D)$ (given by left Kan extension) is left exact functor in that it preserves finite limits.

Then the induced adjunction

$$f_{*}:\mathrm{SPSh}(D{)}_{\mathrm{inj}}^{\mathrm{loc}}\overleftarrow{\to }\mathrm{SPSh}(C{)}_{\mathrm{inj}}^{\mathrm{loc}}:f^{*}$$f_* : SPSh(D)_{inj}^{loc} \stackrel{\leftarrow}{\to} SPSh(C)_{inj}^{loc} : f^*

is a Quillen adjunction for the local injective model structure on presheaves on both sides.

Proof

This is “little fact 5)” on page 10, 11 of

• Jardine, Fields Lectures: Simplicial presheaves (pdf)

Examples

Proposition

Let $C=$ CartSp, $D=$ Diff and let $f:\mathrm{CartSp}\hookrightarrow \mathrm{Diff}$ be the canonical inclusion of a subcategory.

Then there is a Quillen equivalence

$$f_{*}:\mathrm{SPSh}(\mathrm{Diff}{)}_{\mathrm{inj}}^{\mathrm{loc}}\overleftarrow{\to }\mathrm{SPSh}(\mathrm{CartSp}{)}_{\mathrm{inj}}^{\mathrm{loc}}:f^{*}.$$f_* : SPSh(Diff)_{inj}^{loc} \stackrel{\leftarrow}{\to} SPSh(CartSp)_{inj}^{loc} : f^* \,.

Proof

Urs Schreiber: this is what I came up with, check

The functor

$$f_{*}:\mathrm{PSh}(\mathrm{Diff})\to \mathrm{PSh}(\mathrm{CartSp})$$f_* : PSh(Diff) \to PSh(CartSp)

that simply restricts a sheaf on the category of all manifolds to those on just cartesian spaces clearly respects the sheaf condition.

Its left adjoint is given by the left Kan extension formula

$$f^{*}A:(X\in \mathrm{Diff})\mapsto \underrightarrow{\lim }{}_{(X\to {ℝ}^n)}A({ℝ}^n).$$f^* A : (X \in Diff) \mapsto \lim_{\to}_{(X \to \mathbb{R}^n)} A(\mathbb{R}^n) \,.

Since CartSp has terminal object also the comma category $(X/f)$ has a terminal object and is hence a filtered category. Therefore this is a filtered colimit and therefore commutes with finite limits. Since limits of sheaves are computed objectwise (see limits and colimits by example) it follows that $f^{*}$ preserves finite limits.

So $f$ does induce a morphism of sites and thus satisfies the assumptions of the above theorem, which tells us that

$$f_{*}:\mathrm{SPSh}(\mathrm{Diff}{)}_{\mathrm{inj}}^{\mathrm{loc}}\overleftarrow{\to }\mathrm{SPSh}(\mathrm{CartSp}{)}_{\mathrm{inj}}^{\mathrm{loc}}:f^{*}$$f_* : SPSh(Diff)_{inj}^{loc} \stackrel{\leftarrow}{\to} SPSh(CartSp)_{inj}^{loc} : f^*

is a Quillen adjunction.

But we know more: as discussed in

• Daniel Dugger, Sheaves and Homotopy Theory (web, pdf)

the Grothendieck topos $\mathrm{Sh}(\mathrm{Diff})$ has enough topos points,

$$(p_n{)}_{*}:\mathrm{Set}\overleftarrow{\to }\mathrm{Sh}(\mathrm{Diff}):(p_n{)}^{*}$$(p_n)_* : Set \stackrel{\leftarrow}{\to} Sh(Diff) : (p_n)^*

one for each $n\in \mathbb{N}$, which are given by taking colimits over the evaluation of simplicial presheaves on the poset of open $n$-disks $D(r)=\{x\in {ℝ}^n\mid \mid x\mid <r\}$ centered at the origin of ${ℝ}^n$

$$(p_n{)}^{*}A=\underrightarrow{\lim }{}_{r\in ℝ}A(D(r)).$$(p_n)^* A = \lim_\to_{r \in \mathbb{R}} A(D(r)) \,.

Since the open $n$-disk is diffeomorphic to ${ℝ}^n$, we may think of these stalks actually as colimits over diagrams in CartSp. It follows that the functor $f_{*}:\mathrm{SPSh}(\mathrm{Diff}{)}_{\mathrm{inj}}^{\mathrm{loc}}\to \mathrm{SPSh}(\mathrm{CartSp}{)}_{\mathrm{inj}}^{\mathrm{loc}}$ preserves all weak equivalences.

By the same logic, for $A\in \mathrm{SPSh}(\mathrm{Diff})$ and $B\in \mathrm{SPSh}(\mathrm{CartSp})$ we find that if a morphism

$$f^{*}A\to B$$f^* A \to B

is a weak equivalence precisely if it is one on all the above disk-stalks, which is the same condition that its adjunct

$$A\to f_{*}B$$A \to f_* B

is a weak equivalence, simply because testing on the topos points $p_n$ takes place entirly in CartSp.

