+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[model structure on simplicial algebras]] for [[simplicial Lie algebras]].

## Details
 {#Details}

+-- {: .num_defn #CategoryOfSimplicialLieAlgebras}
###### Definition

For $k$ some [[field]], write $LieAlg_k$ for the [[category]] of [[Lie algebras]] over $k$, and write

$$
  ((LieAlg)_k)^{\Delta^{op}}
  \in sSetCat
$$

for the category of [[simplicial objects]] in that category, naturally regarded as a [[simplicial category]]. Say a morphism in that category is a _weak equivalence_ or _fibration_, if its underlying morphism of [[simplicial sets]] is a weak equivalence or fibration, respectively, in the [[classical model structure on simplicial sets]]. Write $((LieAlg)_k)^{\Delta^{op}}_{proj}$ for the category equipped with these classes of morphism.

=--

+-- {: .num_prop #ProjectiveModelStructureOnSimplicialLieAlgebras}
###### Proposition

The category $((LieAlg)_k)^{\Delta^{op}}_{proj}$ from def. \ref{CategoryOfSimplicialLieAlgebras} is a [[simplicial model category]].

=--

Since [[Lie algebras]] are algebras over a [[Lawvere theory]], this is a special case of ([Reedy 74, theorem I](https://ncatlab.org/nlab/show/model+structure+on+simplicial+algebras#Reedy74)) discussed at _[[model structure on simplicial algebras]]_. Without the simplicial enrichment and on [[reduced simplicial sets|reduced]] simplicial Lie algebras this model structure is a special case of ([Quillen 67, II.4 theorem 4](#Quillen67)) and also left as an exercise for the reader in ([Quillen 69, below theorem 4.7](#Quillen69)).

## Properties

### Relation to dg-Lie algebras

+-- {: .num_defn #dgLieAlgebraOfASimplicialLieAlgebra}
###### Definition

Let $(\mathfrak{g}, [-,-]$ be a [[simplicial Lie algebra]]. Then the [[normalized chains complex]] $N \mathfrak{g}$ of the underlying [[simplicial abelian group]] becomes a [[dg-Lie algebra]] by equipping it with the [[Lie bracket]] given by the following [[composition|composite]] morphisms

$$
  [-,-]_{N \mathfrak{g}}
  \;\colon\;
  (N \mathfrak{g}) \otimes_k (N \mathfrak{g})
    \overset{\nabla}{\longrightarrow}
  N (\mathfrak{g} \otimes_k \mathfrak{g})
    \overset{N([-,-])}{\longrightarrow}
  N (\mathfrak{g})
$$

where the first morphism is the [[Eilenberg-Zilber map]].

This construction extends to a [[functor]]

$$
  N
    \;\colon\;
  LieAlg_k^{\Delta^{op}}
    \longrightarrow
  dgLieAlg_k
$$

from simplical Lie algebras to [[dg-Lie algebras]].

=--

([Quillen 69, (4.3)](#Quillen69))

+-- {: .num_prop #NormalizedChainsFromSimplicialLieAlgtodgLieAlgHasLeftAdjoint}
###### Proposition

The functor $N$ from simplicial Lie algebras to dg-Lie algebras from def. \ref{dgLieAlgebraOfASimplicialLieAlgebra} has a [[left adjoint]]

$$
  (N^* \dashv N)
    \;\colon\;
  LieAlg_k^{\Delta^{op}}
    \underoverset
      {\underset{N}{\longrightarrow}}
      {\overset{N^*}{\longleftarrow}}
      {\bot}
  dgLieAlg_k
  \,.
$$

=--

This is ([Quillen 69, prop. 4.4](#Quillen69)).

+-- {: .num_prop #QuillenAdjunctionTodgLieAlgebras}
###### Proposition

With respect to the projective model structure on simplicial Lie algebras from prop. \ref{ProjectiveModelStructureOnSimplicialLieAlgebras} and the projective [[model structure on dg-Lie algebras]] ([this prop.](model+structure+on+dg-Lie+algebras#ProjectiveModelStructureOndgLieAlgebras)) the [[adjunction]] from prop. \ref{NormalizedChainsFromSimplicialLieAlgtodgLieAlgHasLeftAdjoint} is a [[Quillen adjunction]]

$$
  (N^* \dashv N)
    \;\colon\;
  (LieAlg_k^{\Delta^{op}})_{proj}
    \underoverset
      {\underset{N}{\longrightarrow}}
      {\overset{N^*}{\longleftarrow}}
      {\bot}
  (dgLieAlg_k)_{proj}
  \,.
$$

=--

+-- {: .proof}
###### Proof

By the standard [[Dold-Kan correspondence]] $N$ preserves fibrations and weak equivalences  ([this prop.](Dold-Kan+correspondence#QuillenEquivalenceBetweensAbAndCh), [Schwede-Shipley 02, section 4.1](#SchwedeShipley), [p.17](http://www.math.uic.edu/~bshipley/monoidalequi.final.pdf#page=17)).

=--

+-- {: .num_prop }
###### Proposition

The [[adjunction]] between [[homotopy categories]] of the [[derived functors]] of the [[Quillen adjunction]] in prop. \ref{QuillenAdjunctionTodgLieAlgebras}

$$
  Ho(LieAlg_k^{\Delta^{op}})
    \underoverset
      {\underset{\mathbb{R}N}{\longrightarrow}}
      {\overset{\mathbb{L}N^*}{\longleftarrow}}
      {\bot}
  Ho(dgLieAlg_k)
$$

restricts on connected simplicial algebras and connected [[dg-Lie algebras]] (meaning: having the trivial Lie algebra in degree 0) to an [[equivalence of categories]]:

$$
  Ho(LieAlg_k^{\Delta^{op}})_{\geq 1}
    \underoverset
      {\underset{\mathbb{R}N}{\longrightarrow}}
      {\overset{\mathbb{L}N^*}{\longleftarrow}}
      {\simeq}
  Ho(dgLieAlg_k)_{\geq 1}
  \,.
$$

=--

([Quillen 76, section II.4, see p. 224 (20 of 92)](#Quillen76), see also figure 2 on p. 211 (8 of 92))

## Related concepts

* [[model structure on dg-Lie algebras]]

* [[model structure for L-infinity algebras]]

* [[model structure on dg-coalgebras]]

## References

The model structure for for simplicial %T%Palgebras is originally due to

* {#Quillen67} [[Dan Quillen]], _Homotopical Algebra_, Lectures Notes in Mathematics 43, Springer Verlag, Berlin, (1967)

and specifically for [[reduced simplicial set|reduced]] simplicial Lie algebras due to

* {#Quillen69} [[Dan Quillen]], _Rational homotopy theory_, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([JSTOR](http://www.jstor.org/stable/1970725), [pdf](http://www.math.northwestern.edu/~konter/gtrs/rational.pdf))

On the [[homotopy theory]] of simplicial Lie algebras see also

* [[Stewart Priddy]], _On the homotopy theory of simplicial Lie algebras_, ([pdf](http://www.ams.org/journals/proc/1970-025-03/S0002-9939-1970-0270371-9/S0002-9939-1970-0270371-9.pdf))

* [[Graham Ellis]], _Homotopical aspects of Lie algebras_ Austral. Math. Soc. (Series A) 54 (1993), 393-419 ([web](http://anziamj.austms.org.au/JAMSA/V54/Part3/Ellis.html))

See also

* {#SchwedeShipley02} [[Stefan Schwede]], [[Brooke Shipley]], _Equivalences of monoidal model categories_, Algebr. Geom. Topol. 3 (2003), 287--334 ([arXiv:math.AT/0209342](http://arxiv.org/abs/math.AT/0209342))

[[!redirects model structures on simplicial Lie algebras]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

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

Recall that

* a [[model category]] is a way to [[presentable (infinity,1)-category|present]] an [[(∞,1)-category]];

* in the context of [[(∞,1)-categories]] [[presheaf|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
[[presentable (infinity,1)-category|presented]]
by a [[model category]] structure on the ordinary
[[functor category]]

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

-- the category of [[simplicial presheaf|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 [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-category of ∞-stacks]] is given it is shown -- proposition 6.5.2.1 --
that, indeed, the Brown--Joyal--Jardine model is a [[presentable (infinity,1)-category|presentation]] of that.

