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 $(\infty,1)$-category $C$ are given by (∞,1)-functors $C^{op} \to$ 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

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

– 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 J. F. 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 homotopy 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:

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

Here

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

• “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 $SPSh(C)$ 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 the those on simplicial presheaves. (For the injective structure this is in Jardine, for the projective one in Blander, theorem 2.1, 2.2).

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.

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

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

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 $Sh(X,SGrpd)\leftrightarrow PSh(X, SGrpd)$)

Reedy and intermediate model structures

To some extent the injective and projective model structures on simplicial presheaves are the two extremes of a larger family of model structures on simplicial presheaves that all have the same weak equivalences but different classes of cofibrations.

Notably if the domain $C$ has the special property that it is a Reedy category there is the Reedy model structure on $[C, sSet]$. Its class of cofibrations is intermediate that of the projective and the injective model structure on functors and we have Quillen equivalences

$$[C,sSet]_{proj} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C,sSet]_{Reedy} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C,sSet]_{inj} \,.$$

For general $C$, there is still a whole family of model structures on $[C^{op}, sSet]$ that interpolates between the injective and the projective model structure. See intermediate model structure.

Dependency 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 $f_* : PSh(D) \to PSh(C)$ preserves sheaves and its left adjoint $f^* : PSh(C) \to PSh(D)$ (given by left Kan extension) is left exact functor in that it preserves finite limits.

Then the induced adjunction

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

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 (JardineLectures).

Proposition

Let $C$ be a site and $f : D \hookrightarrow$ a full dense sub-site. Then right Kan extension $f_* : [D^{op}, sSet] \to [C^{op}, sSet]$ along $f$ yields a simplicial Quillen adjunction

$$(f^* \dashv f_*) : [D^{op}, sSet]_{inj,loc} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} [C^{op}, sSet]_{inj,loc}$$

between the left Bousfield localizations of the projective model structures at the sieve inclusions $S(\{U_i\}) \to U$ for each covering family $\{U_i \to U\}$.

Proof

It is immediate that we have a simplicial Quillen adjunction on the global injective model structure: by definition of right Kan extension we have an sSet-adjunction and the left adjoint restriction functor $f^*$ trivially preserves injective cofibrations and acyclic cofibrations.

Since we have left proper model categories it is sufficient (by the discussion at recognition of simplicial Quillen adjunctions) for deducing that the Quillen adjunction descends to the local strucuture to check that $f_*$ preserves locally fibrant objects, which in turn by properties of left Bousfield localization is equivalent to checking that $f^*$ sends covering sieve inclusions to weak equivalences in $[D^{op}, sSet]_{proj,loc}$.

By the result on generalized covers, for this it is sufficient to check that for every covering sieve $S(\{U_i\}) \to X$ and every representable $K \in D$ and morphism $K \to f^* X$, there is a covering $\{K_j \to K\}$ in $D$ and local lifts

$$\array{ K_j &\to& f^*(S(\{U_i\})) \\ \downarrow && \downarrow \\ K &\to& f^* X } \,.$$

This follows directly from the single defining condition on a coverage on $C$.

Presentation of $(\infty,1)$-toposes

Definition

Let $C$ be a site. Let $[C^{op}, sSet]_{proj}$ be the projective model structure on simplicial presheaves over $C$.

Let $W = \{C(\{U_i\}) \to U\}$ be the set of Čech nerve projections in $[C, sSet]$ for each covering $\{U_i \to U\}$ in $C$.

Write

$$(Id \dashv Id) : [C^{op}, sSet]_{proj,loc} \stackrel{\overset{Id}{\leftarrow}}{\underset{Id}{\to}} [C^{op}, sSet]_{proj}$$

for the left Bousfield localization at $W$.

Write $([C^{op}, sSet]_{proj})^\circ$ for the full sub-simplicially enriched category on the fibrant-cofibrant objects, similarly for $([CartSp^{op}, sSet]_{proj,loc})^\circ$.