Remark

This fact is noteworthy for the following reason:

By the result of

• Daniel Dugger, Sharon Hollander, Daniel C. Isaksen, Hypercovers and simplicial presheaves (web)

which is described at descent for simplicial presheaves, the fibrant objects in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$ are those that are objectwise Kan complexes and satisfy descent along all hypercovers of representables. But descent on the contractible ${ℝ}^n$s is a drastically simpler condition than on an arbitrary manifold $X$.

For instance, let $G$ be a Lie group and write $BG$ for its corresponding degreewise representable simplicial presheaf $(BG{)}_n=G^{\times n}$.

Then regarded as an object of $\mathrm{SPSh}(\mathrm{Diff}{)}_{\mathrm{proj}}^{\mathrm{loc}}$ the object $BG$ of course does not satisfy descent. Instead, its fibrant replacement is (the recified version of) $G\mathrm{Bund}$, the stack of $G$-principal bundles.

But regarded as an object in $\mathrm{SPSh}(\mathrm{CartSp}{)}_{\mathrm{proj}}^{\mathrm{loc}}$ the object $BG$ does satisfy descent, because every $G$-principal bundle on ${ℝ}^n$ is isomorphic to the trivial one, and the automorphism group of the trivial $G$-bundle is just $C({ℝ}^n,G)$.

So there are considerably more fibrant objects in $\mathrm{SPSh}(\mathrm{CartSp}{)}_{\mathrm{proj}}^{\mathrm{loc}}$ than there are in $\mathrm{SPSh}(\mathrm{Diff}{)}_{\mathrm{proj}}^{\mathrm{loc}}$. Accordingly, there must be less cofibrant objects in $\mathrm{SPSh}(\mathrm{CartSp}{)}^{\mathrm{loc}}$ to compensate this.

Indeed, notice that every representable in any of the model structures on $\mathrm{SPSh}(C)$ is cofibrant. So an arbitrary manifold $X$ is cofibrant in $\mathrm{SPSh}(\mathrm{Diff}{)}_{\mathrm{proj}}^{\mathrm{loc}}$ and therefore there we have

$${\mathrm{Ho}}_{\mathrm{SPSh}(\mathrm{Diff}{)}^{\mathrm{loc}}}(X,BG)\simeq {\pi }_0\mathrm{SPSh}(X,G\mathrm{Bund})\simeq G\mathrm{Bund}(X{)}_{\sim }$$Ho_{SPSh(Diff)^{loc}}(X, \mathbf{B}G ) \simeq \pi_0 SPSh(X, G Bund) \simeq G Bund(X)_\sim

as expected. In $\mathrm{SPSh}(\mathrm{CartSp}{)}_{\mathrm{proj}}^{\mathrm{loc}}$, however, $X$ is in general not representable, hence in general not cofibrant. But by the proposition below, that all objects which are degreewise coproducts of representables are cofibrant in all the model structures, we have that the Čech nerve $C(U)$ of any good cover $U={\coprod }_i{U}_i$ of $X$ (one for which each pathc and all intersections and higher intersections are contractible) is cofibrant. Hence here we find the above result by a different intermediate step

$${\mathrm{Ho}}_{\mathrm{SPSh}(\mathrm{Cart}{)}^{\mathrm{loc}}}(X,BG)\simeq {\pi }_0\mathrm{SPSh}(C(U),BG)\simeq G\mathrm{Bund}(X{)}_{\sim }.$$Ho_{SPSh(Cart)^{loc}}(X, \mathbf{B}G ) \simeq \pi_0 SPSh(C(U), \mathbf{B}G) \simeq G Bund(X)_\sim \,.

Fibrant and cofibrant objects

Fibrant objects

The fibrant objects in the local model structure on simplicial presheaves are those that

• are fibrant with respect to the respective global model structure

• and satisfy descent for simplicial presheaves. See there for more details.

This descent condition is the analog in this model of the sheaf-condition and the stack-condition. In fact, it reduces to these for truncated simplicial presheaves.

Since the fibrancy condition in the global projective model structure is simple – it just requires that the simplicial presheaf is in fact a presheaf of Kan complexes – the local projective model structure has slightly more immediate characterization of fibrant objects than the local injective model structures. (In fact, for suitable choices of sites it may become very simple, as the above discussion of site dependence of the model structure shows).

On the other hand the cofibrancy condition on objects is entirly trivial in the global and local injective model structure: since a cofibration there is just an objectwise cofibration, and since every simplicial set is cofibrant, every object is injective cofibrant.

But the cofibrant objects in the projective structure are not too nasty either: every object that is degreewise a coproduct of representables is cofibrant. In particular the Čech nerves of any good cover (see below for more details) is a projectively cofibrant object.

A cofibrant replacement functor in the local projective structure is discussed in

• Daniel Dugger, Universal homotopy theories (pdf)

Something related to a fibrant replacement functor (”$\mathrm{\infty }$-stackification”) is discussed in section 6.5.3 of

• Jacob Lurie, Higher Topos Theory

Cofibrant objects

In the injective local model structure on simplicial presheaves all objects are cofibrant. For the projective local structure we have the following useful statement.