More precisely

* the [[global model structure on simplicial presheaves]] on a category is a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-presheaves]];

* the [[Čech model structure on simplicial presheaves]] on a [[site]] is a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-sheaves]];

* the [[local model structure on simplicial presheaves]] on a [[site]] is a [[presentable (infinity,1)-category|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 equivalence|Quillen equivalent]] to the injective [[local model structure on simplicial presheaves]].

By repackaging [[Kan complex]]es 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 [[point of a topos|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 category|homotopical categories]] define equivalent [[homotopy category|homotopy categories]].

By Lurie's result these are in each case in turn equivalent to the [[homotopy category of an (infinity,1)-category|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 complex]]es 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 [[presentable (infinity,1)-category|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 [[stalk]]wise acyclic fibrations if the [[point of a topos|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 on simplicial presheaves|global model structure]] and this is a [[Quillen equivalence]].

The [[local model structure on simplicial sheaves|local model structures on simplicial sheaves]] are just the restrictions of the [[local model structure on simplicial presheaves|those on simplicial presheaves]]. (For the injective structure this is in [Jardine](#JardineLecture), for the projective one in [Blander, theorem 2.1, 2.2](#Blander)).

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 [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category of (∞,1)-presheaves]]

while

* the local model structures [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category of (∞,1)-sheaves]] (i.e. [[∞-stacks]]).

The following diagram collection [[model category|model categories]] that are [[presentable (infinity,1)-category|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 equivalence]]s

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

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 {#DependencyOnSite}

+-- {: .num_prop #SiteDependence}
###### Proposition

Let $C,D$ be [[site]]s and let $f : C \to D$ be a [[functor]] that induces a
morphism of [[site]]s in that $f_* : PSh(D) \to PSh(C)$ preserves [[sheaf|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 limit]]s.

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}
###### Proof

This is "little fact 5)" on page 10, 11 of ([JardineLectures](#JardineLecture)).

=--

+-- {: .num_prop}
###### Proposition

Let $C$ be a [[site]] and $f : D \hookrightarrow$ a [[full subcategory|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 [[Bousfield localization of model categories|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}
###### 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 [[Bousfield localization of model categories|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](#GeneralizedCover), 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 {#PresentationOfInfiniToposes}

+-- {: .num_def}
###### 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 [[Bousfield localization of model categories|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
$([CartSp^{op}, sSet]_{proj,loc})^\circ$.

=--

+-- {: .num_prop #PresentationOfTheInfinTopos}
###### 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 functor]]s of the identity functors, as discussed at [[simplicial Quillen adjunction]].

=--

+-- {: .proof}
###### Proof

This follows using the arguments in the proof of [[Higher Topos Theory|HTT, 6.5.2.14]] and [[Higher Topos Theory|HTT, prop. A.3.7.6]].

=--

## Fibrant and cofibrant objects {#FibAndCofibObjects}

### 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 on simplicial presheaves|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 complex]]es -- 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 [[site]]s 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 nerve]]s 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](#Dugger01).

Something related to a **fibrant replacement** functor ("$\infty$-stackification") is discussed in section 6.5.3 of

* [[Jacob Lurie]], [[Higher Topos Theory]]

### Cofibrant objects
 {#CofibrantObjects}

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:

* {#Garner13} [[Richard Garner]], [MO comment](https://mathoverflow.net/a/127187/381)

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

+-- {: .num_defn }
###### 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 [[subobject|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).

+-- {: .num_prop }
###### Proposition

In the _projective_ [[local model structure on simplicial presheaves|local model structure]] all objects that are

1. degreewise [[coproducts]] of [[representable functor|representables]]

1. and whose degenerate cells split off

are cofibrant.

=--

This is in [Dugger 01, Cor. 9.4](#Dugger01).

+-- {: .num_example }
###### 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 nerve]]s $C(\{U_i\})$ 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 cover]]s (which is precisely the special case for $C = $ [[CartSp]]). This is then a split hypercover of _height_ 0.

=--

+-- {: .num_defn }
###### 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 [[representable functor|representable]]s.

=--

+-- {: .num_remark }
###### 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 {#CofibrantReplacement}

In

* [[Daniel Dugger]], _[[DuggerUniv.pdf:file]]_

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

+-- {: .num_defn }
###### 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

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

=--

+-- {: .num_prop }
###### 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 [[point of a topos|enough points]], then local fibrations of simplicial presheaves are equivalently those that are  [[stalk]]wise fibrations of simplicial sets.

This is discussed in ([Jardine 96](#Jardine96)).

## Localization and descent {#Descent}

### Čech localization at Grothendieck (pre)topologies {#ČechLocalization}

We discuss some aspects of the [[Bousfield localization of model categories|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 localization]]s leading to [[(∞,1)-categories of (∞,1)-sheaves]].

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

+-- {: .num_theorem}
###### Theorem

Let $C$ be a [[site]] given by a [[Grothendieck topology]]. The left [[Bousfield localization of model categories|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]] [[subfunctor]]s $R \hookrightarrow j(X)$;

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

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

1. the class of all bounded [[hypercover]]s $U \to X$;

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

1. if the topology is generated from a [[basis for a topology|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](http://front.math.ucdavis.edu/0205.5027).

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

+-- {: .num_def #GeneralizedCover}
###### 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 functor|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](#DuggerHollanderIsaksen))

+-- {: .num_prop}
###### 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](#DuggerHollanderIsaksen).

$\,$

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

  {#DescentForStrictInf}

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 ∞-groupoid]]s. 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 complex]]es of [[abelian group]]s with strict and strictly [[symmetric monoidal (infinity,1)-category|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 complex]]es of groupoids as precisely the [[strict omega-groupoid|strict ∞-groupoid]]s. 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 complex|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](#Verity).

We state a useful theorem for the computation of [[descent]] for presheaves with values in [[strict ∞-groupoid]]s. 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 [[copower|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 [[isomorphism]]s), where in the second but last line we again used the [[copower|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 ∞-groupoid]]s. 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 $Str\infty Grpd$:

+-- {: .num_defn }
###### 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](#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.

+-- {: .num_theorem }
###### 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 complex]]es.

=--

This is proven in [Verity09](#Verity).

+-- {: .num_corollary }
###### 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}
###### 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.