Proposition

We have an equivalence of (∞,1)-categories

$$\array{ Sh_{(\infty,1)}(C) &\stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}& PSh_{(\infty,1)}(C) \\ \uparrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}} \\ ([C^{op}, sSet]_{proj,loc})^\circ & \stackrel { \overset{\mathbb{L} Id}{\leftarrow} } { \underset{\mathbb{R}Id}{\to} } & ([C^{op}, sSet]_{proj})^\circ } \,,$$

where at the bottom we have the left and right derived functors of the identity functors, as discussed at simplicial Quillen adjunction.

Proof

This follows using the arguments in the proof of HTT, 6.5.2.14 and HTT, prop. A.3.7.6.

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 entirely 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 Dugger 01.

Something related to a fibrant replacement functor (“$\infty$-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

necessary and sufficient conditions are given here:

• Richard Garner, MO comment

More specifically, there is the following useful statement (see also projectively cofibrant diagram) (see also Low, remark 8.2.3).

Definition

A simplicial presheaf $X \in sPSh(C)$ is said to have free degeneracies or the degenerate cells split off if in each degree there is a sub-presheaf $N_k \hookrightarrow X_k$ such that the canonical mophism

$$\coprod_{\underset{surj.}{\sigma : [k] \to [n]}} N_n \stackrel{\coprod_{\underset{surj.}{\sigma : [k] \to [n]}} \sigma^*}{\to} F_k$$

is an isomorphism.

So if degenerate cells split off we have in particular that

$$X_k = X_k^{nd} \coprod X_k^{dg} \,,$$

where $X_k^{nd}$ is the presheaf of non-degenerate $k$-cells and $X_k^{dg}$ is a separate presheaf containing all the degenerate cells (and itself a coproduct over separate presheaves for each degree and order of degeneracy).

Proposition

In the projective local model structure all objects that are

1. degreewise coproducts of representables

2. and whose degenerate cells split off

are cofibrant.

This is in Dugger 01, Cor. 9.4.

Example

(split hypercovers)

If $Y \to X$ is an acyclic fibration in the local projective model structure with $X$ a representable and $Y$ cofibration in the above way, it is called a split hypercover .

All Čech nerves $C(\{U_i\})$ coming from an open cover have split degeneracies. The condition that the Čech nerve be degreewise a coproduct of representables is a condition akin to that of good open covers (which is precisely the special case for $C =$ CartSp). This is then a split hypercover of height 0.

Definition

(good cover)

A Čech nerve $U$ with a weak equivalence $U \stackrel{\simeq}{\to} X$ in $SPSh(C)^{loc}$ 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.

Cofibrant replacement

In

• Daniel Dugger, Universal homotopy theories

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

Definition

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

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

where the $U_k$ range over the representables, i.e. the objects in $C \hookrightarrow SPSh(C)$. 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 $A \in SPSh(C)$ an arbitrary simplicial presheaf let $Q A$ be the diagonal of the bisimplicial presheaf obtained by applying $\tilde Q$ degreewise

$$Q A = \left( \cdots \coprod_{U_1 \to U_0 \to A_1} U_1 \stackrel{\to}{\to}\coprod_{U_0 \to A_0} U_0 \right) \,.$$

Proposition

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

This is in prop 2.8 of

• Daniel Dugger, Universal Homotopy Theories

Local fibrations

A local fibration or local weak equivalence of simplicial (pre)sheaves is defined to be one whose lifting property is satisfied after refining to some cover.

Warning. Notice that this is a priori unrelated to equivalences and fibrations with respect to any local model structure.

If the site $C$ has enough points, then local fibrations of simplicial presheaves are equivalently those that are stalkwise fibrations of simplicial sets.

This is discussed in (Jardine 96).

Localization and descent

Čech localization at Grothendieck (pre)topologies

We discuss some aspects of the left Bousfield localization of the projective global model structure on simplicial presheaves at Grothendieck topologies and covering families. By the discussion at topological localization these are models for topological localizations leading to (∞,1)-categories of (∞,1)-sheaves.