Proposition

In the projective local model structure all objects that are degreewise coproducts of representables and satisfy a splitness condition (…) are cofibrant.

This is in the proof of lemma 2.7 in section 9 of

* Daniel Dugger, Universal homotopy theories

This splitness condition is in particular satisfied by all Čech nerves of covers by coproducts of representables.

Definition

(good cover)

A Čech nerve $U$ with a weak equivalence $U\stackrel{\simeq }{\to }X$ in $\mathrm{SPSh}(C{)}^{\mathrm{loc}}$ is a good cover if it is degreewise a coproduct of representables.

Remark

This reduces to the ordinary notion of good cover as an open cover by contractible spaces such that all finite intersections of these are again contractibe when using a site like $C=$ CartSp.

Corollary

Any good cover $U\stackrel{\simeq }{\to }X$ is a cofibrant replacement for $X$.

Cofibrant replacement

In

• Daniel Dugger, Universal homotopy theories

a useful cofibrant replacement functor for the projective local model structure is discussed.

Definition

For $A\in \mathrm{PSh}(C)\hookrightarrow \mathrm{SPSh}(C)$ an ordinary presheaf (simplicially discrete simplicial presheaf) let $\tilde{Q}A$ be the simplicial presheaf which in degree $k$ is

$$(\tilde{Q}A{)}_k:=\coprod _{U_k\to U_{k-1}\to \cdots \to U_0\to A}{U}_k,$$(\tilde Q A)_k := \coprod_{U_k \to U_{k-1} \to \cdots \to U_0 \to A} U_k \,,

where the $U_k$ range over the representables, i.e. the objects in $C\hookrightarrow \mathrm{SPSh}(C)$. The face and degeneracy maps are the obvious ones coming from composing maps and inserting identity maps in the labels over which the coproduct ranges.

For $A\in \mathrm{SPSh}(C)$ an arbitrary simplicial presheaf let $QA$ be the diagonal of the bisimplicial presheaf obtained by applying $\tilde{Q}$ degreewise

$$QA=(\cdots \coprod _{U_1\to U_0\to A_1}{U}_1\overrightarrow{\to }\coprod _{U_0\to A_0}{U}_0).$$Q A = \left( \cdots \coprod_{U_1 \to U_0 \to A_1} U_1 \stackrel{\to}{\to}\coprod_{U_0 \to A_0} U_0 \right) \,.

Proposition

For all $A\in \mathrm{SPSh}(C)$ the object $QA$ is cofibrant and is weakly equivalent to $A$ in $\mathrm{SPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}$.

This is in prop 2.8 of

• Daniel Dugger, Universal homotopy theories

Homotopy (co)limits

Properties of homotopy limits and homotopy colimits of simplicial presheaves are discussed at

• Homotopy (co)limits of simplicial (pre)sheaves

References

A nicely helpful introduction and survey is provided in the notes

• Dan Dugger, Sheaves and homotopy theory (web, dvi, pdf)

This in particular gives a detailed account on the relation and difference between the “Čech model structure” (section 3) which localizes (only) at Čech covers, and the Jardine model structure (section 4), which localizes at all stalkwise weak equivalences (hypercovers). The latter is what in HTT is called the hypercompletion .

The standard lecture notes are

• Jardine07 J. Jardine, Field Lectures: Simplicial presheaves (pdf)

based on

• Jardine01 J. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology, homotopy and applications, vol. 3 (2), 2001, pp.361–384

and …

See also

• Benjamin Blander, Local projective model structures on simplicial presheaves, K-Theory, Volume 24, Number 3, November 2001 , pp. 283–301(19) (journal)

A detailed study of descent for simplicial presheaves is given in

• DHI04 Daniel Dugger, Sharon Hollander, Daniel Isaksen, Hypercovers and simplicial presheaves (web)

• DI02 Daniel Dugger, Daniel Isaksen, Weak equivalences of simplicial presheaves (arXiv)

A survey of many of the model structures together with a treatment of the left local projective one is in

• Benjamin Blander, Local projective model structure on simplicial presheaves (pdf)

See also

• Daniel Isaksen, Flasque model structure for simplicial presheaves (web, pdf)

The characterization of the model category of simplicial presheaves as the canonical presentation of the (hypercompletion of) the (∞,1)-category of (∞,1)-sheaves on a site is in

• proposition 6.5.2.1 of

  • Jacob Lurie, Higher Topos Theory

A set of lecture notes on simplicial presheaves with an eye towrads algebraic sites and derived algebraic geometry is

• Bertrand Toen, Simplicial presheaves and derived algebraic geometry , lecture at Simplicial methofs in higher categories (pdf)

Last not least, it is noteworthy that the idea of localizing simplicial sheaves at stalkwise weak equivalences is already described and applied in

• Kenneth Brown?, Abstract Homotopy Theory and Generalized Sheaf cohomology,

using instead of a full model category structure the more lightweight one of a Brown category of fibrant objects.

A comparison between Brown-Gersten and Joyal-Jardine approach:

• V. Voevodsky, Homotopy theory of simplicial presheaves in completely decomposable topologies, arxiv/0805.4578