=--

+-- {: .num_corollary }
###### 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}
###### 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
 {#Properness}

The global model structures on simplicial presheaves are
all [[left proper model categories|left]] and [[right proper model categories]].
Since left [[Bousfield localization of model categories]] preserves
left properness (as discussed [there](Bousfield+localization+of+model+categories#ExistenceForLeftProperCombinatorialSimplicialModelCategories)), 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. \ref{StalkwiseWeakEquivalencesImpliesRightProperness}.

The injective local model structures which are right proper are [[locally cartesian closed model categories]] and [[presentable (infinity,1)-category|present]] [[locally cartesian closed (infinity,1)-categories]] (by the discussion [there](locally+cartesian+closed+infinity,1-category#PresentationTheorem)).

+-- {: .num_prop #OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks}
###### Proposition

Let $C$ be a [[site]] with [[point of a topos|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](http://www.math.uiuc.edu/K-theory/0175/))

+-- {: .num_prop #StalkwiseWeakEquivalencesImpliesRightProperness}
###### 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 [[simplicial weak equivalence|weak equivalences of simplicial sets]].

=--

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

By Prop. \ref{OverSiteWithEnoughPointsWeakEquivalencesDetectedOnStalks} this sufficient condition holds, for instance, for the injective Jardine model structure when $C$ has [[point of a topos|enough points]].

+-- {: .proof}
###### Proof

The key is that forming [[stalks]] is, being the [[inverse image]] of a [[point of a topos|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 {#MonoidalStructure}

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](http://www.math.univ-toulouse.fr/~toen/crm-2008.pdf#page=24).

+-- {: .num_lemma}
###### 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}
###### 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 [[cofibrantly generated model category|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.

=--

+-- {: .num_lemma}
###### Lemma

Let $C$ be a [[site]] with [[product]]s and let $[C^{op}, sSet]_{proj,cov}$ be the left [[Bousfield localization of model categories|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}
###### 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 [[proper model category|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 complex]]es.

Writing $C(\{U_i\}) = \int^{[n]} \Delta[n] \cdot \coprod U_{i_0, \cdots, i_n}$ and using that the [[hom-functor]] preserves [[end]]s, 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 {#HomotopyLimits}

Properties of [[homotopy limit]]s and [[homotopy colimit]]s of simplicial presheaves are discussed at

* [Homotopy (co)limits of simplicial (pre)sheaves](http://ncatlab.org/nlab/show/homotopy+limit#SimpSheaves)

Let $C$ be a [[site]].

+-- {: .num_prop}
###### Proposition

Let $F : D \to [C^{op}, sSet]$ be a [[finite limit|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}
###### 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)-limit]]s:

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

Here $L \simeq \mathbb{L} Id$ is the left [[derived functor]] of the identity for the [above](#PresentationOfTheInfinTopos) 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 {#InclusionOfChainComplexes}

We discuss how [[chain complex]]es of presheaves of [[abelian group]]s
embed into the model structure on simplicial presheaves. Under passing to the intrinsic [[cohomology]] of the [[(∞,1)-topos]]
[presented by](#PresentationOfInfiniToposes) 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 group]]s 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]]

[[!include locally presentable categories - table]]

## References

Introductions:

* {#Dugger98} [[Dan Dugger]], _Sheaves and homotopy theory_, (1998) [[web](http://www.uoregon.edu/~ddugger/cech.html), [dvi](http://www.uoregon.edu/~ddugger/cech.dvi), [pdf](http://ncatlab.org/nlab/files/cech.pdf), [[DuggerSheavesAndHomotopyTheory.pdf:file]]]

* {#Jardine07} [[John F. Jardine]], *Simplicial Presheaves*, lecture notes, Fields Institute (2007) [[pdf](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf), [webpage](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields_lectures.html)]

Detailed discussion of the injective model structures on simplicial presheaves:

* {#JardineLecture} [[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](https://projecteuclid.org/euclid.hha/1139840259))

* {#Jardine96} [[John F. Jardine]], _Boolean localization, in practice_, Documenta Mathematica **1** (1996), 245-275 ([documenta:vol-01/13](https://www.math.uni-bielefeld.de/documenta/vol-01/13.html))

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

The projective model structure is discussed in

* {#Dugger01} [[Daniel Dugger]], _Universal homotopy theories_, Advances in Mathematics Volume 164, Issue 1, (2001) Pages 144-176  ([pdf](http://hopf.math.purdue.edu/Dugger/dduniv.pdf), [arXiv:math/0007070](https://arxiv.org/abs/math/0007070), [doi:10.1006/aima.2001.2014](https://doi.org/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](http://dx.doi.org/10.1023/a:1013302313123).

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](https://core.ac.uk/download/pdf/82652265.pdf)]

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

* {#Olsson} Martin Olsson, _Towards non-abelian $p$-adic Hodge theory in the good reduction case_ ([pdf](http://math.berkeley.edu/~molsson/PHT3-24-08.pdf))

A detailed study of [[descent]] for simplicial presheaves is given in

* {#DuggerHollanderIsaksen} [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], _Hypercovers and simplicial presheaves_, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 9–51 ([web](http://www.math.uiuc.edu/K-theory/0563/))

* [[Daniel Dugger]], [[Daniel Isaksen]], _Weak equivalences of simplicial presheaves_ ([arXiv](http://arxiv.org/abs/math/0205025))

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

* {#Blander} [[Benjamin Blander]], _Local projective model structure on simplicial presheaves_ ([pdf](http://www.math.uiuc.edu/K-theory/0462/combination2.pdf))

See also

* [[Daniel Isaksen]], _Flasque model structure for simplicial presheaves_  ([web](http://www.math.uiuc.edu/K-theory/0679/), [pdf](http://www.math.uiuc.edu/K-theory/0679/flasque.pdf))

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

* [proposition 6.5.2.1](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf#page=528) 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](http://www.crm.es/HigherCategories/)  ([pdf](http://www.crm.cat/HigherCategories/hc1.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]], _[[BrownAHT|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](http://arxiv.org/abs/0805.4578)

The proposal for descent objects for strict $\infty$-groupoid-valued presheaves discussed in [Descent for strict infinity-groupoids](#DescentForStrictInf) appeared in

* [[Ross Street]], _Categorical and combinatorial aspects of descent theory_ ([arXiv](http://arxiv.org/abs/math/0303175)) {#Street03}

The relation to the general descent conditionF is discussed in

* [[Dominic Verity]], _[[Verity on descent for strict omega-groupoid valued presheaves|Relating descent notions]]_ {#Verity}

A useful collection of facts is in

* {#Low} [[Zhen Lin Low]], _[[Notes on homotopical algebra]]_

[[!redirects model structures on simplicial presheaves]]

[[!redirects model category of simplicial presheaves]]
[[!redirects model categories of simplicial presheaves]]

[[!redirects projective model structure on simplicial presheaves]]
[[!redirects projective model structures on simplicial presheaves]]

[[!redirects projective global model structure on simplicial presheaves]]
[[!redirects projective global model structures on simplicial presheaves]]

[[!redirects injective model structure on simplicial presheaves]]
[[!redirects injective model structures on simplicial presheaves]]

[[!redirects injective global model structure on simplicial presheaves]]
[[!redirects injective global model structures on simplicial presheaves]]

[[!redirects intermediate model structure on simplicial presheaves]]
[[!redirects intermediate model structures on simplicial presheaves]]


model structure on simplicial sets

+-- {: .rightHandSide}
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###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[simplicial set|Simplicial sets]] are the archetypical combinatorial "[[model category|model]]" for the [[(∞,1)-category]] of (compactly generated weakly Hausdorff) [[topological space]]s and equivalently that of [[∞-groupoid]]s, as well as a standard model for the [[(∞,1)-category of (∞,1)-categories]] [[(∞,1)Cat]] itself.