The central reference is (DuggerHollanderIsaksen) with the central theorem being this one:

Theorem

Let $C$ be a site given by a Grothendieck topology. The left Bousfield localization of $sPSh(C)_{proj}$ and $sPSh(C)_{inj}$, respectively, at the following classes of morphisms exist and coincide:

1. the set of all covering sieve subfunctors $R \hookrightarrow j(X)$;

2. the set of all morphisms $hocolim_R \to U \to X$ for $R$ a covering sieve of $X$;

3. the set of all Čech nerve projections $C(\{U_i\}) \to X$ for $\{U_i \to X\}$ a covering sieve;

4. the class of all bounded hypercovers $U \to X$;

5. the class of morphisms $F \to \bar F$ from a simplicial presheaf $F$ to the simplicial sheaf obtained by degreewise sheafification.

6. if the topology is generated from a basis, then: the set of covering sieve subfunctors $R_U \to X$ for each covering family $\{U_i \to X\}$ in the basis.

This is theorem A5 in DugHolIsak.

This localization $sPSh(C)_{proj,cov}$ is the Čech localization of $sPSh(C)$ with respect to the given Grothendieck topology. It is a presentation of topological localization of an (∞,1)-category of (∞,1)-presheaves to an (∞,1)-category of (∞,1)-sheaves.

$$\array{ Sh_{(\infty,1)}(C) &\stackrel{\overset{L}{\to}}{\hookrightarrow}& Psh_{(\infty,1)}(C) \\ \uparrow^{\mathrlap{\simeq}} && \uparrow^{\mathrlap{\simeq}} \\ (sPSh(C)_{proj,cov})^\circ &\stackrel{\overset{left. Bousf.}{\leftarrow}}{\underset{}{\to}}& (sPSh(C)_{proj})^\circ } \,.$$

The following definition and proposition provides information on what the general morphisms are which become weak equivalences after localization at

Defintion

(generalized cover)

Let $C$ be a site. A local epimorphism (or generalized cover) in $sPSh(C)$ is a morphism $f : E \to B$ of simplicial presheaves with the property that for every representable $U$ and every morphism $j(U) \to B$ there exists a covering sieve $\{U_i \to U\}$ such that for every $U_i \to U$ the composite $U_i \to U \to B$ has a lift $\sigma$ through $f$

$$\array{ j(U_i) &\stackrel{\exists \sigma}{\to}& E \\ \downarrow && \downarrow \\ j(U) &\stackrel{\forall}{\to} & B } \,.$$

(Dugger-Hollander-Isaksen, corollary A.3)

Proposition

For $f : E \to B$ a local epimorphism in $sPSh(C)$ in the above sense, its Čech nerve projection

$$C(E) \to B$$

is a weak equivalence in the projective local model structure $sPSh(C)_{proj, loc}$.

This is Dugger-Hollander-Isaksen, corollary A.3.

$\,$

For values in strict and abelian $\infty$-groupoids

Many simplicial presheaves appearing in practice are (equivalent) to objects in sub-(∞,1)-categories of $Sh_{(\infty,1)}(C)$ of abelian or at least strict ∞-groupoids. These subcategories typically offer convenient and desireable contexts for formulating and proving statements about special cases of general simplicial presheaves.

One well-known such notion is given by the Dold-Kan correspondence. This identifies chain complexes of abelian groups with strict and strictly symmetric monoidal $\infty$-groupoids.

Dropping the condition on symmetric monoidalness we obtain a more general such inclusion, a kind of non-abelian Dold-Kan correspondence:

the identification of crossed complexes of groupoids as precisely the strict ∞-groupoids. This has been studied in particular in nonabelian algebraic topology.

So we have a sequence of inclusions

$$\array{ ChainCplx &\hookrightarrow& CrsCpl &\hookrightarrow& KanCplx \\ \downarrow^{\mathrlap{simeq}} && \downarrow^{\mathrlap{simeq}} && \downarrow^{\mathrlap{simeq}} \\ StrAb Str\infty Grpd &\hookrightarrow& Str \infty Grpd &\hookrightarrow& \infty Grpd }$$

of strict $\infty$-groupoids into all $\infty$-groupoids. See also the cosmic cube of higher category theory.

