model structure on pointed objects

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#Contents#
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## Idea

For $\mathcal{C}$ a [[model category]], there is canonically induced a model category structure on the corresponding [[category of pointed objects]] $\mathcal{C}^{\ast/}$, namely the [[coslice model structure]] under "the point", i.e. under the [[terminal object]] $\ast \in \mathcal{C}$.

## Definition

As a special case of the general discussion at *[[coslice model structures]]*, we have:

\begin{prop}\label{ModelStructureOnSliceCategory}
  Let $\mathcal{C}$ be a [[model category]] with [[terminal object]] dented $\ast \in \mathcal{C}$. Then there is a model category structure on its [[category of pointed objects]] $\mathcal{C}^{\ast/}$, hence on the [[coslice category|category under]] $\ast$, whose classes of morphisms ([[weak equivalences]], [[fibrations]], [[cofibrations]]) are those created by the [[forgetful functor]] $\mathcal{C}^{\ast/} \to \mathcal{C}$.
\end{prop}

(e.g. [Hovey 99, Prop. 1.1.8](#Hovey99))

## Examples

+-- {: .num_example #ClassicalPointedHomotopyCategory}
###### Example
**([[classical model structure on pointed topological spaces]])**

For $\mathcal{C} = Top_{Qu}$, the [[classical model structure on topological spaces]], then the model structure on [[pointed topological spaces]] induced via prop. \ref{ModelStructureOnSliceCategory} we call the _[[classical model structure on pointed topological spaces]]_ $Top_{Qu}^{\ast/}$. Its [[homotopy category of a model category]] is the _classical pointed homotopy theory_ $Ho(Top^{\ast/})$.

=--

## Properties

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

The [[fibrant objects]] in the pointed model structure $\mathcal{C}^{\ast/}$, prop. \ref{ModelStructureOnSliceCategory}, are those that are fibrant as objects of $\mathcal{C}$.

But the [[cofibrant objects]] in $\mathcal{C}^{\ast/}$ are those for which the basepoint inclusion is a [[cofibration]] in $X$.

For $\mathcal{C}^{\ast/} = Top^{\ast/}_{Qu}$ (Ex. \ref{ClassicalPointedHomotopyCategory}), the cofibrant pointed topological spaces are typically referred to as spaces _with non-degenerate basepoints_. Notice that the point itself is cofibrant in $Top_{Qu}$, so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.

=--

+-- {: .num_example #HomotopyCategoryOfPointedModelStructureIsEnrichedInPointedSets}
###### Example

For $\mathcal{C}$ any [[model category]], with $\mathcal{C}^{\ast/}$ its [[slice model structure|pointed model structure]] according to Prop. \ref{ModelStructureOnSliceCategory}, then corresponding [[homotopy category of a model category|homotopy category]] is, (by [this Remark](pointed+object#PointedObjectsHaveZeroObject)), canonically [[enriched category|enriched]] in [[pointed sets]], in that its [[hom-functor]] is of the form

$$
  [-,-]_\ast
  \;\colon\;
  Ho(\mathcal{C}^{\ast/})^\op
  \times
  Ho(\mathcal{C}^{\ast/})
  \longrightarrow
  Set^{\ast/}
  \,.
$$

=--

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

If $\mathcal{C}$ is a [[monoidal model category]] with cofibrant [[tensor unit]], then the pointed model structure on  $\mathcal{C}^{\ast/}$ (Prop. \ref{ModelStructureOnSliceCategory}) is also a [[monoidal model category]], and the [[smash product]]$\dashv$[[mapping space]] adjunction of prop. \ref{SmashProductLeftAdjointToPointedMappingSpace} is a [[Quillen adjunction]]

$$
  \big(
    X \wedge (-)
    \;\dashv\;
    (-)^X
  \big)
  \;\;\colon\;\;
  \mathcal{C}^{\ast/}
  \leftrightarrow
  \mathcal{C}^{\ast/}
  \,.
$$

=--

\begin{prop}\label{InducedQuilleAdjunctionsOnModelCategoriesOfPointedObjects}
**(induced [[Quillen adjunction]] on [[model categories of pointed objects]])** \linebreak
Given a [[Quillen adjunction]] between [[model categories]]
$$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \mathcal{C}
  \,,
$$
there is induced a [[Quillen adjunction]] between the corresponding [[model categories of pointed objects]]
$$
  \mathcal{D}^{\ast\!/}
    \underoverset
      {\underset{R^{\ast\!/}}{\longrightarrow}}
      {\overset{L^{\ast\!/}}{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \mathcal{C}^{\ast\!/}
  \,,
$$
where

* the [[right adjoint]] acts directly as $R$ on the triangular [[commuting diagrams]] in $\mathcal{C}$ that define the morphisms in $\mathcal{C}^{\ast\!/}$;