This statement is made precise by the existence of the structure of a [[model category]] on [[sSet]], called the **[[classical model structure on simplicial sets]]**  that is a [[presentable (infinity,1)-category|presentation]] for the [[(infinity,1)-category]] [[Top]], as well as the **Joyal model structure** which similarly is a presentation of the $(\infty,1)$-category $(\infty,1)Cat$.

## Classical Model Structure

The [[classical model structure on simplicial sets]], $sSet_{Quillen}$, has the following distinguished classes of morphisms:

+-- {: .num_defn}
###### Definition

* The **cofibrations** $C$ are simply the [[monomorphism]]s $f : X \to Y$ which are precisely the levelwise injections, i.e. the morphisms of simplicial sets such that $f_n : X_n \to Y_n$ is an injection of sets for all $n \in \mathbb{N}$.

* The **weak equivalences** $W$ are **weak homotopy equivalences**, i.e., morphisms whose [[geometric realization]] is a weak homotopy equivalence of topological spaces.

=--

* The **fibrations** $F$ are the **[[Kan fibration]]s**, i.e., maps $f : X \to Y$ which have the [[right lifting property]] with respect to all [[horn]] inclusions.
  $$
    \array{
      \Lambda^k[n] &\to& X
      \\
      \downarrow &{}^\exists\nearrow& \downarrow^f
      \\
      \Delta[n] &\to& Y
    }
    \,.
  $$

* A morphism $f : X \to Y$ of fibrant simplicial sets / [[Kan complex]]es is a weak equivalence precisely if it induces an [[isomorphism]] on all [[simplicial homotopy groups]].

* All simplicial sets are cofibrant with respect to this model structure.

* The fibrant objects are precisely the [[Kan complex|Kan complexes]].

+-- {: .num_prop}
###### Proposition

The **acyclic fibrations** (i.e. the maps that are both fibrations as well as weak equivalences) between [[Kan complex]]es are precisely the morphisms $f : X \to Y$ that have the [[right lifting property]] with respect to all inclusions $\partial \Delta[n] \hookrightarrow \Delta[n]$ of boundaries of $n$-simplices into their $n$-simplices
  $$
    \array{
      \partial \Delta[n] &\to& X
      \\
      \downarrow &{}^\exists\nearrow& \downarrow^f
      \\
      \Delta[n] &\to& Y
    }
    \,.
  $$

=--

This appears spelled out for instance as ([Goerss-Jardine, theorem 11.2](#GoerssJardine)).

In fact:

+-- {: .num_prop}
###### Proposition

$sSet_{Quillen}$ is a [[cofibrantly generated model category]] with

* generating cofibrations the [[boundary]] inclusions $\partial \Delta[n] \to \Delta[n]$;

* generating acyclic cofibrations the [[horn]] inclusions $\Lambda^i[n] \to \Delta[n]$.

=--

+-- {: .num_theorem}
###### Theorem

The [[singular simplicial complex]]-functor and [[geometric realization]]

$$
  ({\vert -\vert}\dashv Sing)
  :
  Top_{Quillen}
  \stackrel{\overset{{\vert -\vert}}{\leftarrow}}{\underset{Sing}{\to}}
  sSet_{Quillen}
$$

constitutes a [[Quillen equivalence]] with the standard [[Quillen model structure on topological spaces]].

=--

For more on this see [[homotopy hypothesis]].

### Characterisations of weak homotopy equivalences

+-- {: .num_theorem}
###### Theorem

Let $W$ be the smallest class of morphisms in $sSet$ satisfying the following conditions:

1. The class of monomorphisms that are in $W$ is closed under [[pushout]], [[transfinite composition]], and [[retracts]].
2. $W$ has the [[two-out-of-three]] property in $sSet$ and contains all the [[isomorphisms]].
3. For all natural numbers $n$, the unique morphism $\Delta [n] \to \Delta [0]$ is in $W$.

Then $W$ is the class of weak homotopy equivalences.
=--

+-- {: .proof}
###### Proof

* First, notice that the horn inclusions $\Lambda^0 [1] \hookrightarrow \Delta [1]$ and $\Lambda^1 [1] \hookrightarrow \Delta [1]$ are in $W$.
* Suppose that the horn inclusion $\Lambda^k [m] \hookrightarrow \Delta [m]$ is in $W$ for all $m \lt n$ and all $0 \le k \le m$. Then for $0 \le l \le n$, the horn inclusion $\Lambda^l [n] \hookrightarrow \Delta [n]$ is also in $W$.
* Quillen's [[small object argument]] then implies all the trivial cofibrations are in $W$.
* If $p : X \to Y$ is a trivial Kan fibration, then its right lifting property implies there is a morphism $s : Y \to X$ such that $p \circ s = id_Y$, and the two-out-of-three property implies $s : Y \to X$ is a trivial cofibration. Thus every trivial Kan fibration is also in $W$.
* Every weak homotopy equivalence factors as $p \circ i$ where $p$ is a trivial Kan fibration and $i$ is a trivial cofibration, so every weak homotopy equivalence is indeed in $W$.
* Finally, noting that the class of weak homotopy equivalences satisfies the conditions in the theorem, we deduce that it is the _smallest_ such class.

=--

As a corollary, we deduce that the classical model structure on $sSet$ is the smallest (in terms of weak equivalences) model structure for which the cofibrations are the monomorphisms and the weak equivalences include the (combinatorial) homotopy equivalences.

+-- {: .num_prop}
###### Proposition

Let $\pi_0 : sSet \to Set$ be the connected components functor, i.e. the left adjoint of the constant functor $cst : Set \to sSet$. A morphism $f : Z \to W$ in $sSet$ is a weak homotopy equivalence if and only if the induced map
$$\pi_0 K^f : \pi_0 K^W \to \pi_0 K^Z$$
is a bijection for all _Kan complexes_ $K$.

=--

+-- {: .proof}
###### Proof

One direction is easy: if $K$ is a Kan complex, then axiom SM7 for [[simplicial model categories]] implies  the functor $K^{(-)} : sSet^{op} \to sSet$ is a right [[Quillen functor]], so Ken Brown's lemma implies it preserves all weak homotopy equivalences; in particular, $\pi_0 K^{(-)} : sSet^{op} \to Set$ sends weak homotopy equivalences to bijections.

Conversely, when $K$ is a Kan complex, there is a natural bijection between $\pi_0 K^X$ and the hom-set $Ho (sSet) (X, K)$, and thus by the [[Yoneda lemma]], a morphism $f : Z \to W$ such that the induced morphism $\pi_0 K^W \to \pi_0 K^Z$ is a bijection for all Kan complexes $K$ is precisely a morphism that becomes an isomorphism in $Ho (sSet)$, i.e. a weak homotopy equivalence.