Among the special tools for handling $\infty$-stacks on $C$ that factor at some point through the above inclusion are the following:

• restriction to abelian sheaf cohomology – Using the fact that the objects of $Sh_{(\infty,1)}(C)$ are modeled by simplicial presheaves symmetric monoidal $\infty$-Lie groupoids are identified under the Dold-Kan correspondence with $\mathbb{N}$-graded chain complexes of sheaves. To these the rich set of tools for abelian sheaf cohomology apply.

• descent for strict $\infty$-groupoid valued sheaves – There is a good theory of descent for (presheaves) with values in strict $\infty$-groupoids (more restrictive than the fully general theory but more general than abelian sheaf cohomology). This goes back to Ross Street and its relation to the full theory has been clarified by Dominic Verity in Verity09.

We state a useful theorem for the computation of descent for presheaves with values in strict ∞-groupoids. Recall the standard terminology for descent, i.e. for the $(\infty,1)$-categorical sheaf-condition:

For $U \in C$ a representable, $Y,A \in [C^{op}, sSet]$ simplicial presheaves and $p : Y \to U$ a morphism, we say that $A$ satisfies descent along $p$ or equivalently that $A$ is a $p$-local object if the canonical morphism

$$A(U) \stackrel{=}{\to} [C^{op}, sSet](U,A) \to [C^{op}, sSet](Y,A)$$

is a weak equivalence. Here the first equality is the enriched Yoneda lemma. By the co-Yoneda lemma we may decompose $Y$ into its cells as

$$Y = \int^{[n] \in \Delta} \Delta[n] \cdot Y_n \,,$$

where in the integrand we have the tensoring of $[C^{op}, sSet]$ over sSet. Using that the enriched hom-functor sends coends to ends, the enriched hom-functor on the right we may equivalently write out as an end

$$\begin{aligned} [C^{op}, sSet](Y,A) & = [C^{op}, sSet](\int^{[n] \in \Delta} \Delta[n] \cdot Y_n ,A) \\ & = \int_{[n] \in \Delta}[C^{op}, sSet](\Delta[n] \cdot Y_n ,A) \\ & = \int_{[n] \in \Delta} sSet(\Delta[n], [C^{op}, sSet](Y_n, A)) \\ & = \int_{[n] \in \Delta} sSet(\Delta[n], A(Y_n)) \\ & =:Desc(Y,A) \end{aligned}$$

(equality signs denote isomorphisms), where in the second but last line we again used the tensoring of simplicial presheaves $[C^{op}, sSet]$ over sSet.

In the last line we have the totalization of the cosimplicial simplicial object

$$A(Y_\bullet) : \Delta \to sSet \,,$$

sometimes called the descent object of $A$ relative to $Y$, even though in this case it is really nothing but the hom-object of $Y$ into $A$. If $A$ is fibrant and $Y$ cofibrant, then $Desc(Y,A)$ is a Kan complex: the descent $\infty$-groupoid .

Now suppose that $\mathcal{A} : C^{op} \to Str \infty Grpd$ is a presheaf with values in strict ∞-groupoids. In the context of strict $\infty$-groupoids the standard $n$-simplex is given by the $n$th oriental $O(n)$. This allows to perform a construction that looks like a descent object in $Str\infty Grpd$:

Definition

(Ross Street)

The descent object for $\mathcal{A} \in [C^{op}, Str \infty Grpd]$ relative to $Y \in [C^{op}, sSet]$ is

$$Desc(Y,\mathcal{A}) := \int_{[n] \in \Delta} Str\infty Cat(O(n), \mathcal{A}(Y_n)) \;\in Str \infty Grpd \,,$$

where the end is taken in $Str \infty Grpd$.

This objects had been suggested by Ross Street to be the right descent object for strict $\infty$-category-valued presheaves in Street03

Under the ∞-nerve functor $N_O : Str\infty Grpd \to sSet$ this yields a Kan complex $N_0 Desc(Y,\mathcal{A})$. On the other hand, applying the $\omega$-nerve directly to $\mathcal{A}$ yields a simplicial presheaf $N_O\mathcal{A}$ to which the above simplicial descent applies.

The following theorem asserts that under certain conditions both notions coincide.

Theorem

(Dominic Verity)

If $\mathcal{A} : C^{op}, Str \infty Grpd$ and $Y : C^{op} \to sSet$ are such that $N_O \mathcal{A}(Y_\bullet) : \Delta \to sSet$ is fibrant in the Reedy model structure $[\Delta, sSet_{Quillen}]_{Reedy}$, then

$$N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A})$$

is a weak homotopy equivalence of Kan complexes.

This is proven in Verity09.

Corollary

If $Y \in [C^{op}, sSet]$ is such that $Y_\bullet : \Delta \to [C^{op}, Set] \hookrightarrow [C^{op}, sSet]$ is cofibrant in $[\Delta, [C^{op}, sSet]_{proj}]_{Reedy}$ then for $\mathcal{A} : C^{op} \to Str \infty Grpd$ we have

$$N_O Desc(Y,\mathcal{A}) \stackrel{\simeq}{\to} Desc(Y, N_O \mathcal{A}) \,.$$

Proof

If $Y_\bullet$ is Reedy cofibrant, then by definition the canonical morphisms

$$\lim_{\to}( ([n] \stackrel{+}{\to} [k]) \mapsto Y_k ) \to Y_n$$

are cofibrations in $[C^{op}, sSet]_{proj}$. Since the latter is an $sSet_{Quillen}$ enriched model category and $N_O \mathcal{A}$ is fibrant, it follows that the hom-functor $[C^{op}, sSet](-, N_O \mathcal{A})$ sends cofibrations to fibrations, so that

$$N_O\mathcal{A}(Y_n) \to \lim_{\leftarrow}( [n]\stackrel{+}{\to} [k] \mapsto N_O\mathcal{A}(Y_k))$$

is a Kan fibration. But this says that $N_O \mathcal{A}(Y_\bullet)$ is Reedy fibrant, so that the assumption of Verity’s theorem is met.

Corollary

For $Y$ the Čech nerve of a good open cover $\{U_i \to X\}$ of a manifold $X$ and any $\mathcal{A} : CartSp^{op} \to Str \infty Grpd$ we have that

$$[C^{op}, sSet](Y,N_O \mathcal{A}) \simeq N_O Desc(Y_\bullet, \mathcal{A}) \,.$$

Proof

By the above is sufices to note that $Y_\bullet$ is cofibrant in $[\Delta^{op}, [C^{op}, sSet]_{proj}]_{Reedy}$ if $Y$ is the Čech nerve of a good open cover. By the assumption of good open cover we have that $Y$ is degreewise a coproduct of representables and that the inclusion of all degenerate $n$-cells into all $n$-cells is a full inclusion into a coproduct, i.e. an inlusion of the form

$$\coprod_{i \in I} U_i \to \coprod_j U_{j \in J}$$

induced from an inclusion of subsets $I \hookrightarrow J$. Since all representables are cofibrant in $[C^{op}, sSet]_{proj}$ such an inclusion is a cofibration.

In conclusion we find that for determining the $\infty$-stack condition for strict $\infty$-Lie groupoids we may equivalently use Street’s formula for strict $\infty$-groupid valued presheaves. This is sometimes useful for computations in low categorical degree.

Properness

The global model structures on simplicial presheaves are all left and right proper model categories. Since left Bousfield localization of model categories preserves left properness (as discussed there), all local model structures on simplicial presheaves are also left proper.

But the local model structures are not in general right proper anymore. A sufficient condition for them to be right proper is given in the following Prop. .

The injective local model structures which are right proper are locally cartesian closed model categories and present locally cartesian closed (infinity,1)-categories (by the discussion there).

Proposition

Let $C$ be a site with enough points. Then the weak equivalences in the local model structures on $sPSh(C)$ are the stalk-wise weak homotopy equivalences of simplicial sets.