=--

### Relation to the model structure on strict $\infty$-groupoids

> under construction

Recall the [[model structure on strict omega-groupoids]] and the [[omega-nerve]] operation

$$
  N : Str \infty Grpd \to Kan Complx
  \,.
$$

> this ought to be a Quillen functor, but is it?

As a warmup, let $C, D$ be ordinary [[groupoid]]s and $N(C)$, $N(D)$ their ordinary [[nerve]]s. We'd like to show in detail that

+-- {: .num_prop}
###### Proposition

A [[functor]] $F : C \to D$ is

* [[k-surjective functor|k-surjective]] for all $k$ and hence a surjective [[equivalence of categories]] precisely if under the [[nerve]] $N(F) : N(C) \to N(D)$ it induces an acyclic fibration of Kan complexes;

=--

+-- {: .proof}
###### Proof

We know that both $N(C)$ and $N(D)$ are Kan complexes. By the above theorem it suffices to show that $N(f)$ being a surjective equivalence is the same as having all lifts

$$
  \array{
    \partial \Delta[n] &\to& N(C)
    \\
    \downarrow &{}^\exists\nearrow& \downarrow^{N(F)}
    \\
    \Delta[n] &\to& N(D)
  }
  \,.
$$

We check successively what this means for increasing $n$:

* $n= 0$. In degree 0 the boundary inclusion is that of the empty set into the [[point]] $\emptyset \hookrightarrow {*}$. The lifting property in this case amounts to saying that every point in $N(D)$ lifts through $N(F)$.
  $$
    \array{
      \emptyset &\to& N(C)
      \\
      \downarrow &{}^\exists\nearrow& \downarrow^{N(F)}
      \\
      {*} &\to& N(D)
    }
    \Leftrightarrow
    \array{
      && N(C)
      \\
      &{}^\exists\nearrow& \downarrow^{N(F)}
      \\
      {*} &\to& N(D)
    }
    \,.
  $$
  This precisely says that $N(F)$ is surjective on 0-cells and hence that $F$ is surjective on objects.

* $n=1$. In degree 1 the boundary inclusion is that of a pair of points as the endpoints of the interval
  $\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}$. The lifting property here evidently is  equivalent to saying that for all objects $a,b \in Obj(C)$ all elements in $Hom(F(a),F(b))$ are hit. Hence that $F$ is a [[full functor]].

* $n=2$. In degree 2 the boundary inclusion is that of the triangle as the boundary of a filled triangle. It is sufficient to restrict attention to the case that the map $\partial \Delta[2] \to N(C)$ sends the top left edge of the triangle to an identity. Then the lifting property here evidently is  equivalent to saying that for all objects $a,b \in Obj(C)$ the map $F_{a,b} : Hom(a,b) \to Hom(F(a),F(b))$ is injective. Hence that $F$ is a [[faithful functor]].
  $$
    \left(
    \array{
      && b
      \\
      & {}^{Id_a}\nearrow && \searrow^{f}
      \\
      a &&\stackrel{g}{\to}&& b
    }
    \right)
    \stackrel{N(F)}{\mapsto}
    \left(
    \array{
      && b
      \\
      & {}^{Id_a}\nearrow &\Downarrow^=& \searrow^{F(f)}
      \\
      a &&\stackrel{F(g)}{\to}&& b
    }
    \right)
  $$

=--

### Constructive version

The original [[proofs]] of the existence of the [[classical model structure on simplicial sets]] are based in [[classical mathematics]] as they use the [[principle of excluded middle]] and the [[axiom of choice]], and are hence not valid in [[constructive mathematics]]. This becomes more than a philosophical issue with the relevance of this [[model category]]-[[structure]] in [[homotopy type theory]], where [[internalization]] into the [[type theory]] requires [[constructive mathematics|constructive]] methods for interpreting [[proofs as programs]].

A constructively valid model structure on simplicial sets and coinciding with the [[classical model structure on simplicial sets|classical model structure]] if [[excluded middle]] and [[axiom of choice]] are assumed was found in [Henry 19](constructive+model+structure+on+simplicial+sets#Henry19). Alternative simpler proofs were found in [Gambino-Sattler-Szumiło 19](constructive+model+structure+on+simplicial+sets#GambinoSattlerSzumilo19).

See at _[[constructive model structure on simplicial sets]]_.

## Joyal's Model Structure

There is a second model structure on $sSet$ -- the **[[model structure for quasi-categories]]** $sSet_{Joyal}$ -- which is different (not [[Quillen equivalence|Quillen equivalent]]) to the classical one, due to [[André Joyal]], with the following distinguished classes of morphisms:

* The [[cofibrations]] of $sSet_{Joy}$ are the [[monomorphisms]], hence the degreewise [[injections]],

* The [[weak equivalences]] in $sSet_{Joy}$ are **[[weak categorical equivalences]]**,

  namely those [[maps]] $u \colon A \rightarrow B$ of simplicial sets such that the induced map $u^* \colon X^B \rightarrow X^A$ of [[internal homs]] ([[mapping complexes]]) for all [[quasi-category|quasi-categories]] $X$ induces an [[isomorphism]] when applying the functor $\tau_0$ that takes a simplicial set to the set of isomorphism classes of objects of its fundamental category.

* The fibrations $F$ are called variously **isofibrations** or **[[inner Kan fibration|quasi-fibrations]]**. As always, these are determined by the class of [[acyclic cofibrations]] already defined.

  Quasi-fibrations between weak Kan complexes have a simple description; they are precisely the [[inner Kan fibrations]], the maps that have the right lifting property with respect to the inner [[horn]] inclusions and also the inclusion $j_0 : * \rightarrow J$ where $*$ is the terminal simplicial set and $J$ is the nerve of the groupoid on two objects with one non-trivial isomorphism.

All objects are [[cofibrant object|cofibrant]]. The [[fibrant objects]] are precisely the [[quasi-categories]].

This model structure is cofibrantly generated. The generating cofibrations are the set $I$ described above. There is no known explicit description for the generating trivial cofibrations.

Importantly, this model structure is Quillen equivalent to several alternative model structures for the ''homotopy theory of homotopy theories" such as that on the category of [[simplicially enriched category|simplicially enriched categories]].

### Comparison

Every weak categorical equivalence is a [[simplicial weak homotopy equivalence]] (but not conversely). Since both model structures have the same class of cofibrations (namely the [[monomorphisms]]), it follows that the [[classical model structure on simplicial sets]] is a [[Bousfield localization]] of [[Joyal's model structure]] (namely at the [[outer horn]]-inclusions):

$$
  sSet_{KQ}
  \underoverset
    {\underset{id}{\longrightarrow}}
    {\overset{id}{\longleftarrow}}
    {\;\; \bot \;\;}
  sSet_{Joy}
  \,.
$$

## Fibrant replacement

Fibrant replacement in $sSet_{Quillen}$ models the process of _$\infty$-groupoidification_, of freely inverting all [[k-morphism]]s in a simplicial set. Techniques for fibrant replacements $sSet_{Quillen}$ are discussed at

* [[Kan fibrant replacement]].