(p. 12 here)

Proposition

A sufficient condition for an injective or projective local model structure of simplicial presheaves over a site $C$ to be right proper is that the weak equivalences are precisely the stalk-wise weak equivalences of simplicial sets.

This is mentioned for instance in (Olsson, remark 4.3).

By Prop. this sufficient condition holds, for instance, for the injective Jardine model structure when $C$ has enough points.

Proof

The key is that forming stalks is, being the inverse image of a geometric morphism

$$(x^* \dashv x_*) := Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} Sh(C)$$

an operation that preserves finite limits.

Let therefore $f : X \to A$ be a stalkwise weak equivalence of simplicial presheaves and let $g : A \to B$ be a fibration. Notice that in all the model structures (injective, projective, global, local) the fibrations are always in particular objectwise fibrations.

Then the pullback $g^* f$ in

$$\array{ g^* X &\to& X \\ \big\downarrow {}^{\mathrlap{g^* f}} && \big\downarrow {}^{\mathrlap{f}} \\ A &\stackrel{g}{\to}& B }$$

is a weak equivalence if for all topos points $x$ the stalk $x^* (g^* f)$ is a weak equivalence of simplicial sets. But since stalks preserve finite limits, we have a pullback diagram of simplicial sets

$$\array{ x^*(g^* X) &\to& x^*( X) \\ \downarrow^{\mathrlap{x^*(g^* f)}} && \downarrow^{\mathrlap{x^*(f)}} \\ x^*(A) &\stackrel{x^*(g)}{\to}& x^*(B) } \,.$$

It is now sufficient to observe that $x^* g$ is a Kan fibration, which implies the result by the fact that the classical model structure on simplicial sets is right proper.

To see this, notice that $x^*(g)$ is a Kan fibration precisely if for all $1 \leq k$ and $0 \leq i \leq k$ the morphism

$$(x^* A)^{\Delta[k]} \to (x^* A)^{\Lambda[k]^i} \times_{(x^* B)^{\Lambda[k]^i} } (x^* B)^{\Delta[k]}$$

is an epimorphism of sets. Since stalks commute with finite limits, this is equivalent to

$$x^* \left( A^{\Delta[k]} \to A^{\Lambda[k]^i} \times_{ B^{\Lambda[k]^i} } B^{\Delta[k]} \right)$$

being an epimorphism. Now the morphism in parenthesis is an epimorphism since the fibration $f$ is in particular an objectwise Kan fibration, and left adjoint functors such as $x^*$ preserve epimorphisms.

Closed monoidal structure

If the underlying site has finite products, then both the injective and the projective, the global and the local model structure on simplicial presheaves becomes a monoidal model category with respect to the standard closed monoidal structure on presheaves.

See for instance here.

Lemma

Let $C$ be a category with products. Then the closed monoidal structure on presheaves makes $[C^{op}, sSet]_{proj}$ a monoidal model category.

Proof

It is sufficient to check that the Cartesian product of presheaves

$$\otimes : sPSh(C)_{proj} \times sPSh(C)_{proj} \to sPSh(C)_{proj}$$

is a left Quillen bifunctor. As discussed there, since $sPSh(C)$ is a cofibrantly generated model category for that it is sufficient to check that $\otimes$ satisfies the pushout-prodct axiom? on generating (acyclic) cofibrations.

As discussed at model structure on functors, these are those morphisms of the form

$$Id \times i : U \cdot S \to U \cdot T$$

for $U \in C$ representable and $i : S \to T$ an (acylic) cofibration in $sSet_{Quillen}$. For these morphisms checking the pushout-product axiom amounts to checking it in $sSet$, where it is evident.

Lemma

Let $C$ be a site with products and let $[C^{op}, sSet]_{proj,cov}$ be the left Bousfield localization at the Čech nerve projections.

Then for $X$ any cofibrant object, the closed monoidal structure on presheaves-adjunction

$$(X \times (-) \dashv [X,-]) : [C^{op}, sSet]_{proj,cov} \to [C^{op}, sSet]_{proj,cov}$$

is a Quillen adjunction.