## Related concepts

* [[model structure on topological spaces]], [[model structure on simplicial presheaves]]

* [[model structure on semi-simplicial sets]]

## Properness

The Quillen model structure is both left and right [[proper model category|proper]].  Left properness is automatic since all objects are cofibrant.  Right properness follows from the following argument: it suffices to show that there is a functor $R$ which (1) preserves fibrations, (2) preserves pullbacks of fibrations, (3) preserves and reflects weak equivalences, and (4) lands in a category in which the pullback of a weak equivalence along a fibration is a weak equivalence.  For if so, we can apply $R$ to the pullback of a fibration along a weak equivalence to get another such pullback in the codomain of $R$, which is a weak equivalence, and hence the original pullback was also a weak equivalence.  Two such functors $R$ are

* geometric realization $sSet \to Top$, where $Top$ denotes a sufficiently [[convenient category of topological spaces]] (e.g. the category of [[k-spaces]] suffices) and
* $Ex^\infty : sSet \to Kan$, where $Kan$ is the category of [[Kan complexes]].

This can be found, for instance, in [Goerss-Jardine, Cor. II.9.6](#GoerssJardine).  Another proof can be found in [Moss](#Moss), and a different proof of properness can be found in [Cisinski, Prop. 2.1.5](#Cisinski06).

## References

[[!include classical model structure on simplicial sets -- references]]

### Joyal model structure on simplicial sets

For references on the *[[model structure for quasi-categories]]* see there.


model structure on simplicial sheaves

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[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Andre Joyal|Joyal]]- _[[model category|model structure]] on [[simplicial object|simplicial]] [[sheaves]]_ over a [[site]] is a [[model category]] [[presentable (infinity,1)-category|presentation]] of the [[hypercomplete (∞,1)-topos]] over that site.

It is [[Quillen equivalence|Quillen equivalent]] to the Jardine-local [[model structure on simplicial presheaves]].

There is also a _[[Čech model structure on simplicial sheaves]]_ (see there) modelling not the hypercompletion but just the [[topological localization]] of the [[(∞,1)-category of (∞,1)-presheaves]].

## Definition

Let $C$ be a [[small category|small]] [[site]]. Write $Sh(C)$ for the [[category of sheaves]] on $C$ and $Sh(C)^{\Delta^{op}}$ for the category of [[simplicial object]]s in $Sh(C)$: the category of _simplicial sheaves_ over $C$.

+-- {: .num_theorem }
###### Theorem

There is a [[left proper model category|left proper]] [[simplicially enriched category|simplicially enriched]] [[model category]] structure on $Sh(C)^{\Delta^{op}}$ such that

* cofibrations are precisely the objectwise cofibrations (monomorphisms) of [[simplicial set]]s;

* weak equivalences are the _local weak equivalences_ of the underlying simplicial presheaves as defined at [[model structure on simplicial presheaves]].

=--

This is ([Jardine, theorem 5](#Jardine)).

Call this the **local injective model structure** on simplicial sheaves.

## Properties

+-- {: .num_theorem }
###### Theorem

The [[geometric embedding]] of a [[category of sheaves]] into its [[category of presheaves]]

$$
  (L \dashv i) : Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  PSh(C)
$$

with $L$ given by [[sheafification]] extends to a [[Quillen equivalence]]

$$
  (L \dashv i) : Sh(C)^{\Delta^{op}}_{loc} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  PSh(C)^{\Delta^{op}}_{loc}
$$

between the above local model structure on simplicial sheaves and the injective hyperlocal Jardine-[[model structure on simplicial presheaves]].

=--

This is ([Jardine, theorem 5](#Jardine)).

+-- {: .num_prop #PresentationOfhypercompleteTopos}
###### Proposition

The [[simplicial combinatorial model category]] $Sh(C)^{\Delta^{op}}_{loc}$ is a [[presentable (∞,1)-category|presentation]] for the [[hypercompletion]] $\hat Sh_{(\infty,1)}(c)$ of the [[(∞,1)-category of (∞,1)-sheaves]] on $C$:

$$
  \hat Sh_{(\infty,1)}(C) \simeq (Sh(C)^{\Delta^{op}}_{loc})^\circ
  .
$$

=--

+-- {: .proof}
###### Proof

The proof is spelled out at [[hypercomplete (∞,1)-topos]].

=--

+-- {: .num_cor }
###### Corollary

For $D$ a [[dense sub-site]] of $C$ we have an [[equivalence of (∞,1)-categories]]

$$
  \hat Sh_{(\infty,1)}(C) \simeq \hat Sh_{(\infty,1)}(D)
  \,.

$$

=--

+-- {: .proof}
###### Proof

By the _comparison lemma_ at [[dense sub-site]] we have already an [[equivalence of categories]]

$$
  Sh(C) \simeq Sh(D)
  \,.
$$

This implies the claim with the [above proposition](#PresentationOfhypercompleteTopos).

=--

## Related concepts

* [[Čech model structure on simplicial sheaves]]

* [[model structure on cubical presheaves]]

## References

The local model structure on simplicial sheaves was proposed in

* {#Joyal1984} [[Andre Joyal]], Letter to [[Alexander Grothendieck]], 11.4.1984, ([pdf scan](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf)).

This is,  with [[BrownAHT]], among the first proposals for models for [[∞-stacks]] which eventually came to be used in the theory of [[(∞,1)-toposes]].

Discussion of the model structure:

* [[Sjoerd Crans]], _Quillen closed model structure for sheaves_,  J. Pure Appl. Algebra  101 (1995), 35-57 ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/Crans.pdf))

Jardine's lectures

* {#Jardine} [[John Frederick Jardine]], _Field Lectures: Simplicial presheaves_ ([pdf](https://www.uwo.ca/math/faculty/jardine/courses/fields/fields-01.pdf))

discuss the [[Quillen equivalence]] between the model structure on simplicial sheaves and the [[model structure on simplicial presheaves]].

Wendt discusses "the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves" in

* [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_, [arxiv/1009.2930](http://arxiv.org/abs/1009.2930)

[[!redirects model structures on simplicial sheaves]]
[[!redirects model structure on the category of simplicial sheaves]]


model structure on spectra

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#### Model category theory
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[[!include model category theory - contents]]
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#### Stable homotopy theory
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[[!include stable homotopy theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] structure on a category of [[spectra]] presents a [[stable (∞,1)-category]] of [[spectrum objects]].

Typically and naturally, a model structure on spectra forms a [[stable model category]]. In good cases it also forms a [[symmetric monoidal model category]] with respect to the [[smash product of spectra]], see at _[[symmetric monoidal smash product of spectra]]_.

## Examples

A classical

* [[model structure on sequential spectra]]

in [[simplicial sets]], not however with a symmetric monoidal smash product, is ([Bousfield-Friedlander 78](#BousfieldFriedlander78)) the

* _[[Bousfield-Friedlander model structure]]_,

with its analogue in [[topological spaces]], the

* [[model structure on topological sequential spectra]].

These are related by a [[zig-zag]] of [[Quillen equivalences]] to the

* _[[model structure on combinatorial spectra]]_.

A [[Quillen equivalence|Quillen equivalent]] model structure to the model structures on sequential spectra that does carry a [[symmetric monoidal smash product of spectra]] is the

* _[[model structure for excisive functors]]_.