Proof

The above lemma implies that the left adjoint $X \times (-)$ preserves cofibrations. As discussed in the section on sSet-enriched adjunctions at Quillen adjunction since the adjunction is $sSet$-enriched and since $[C^{op}, sSet]_{proj,cov}$ is a left proper simplicial model category it suffices to check that $[X,-]$ preserves fibrant objects.

For that let $\{U_i \to U\}$ be a covering family and $C(\{U_i\})$ the corresponding Čech nerve. We need to check that if $A \in [C^{op}, sSet]_{proj,cov}$ is fibrant, then

$$[C^{op}, sSet](U, [X,A]) \to [C^{op},sSet](C(\{U_i\}), [X,A])$$

is an equivalence of Kan complexes.

Writing $C(\{U_i\}) = \int^{[n]} \Delta[n] \cdot \coprod U_{i_0, \cdots, i_n}$ and using that the hom-functor preserves ends, this is eqivalent to

$$[C^{op},sSet]( X \times C(\{U_i\}) \to X \times U , A)$$

being an equivalence. Now we observe that $X \times C(\{U_i\}) \to X\times U$ is a local epimorphism in the above sense, namely a morphism such that for every morphism $V \to X \times U$ out of a representable, there is a lift $\sigma$

$$\array{ && X \times C(\{U_i\}) \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow \\ V &\to& X \times U } \,.$$

By the above discussion of the Čech-localization of $[C^{op}, sSet]_{proj}$, this is a local morphism, hence does produce an equivalence when hommed into the fibrant object $A$.

Homotopy (co)limits

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

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

Let $C$ be a site.

Proposition

Let $F : D \to [C^{op}, sSet]$ be a finite diagram.

Write $\mathbb{R}_{glob}\lim_{\leftarrow} F \in [C^{op}, sSet]$ for any representative of the homotopy limit over $F$ computed in the global model structure $[C^{op}, sSet]_{proj}$, well defined up to isomorphism in the homotopy category.

Then $\mathbb{R}_{glob}\lim_{\leftarrow} F \in [C^{op},sSet]$ presents also the homotopy limit of $F$ computed in the local model structure $[C^{op}, sSet]_{proj,loc}$.

Proof

By the discussion at (∞,1)-limit the homotopy limit $\mathbb{R}\lim_{\leftarrow}$ computes the corresponding (∞,1)-limit and (∞,1)-sheafification $L$ is a left exact (∞,1)-functor and preserves these finite (∞,1)-limits:

$$\array{ ([D, [C^{op}, sSet]_{proj, loc}]_{inj})^\circ &\stackrel{L_*}{\leftarrow}& ([D, [C^{op}, sSet]_{proj}]_{inj})^\circ \\ \downarrow^{\mathrlap{\mathbb{R} \lim_\leftarrow}} && \downarrow^{\mathrlap{\mathbb{R} \lim_\leftarrow}} \\ ([C^{op}, sSet]_{proj,loc})^\circ &\stackrel{L \simeq \mathbb{L} Id}{\leftarrow}& ([C^{op}, sSet]_{proj})^\circ } \,.$$

Here $L \simeq \mathbb{L} Id$ is the left derived functor of the identity for the above left Bousfield localization. Since left Bousfield localization does not change the cofibrations and includes the global weak equivalences into the local weak equivalences, the postcomposition of the diagram $F$ with $\mathbb{L} Id$ is given by cofibrant replacement in the local structure, too. But the homotopy limit of the diagram is invariant, up to equivalence, under cofibrant replacement, and hence a finite homotopy limit diagram in the global structure is also one in the local structure.

Inclusion of chain complexes of sheaves

We discuss how chain complexes of presheaves of abelian groups embed into the model structure on simplicial presheaves. Under passing to the intrinsic cohomology of the (∞,1)-topos presented by by $[C^{op}, sSet]_{loc}$, this realizes traditional abelian sheaf cohomology over $C$ and generalizes it to general base objects.