This models spectra as [[enriched functors]] on the site of [[pointed homotopy type|pointed]] [[finite homotopy types]]. Restricting that to smaller sub-sites, yields model structures for "[[highly structured spectra]]" with a [[symmetric monoidal smash product of spectra]]: the

* [[model structure on orthogonal spectra]],

* [[model structure on symmetric spectra]],

A unified treatment and comparison of these is in

* [[Model categories of diagram spectra]]

Then there is also the

* [[model structure on S-modules]].

## Related concepts

* [[model structure for connective spectra]]

* [[model structure on presheaves of spectra]]

* [[model structure for ring spectra]]

* [[model structure for parameterized spectra]]

* [[model structure for dendroidal left fibrations]]

* [[higher algebra]], [[brave new algebra]], [[higher linear algebra]]

* [[stable (infinity,1)-category]]

* [[Spectra]]

## References

The [[Bousfield-Friedlander model structure]] on [[sequential spectra]] in [[simplicial sets]] is due to

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf))

(a quick review of this is in [Lydakis 98, section 10](#Lydakis98)).

The Quillen equivalent [[model structure on excisive functors]] on pointed simplicial sets is due to

* {#Lydakis98} Lydakis, section 10 of _Simplicial functors and stable homotopy theory_ Preprint, available via Hopf archive, 1998 ([pdf](http://hopf.math.purdue.edu/Lydakis/s_functors.pdf))

and a similar model structure for functors on topological spaces has been given in

* [[William Dwyer]], _Localizations_, In _Axiomatic, enriched and motivic homotopy theory_, volume 131 of NATO Sci. Ser. II Math. Phys. Chem., pages 3–28. Kluwer Acad. Publ., Dordrecht, 2004

See also

* [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf))

A discussion of model structures on spectra in general ambient model categories (general [[spectrum objects]], including e.g. [[motivic spectra]]) is in

* {#Hovey00} [[Mark Hovey]], _Spectra and symmetric spectra in general model categories_ ([arXiv:0004051](http://arxiv.org/abs/math/0004051))

and for the [[Bousfield-Friedlander model structure|Bousfield-Friedlander-type model structure]] in

* {#Schwede97} [[Stefan Schwede]], _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 77-104 ([pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf))

and for the [[model structure for excisive functors|excisive-functor-type model structure]] in

* [[Bjørn Ian Dundas]], [[Oliver Röndigs]], [[Paul Arne Østvær]], _Enriched functors and stable homotopy theory_,  Doc. Math., 8:409–488, 2003 ([EuDML](https://eudml.org/doc/123650))

Review:

* [[Daniel Dugger]], *Stable categories and spectra via model categories* ([pdf](https://pages.uoregon.edu/ddugger/models_chapter.pdf)), in:  [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*, MSRI Book Series, Cambridge University Press

[[!redirects model structures on spectra]]

[[!redirects model category of spectra]]
[[!redirects model categories of spectra]]

[[!redirects model structures of spectra]]

[[!redirects model structure for spectra]]
[[!redirects model structures for spectra]]


model structure on sSet-categories

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Depending on the chosen [[model category]] structure, the [[category]] [[sSet]] of [[simplicial set]]s may model [[∞-groupoids]] (for the standard [[model structure on simplicial sets]]) or [[(∞,1)-categories]] in the form of [[quasi-categories]] (for the [[Joyal model structure]]) on [[SSet]].

Accordingly, there are [[model category]] structures on [[sSet-categories]] that similarly model [[(n,r)-categories]] with $r$ shifted up by 1:

* there is a model structure on $SSet$-enriched categories whose fibrant objects are [[∞-groupoid]]/[[Kan complex]]-enriched categories and which models [[(∞,1)-categories]];

This we discuss [below](ModelForInfinityCategories).

* there is another model structure on [[sSet-categories]] whose fibrant objects are [[(∞,1)-category]]/[[quasi-category]]-enriched categories, and which model [[(∞,2)-categories]].

  For more on this see [elsewhere](%28infinity%2C2%29-category#TotMod)

Both are special cases of a [[model structure on enriched categories]].

## Model for $(\infty,1)$-categories
 {#ModelForInfinityCategories}

Here we describe the model category structure on [[SSet Cat]] that  makes it a model for the [[(∞,1)-category of (∞,1)-categories]].

\begin{definition}\label{DwyerKanEquivalences}
**(Dwyer-Kan equivalences)**
\linebreak
An [[sSet]]-[[enriched functor]] $F : C \to D$ between [[sSet-categories]] is called a **weak equivalence** precisely if

* it is *essentially surjective* in that the induced functor of [[homotopy categories]] is an ordinary [[essentially surjective functor]];

* it is an $\infty$-[[full and faithful functor]] in that for all objects $x,y \in C$ the morphism

  $$
    F_{x,y} : C(x,y) \to D(F(x), F(y))
  $$

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

\end{definition}

This notion is due to [Dwyer & Kan (1980 FuncComp), §2.4](#DwyerKan80FunctionComplexes) and now known as *[[Dwyer-Kan equivalences]]*.

+-- {: .num_prop}
###### Proposition

A [[Quillen equivalence]] $D \leftrightarrows C$ between [[model categories]] induces a Dwyer-Kan-equivalence $L C \leftrightarrow L D$ between their [[simplicial localizations]].

=--

This is made explicit in [Mazel-Gee 15, p. 17](#MazelGee15) to follow from [Dwyer & Kan 80, Prop. 4.4 with 5.4](#DwyerKan80FunctionComplexes).

The analogous statement under further passage to [[model structure on quasi-categories|Joyal equivalences]] of [[quasi-categories]] is [Lurie (2009), Cor. A.3.1.12](#Lurie09), under the additional assumptions that the model categories are [[simplicial model category|simplicial]], that every object of $C$ is [[cofibrant object|cofibrant]] and that the right adjoint is an sSet [[enriched functor]].

+-- {: .num_prop #BergnerModelStructure}
###### Proposition

The category [[SSet Cat]] of [[small category|small]] [[sSet-enriched categories]] carries the structure of a [[model category]] with

* [[weak equivalences]] the [[Dwyer-Kan equivalences]] (Def. \ref{DwyerKanEquivalences});

* [[fibrations]] those [[sSet]]-[[enriched functors]] $F \colon C \to D$ such that

  1. for all $x, y \in C$ the morphism $F_{x,y} \colon C(x,y) \to D\big(F(x), F(y)\big)$ is a fibration in the [[classical model structure on simplicial sets]];

  1. the induced functor $\pi_0(F) \colon Ho(C) \to Ho(D)$ on [[homotopy categories]] is an [[isofibration]].

=--

[[Bergner (2004)](#Bergner04), [Lurie (2009), thm. A.3.2.4](#Lurie09)]

+-- {: .num_remark}
###### Remark

In particular, the [[fibrant objects]] in this model structure are the [[Kan complex]]-[[enriched categories]], i.e. the strictly [[∞-groupoid]]-enriched ones (see *[[(n,r)-category]]*).

=--

\begin{proposition}
\label{DKBCofibrationsAreInParticularHomSpaceWiseMonos}
The [[cofibrations]] in the Dwyer-Kan-Bergner model structure (Prop. \ref{BergnerModelStructure}) are in particular [[hom-object]]-wise cofibrations in the [[classical model structure on simplicial sets]], hence [[hom-object]] wise [[monomorphism]] of [[simplicial sets]]:
$$
  \left.
  \begin{array}{l}
    \mathbf{D}, \, \mathbf{C} \,\in\, sSet\text{-}Cat,
    \\
    F \colon \mathbf{D}
     \underset{\in Cof_{DKB}}{\longrightarrow} \mathbf{C},
    \\
    X,Y \,\in\, \mathbf{D}
  \end{array}
  \;\;\;
  \right\}
  \;\;\;\;\;\;\;
    \vdash
  \;\;\;\;\;\;\;
  F_{X,Y}
    \colon
  \mathbf{D}(X,Y)
    \underset{\in Cof_{KQ}}{\hookrightarrow}
  \mathbf{C}\big(F(X), F(Y)\big)
  \,.
$$
\end{proposition}
[[Lurie (2009), Rem. A.3.2.5](#Lurie09)]

## Properties

### Further model category properties

+-- {: .num_prop #RightProperness}
###### Proposition

The Bergner model structure of prop. \ref{BergnerModelStructure} is a [[proper model category]].

=--

A reference for right properness is ([Bergner 04, prop. 3.5](#Bergner04)). A reference for left properness is ([Lurie, A.3.2.4](#Lurie09)) and ([Lurie, A.3.2.25](#Lurie09)).

\begin{remark}\label{FailureOfCartesianClosedModelStructure}
**(failure of cartesian closed model structure)**
\linebreak
  While the [[underlying]] category [[VCat|$sSet Cat$]] of [[sSet-enriched categories]] is [[cartesian closed category|cartesian closed]] (on general grounds, see [here](category+of+V-enriched+categories#MonoidalStructure), also cf. [Joyal (2008), §51.2](#Joyal2008)), the construction of [[enriched product category|$sSet$-enriched product categories]] and [[enriched functor categories|$sSet$-enriched functor categories]] does *not* make the Bergner model structure of prop. \ref{BergnerModelStructure} into a [[monoidal model category]] hence *not* into a [[cartesian closed model category]].

This is in contrast to the [[model structure on quasi-categories]] (see [there](model+structure+for+quasi-categories#CartesianMonoidalModelStructure)) to which the Bergner structure is [[Quillen equivalence|Quillen equivalent]] (see [below](#RelationToQuasiCategories)).
\end{remark}

### Relation to simplicial groupoids
 {#RelationToSimplicialGroupoids}

\begin{proposition}
The canonical inclusion functor
$$
  \iota
    \,\colon\,
  sSet\text{-}Grpd
   \xhookrightarrow{\phantom{--}}
  sSet\text{-}Cat
$$
of the category of [[sSet]]-[[enriched groupoids]] (aka *Dwyer-Kan [[simplicial groupoids]]*) into that of [[sSet-enriched categories]]

1. has a [[left adjoint]], given degreewise by the [[free groupoid]]-construction ([[localization of a category|localization]] at the class of all morphisms)

1. evidently preserves fibrations and weak equivalences between the above Bergner-model structure (Prop. \ref{BergnerModelStructure})
and the Dwyer-Kan [[model structure on simplicial groupoids]]

hence we have a [[Quillen adjunction]]:
$$
  sSet\text{-}Grpd_{DK}
  \underoverset
     {\underset{\iota}{\hookrightarrow}}
     {\overset{F}{\longleftarrow}}
     {\;\;\; \bot_{\mathrm{Qu}} \;\;\;}
  sSet\text{-}Cat_{Berg}
  \,.
$$
\end{proposition}

(see also [Minichiello, Rivera & Zeinalian (2023), Prop. 2.8](model+structure+on+simplicial+groupoids#MinichielloRiveraZeinalian23))

### Relation to quasi-categories
 {#RelationToQuasiCategories}

The operations of forming [[homotopy coherent nerves]], $N$, and of [[rigidification of quasi-categories]], $\mathfrak{C}$, constitute a [[Quillen equivalence]] between the Bergner model structure on $sSet Cat$ (Prop. \ref{BergnerModelStructure}) and the [[model structure for quasi-categories]].

For more see at *[[relation between sSet-enriched categories and quasi-categories]]*.

## Related concepts

* [[model structure on simplicial groupoids]]

* [[model structure for quasi-categories]]

* [[relation between quasi-categories and simplicial categories]]

* [[simplicial localization]]

[[!include table - models for (infinity,1)-operads]]

## References

A model category structure on the category of $sSet$-categories with a fixed collection of objects was first given in:

* {#DwyerKan1980Localizations} [[William Dwyer]], [[Daniel Kan]],  §7.1 in: *Simplicial localizations of categories*, J. Pure Appl. Algebra **17** 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3]

The notion of what is now called *[[Dwyer-Kan equivalences]]* appears in

* {#DwyerKan80FunctionComplexes} [[William Dwyer]], [[Daniel Kan]], §2.4 in: _Function complexes in homotopical algebra_, Topology 19 (1980), 427-440 [doi:10.1016/0040-9383(80)90025-7, [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dwyer-kan-3.pdf)]

Dywer, Spalinski and later [[Charles Rezk|Rezk]] then pointed out that there ought to exist a model category structure on the collection of all $sSet$-categories that models the [[(∞,1)-category of (∞,1)-categories]]. This was then constructed in:

* {#Bergner04} [[Julie Bergner]], *A model category structure on the category of simplicial categories*,  Trans. Amer. Math. Soc. **359** (2007) 2043-2058 [[arXiv:0406507](http://arxiv.org/abs/math/0406507), [doi:10.1090/S0002-9947-06-03987-0](https://doi.org/10.1090/S0002-9947-06-03987-0)]

Also:

* {#Lurie09} [[Jacob Lurie]], Section A.3.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press (2009) [[pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf)]

* [[Jacob Lurie]], _$(\infty,2)$-categories and the Goodwillie calculus_, [GoodwillieI.pdf](http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf)

Survey and review:

* [[Julie Bergner]], Section 3 of: _A survey of $(\infty,1)$-categories_ ([arXiv:math/0610239](http://arxiv.org/abs/math/0610239))

* [[Emily Riehl]], Section 16 of: _[[Categorical Homotopy Theory]]_, Cambridge University Press, 2014 ([pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf), [doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457))

See also

* {#Joyal2008} [[André Joyal]], *Notes on quasi-categories* (2008) [[pdf](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf), [[JoyalNotesOnQuasiCategories.pdf:file]]]

* {#MazelGee15} [[Aaron Mazel-Gee]], _Quillen adjunctions induce adjunctions of quasicategories_, New York Journal of Mathematics Volume 22 (2016) 57-93 ([arXiv:1501.03146](https://arxiv.org/abs/1501.03146), [publisher](http://nyjm.albany.edu/j/2016/22-4.html))

Recall the slight but crucial difference between the two notions of "[[simplicial categories]]", the other being an [[internal category]] in [[sSet]]. But also for this latter concept there is a [[model category]] structure which presents [[(infinity,1)-categories]], see

* {#Horel14} [[Geoffroy Horel]], _A model structure on internal categories_, Theory and Applications of Categories, Vol. 30, 2015, No. 20, pp 704-750 ([arXiv:1403.6873](http://arxiv.org/abs/1403.6873)).

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