Observe from the discussion at model structure on simplicial abelian groups that the degreewise free functor-forgetful functor adjunction $(F \dashv U) : Ab \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Set$ (see algebra over a Lawvere theory for details) induces a Quillen adjunction

$$(F \dashv U) : sAb_{Quillen } \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet_{Quillen}$$

between the model structure on simplicial abelian groups and the classical model structure on simplicial sets, which exhibits $sAb_{Quillen}$ as the corresponding transferred model structure.

Moreover, the Dold-Kan correspondence constitutes in particular a Quillen equivalence

$$(N_\bullet \dashv \Gamma) : Ch_\bullet^+_{proj} \stackrel{\overset{N_\bullet}{\leftarrow}}{\underset{\Gamma}{\to}} sAb_{Quillen}$$

between the projective model structure on chain complexes of abelian groups in non-negative degree and simplicial abelian groups.

We write

$$(N_\bullet F \dashv \Xi) : Ch_\bullet^+_{proj} \stackrel{\overset{N_\bullet}{\leftarrow}}{\underset{\Gamma}{\to}} sAb_{Quillen} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet_{Quillen}$$

for the composite Quillen adjunction. For $C$ any category, postcomposition with $\Xi$ induces a Quillen adjunction

$$(N_\bullet F \dashv \Xi) : [C^{op}, Ch_\bullet^+_{proj}]_{proj} \stackrel{\overset{N_\bullet F}{\leftarrow}}{\underset{\Xi}{\to}} [C^{op}, sSet]_{proj}$$

between the projective model structure on functors $[C^{op}, Ch_\bullet^+_{proj}]_{proj}$ and the global projective model structure on simplicial presheaves, which by convenient abuse of notation we denote by the same symbols.

Related concepts

• local fibration

• Brown-Gersten property

model topos

• model structure on simplicial presheaves

• model structure on simplicial sheaves

• model structure on sSet-enriched presheaves

• model structure on cubical presheaves

• model structure on presheaves of spectra

• model structure for (2,1)-sheaves

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

References

Introductions:

• Dan Dugger, Sheaves and homotopy theory, (1998) [web, dvi, pdf, pdf]

• John F. Jardine, Simplicial Presheaves, lecture notes, Fields Institute (2007) [pdf, webpage]

Detailed discussion of the injective model structures on simplicial presheaves:

• John F. Jardine, Simplicial presheaves, Journal of Pure and Applied Algebra 47 (1987) 35-87 [doi:10.1016/0022-4049(87)90100-9]

• John F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology, homotopy and applications, vol. 3 (2), 2001, pp.361–384 (euclid:hha/1139840259)

• John F. Jardine, Boolean localization, in practice, Documenta Mathematica 1 (1996), 245-275 (documenta:vol-01/13)

• John F. Jardine, Local homotopy theory, Springer Monographs in Mathematics (2015) [doi:10.1007/978-1-4939-2300-7]

The projective model structure is discussed in

• Daniel Dugger, Universal homotopy theories, Advances in Mathematics Volume 164, Issue 1, (2001) Pages 144-176 (pdf, arXiv:math/0007070, doi:10.1006/aima.2001.2014)

See also

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

On derived hom-spaces (function complexes) in the projective model structure on simplicial presheaves:

• William Dwyer, Daniel Kan, Function complexes for diagrams of simplicial sets, Indagationes Mathematicae (Proceedings) 86 2 (1983) 139-147 [doi:10.1016/1385-7258(83)90051-3, pdf]

A brief review in the context of nonabelian Hodge theory is in section 4 of

• Martin Olsson, Towards non-abelian $p$-adic Hodge theory in the good reduction case (pdf)

A detailed study of descent for simplicial presheaves is given in

• Daniel Dugger, Sharon Hollander, Daniel Isaksen, Hypercovers and simplicial presheaves, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51 (web)

• 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 towards 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:

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

The proposal for descent objects for strict $\infty$-groupoid-valued presheaves discussed in Descent for strict infinity-groupoids appeared in

• Ross Street, Categorical and combinatorial aspects of descent theory (arXiv)

The relation to the general descent conditionF is discussed in

• Dominic Verity, Relating descent notions

A useful collection of facts is in

• Zhen Lin Low, Notes on homotopical algebra