* the [[left adjoint]] is the [[composition|composite]] of the corresponding direct application of $L$ followed by [[pushout]] along the [[adjunction counit]] $L(\ast) \simeq L \circ R(\ast) \xrightarrow{ \;\epsilon_\ast \; } \ast$ (using that $R(\ast) \simeq \ast$ since [[right adjoints preserve limits]] and hence [[terminal objects]]):

  $$
    L^{\ast\!/}
    \;\colon\;
    \mathcal{C}^{\ast\!/}
    \xrightarrow{ \;\; L  \;\; }
    \mathcal{D}^{L(\ast)\!/}
    \;\simeq\;
    \mathcal{D}^{L\circ R(\ast)\!/}
    \xrightarrow{ \;\; (-) \sqcup \epsilon_\ast \;\; }
    \mathcal{D}^{\ast\!/}
    \,.
  $$

\end{prop}
\begin{proof}
  It is fairly straightforward to check this directly (e.g. [Hovey 1999, Prop. 1.3.5](#Hovey99)), but it is also a special case of [this general prop.](slice+model+structure#SlicedQuillenAdjunction) about [[slice model categories]] --- to make this explicit, notice that passing to [[opposite categories]] with their [[opposite model structures]] turns the original Quillen adjunction into the [[opposite Quillen adjunction]]:

$$
  \mathcal{C}^{op}
    \underoverset
      {\underset{L^{op}}{\longrightarrow}}
      {\overset{R^{op}}{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \mathcal{D}^{op}
  \,.
$$

Now the passage to [[pointed objects]] corresponds to [[slice category|slicing]] (instead of [[coslice category|co-slicing]]), since

\[
  \label{PointedObjectsIsOppositeOfCopointedObjectsInOpposite}
  \mathcal{C}^{\ast\!/}
  \;\;
  \simeq
  \big(
    \mathcal{C}^{op}_{/\ast}
  \big)^{op}
  \,,
  \;\;\;\;\;\;\;
  \text{similarly}
  \;\;\;
  \mathcal{D}^{\ast\!/}
  \;\;
  \simeq
  \big(
    \mathcal{D}^{op}_{/R(\ast)}
  \big)^{op}
  \,,
\]

whence item (1) in [that Prop.](slice+model+structure#SlicedQuillenAdjunction) says that there is a Quillen adjunction of the form

$$
  \mathcal{C}^{op}_{/R(\ast)}
    \underoverset
      {\underset{L^{op}_{/\ast}}{\longrightarrow}}
      {\overset{R^{op}_{/\ast}}{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \mathcal{D}^{op}_{/\ast}
  \,,
$$

hence with [[opposite Quillen adjunction]] of the required form

$$
  \mathcal{D}^{\ast\!/}
  \simeq
  \big(\mathcal{D}^{op}_{/R(\ast)}\big)^{op}
    \underoverset
      {\underset{
        R^{\ast\!/} \,\coloneqq\, \big( R^{op}_{/\ast} \big)^{op}
      }{\longrightarrow}}
      {\overset{
        L^{\ast\!/} \,\coloneqq\, \big( L^{op}_{/\ast} \big)^{op}
      }{\longleftarrow}}
      {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;}
  \big(\mathcal{C}^{op}_{/\ast}\big)^{op}
  \simeq
  \mathcal{C}^{\ast\!/}
  \,,
$$

with $R^{op}$ acting directly as $R$ on underlying diagrams, and with $L^{op}$ acting as the composite of $L$ following by [[pullback]] -- in $\mathcal{C}^{op}$ -- along the [[adjunction unit]] of $(R^{op} \dashv L^{op})$. Since the component morphism of the [[adjunction unit|unit]] of the [[opposite adjunction]] $(R^{op} \dashv L^{op})$ is that of the adjunction unit of $(L \dashv R)$, and since [[pullback]] in an [[opposite category]] is [[pushout]] in the original category, this implies the claim.
\end{proof}

## References

Textbook accounts:

* {#Hovey99} [[Mark Hovey]], Prop. 1.1.8 in: _[[Model Categories]]_, Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

[[!redirects model category of pointed objects]]
[[!redirects model categories of pointed objects]]


model structure on presheaves of simplicial groupoids

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#### Homotopy theory
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## Idea

The model structure on [[presheaves]] of (Dwyer-Kan) [[simplicial groupoids]] ([[sSet]]-[[enriched groupoids]]) is one of the [[models for ∞-stack (∞,1)-toposes]]. It is a slight variant on the [[model structure on simplicial presheaves]]. (At that link more general information is collected).

For various applications it is useful to

* use [[simplicial groupoids]] instead of [[Kan complexes]] or [[simplicial sets]];

* use a model structure on (pre)sheaves with values in [[simplicial groupoids]] instead of the [[model structure on simplicial presheaves]] (see there for motivation on why to consider these models in the first place).

An example is the discussion of [[principal infinity-bundles]] in section 3 of ([Jardine & Luo](#JardineLuo))

## Definition/Properties

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

Write $(G \dashv \bar W) \colon Grpd^\Delta \leftrightarrow sSet$ for the [[Quillen adjunction]] discussed at *[[model structure on simplicial groupoids]]*. This directly prolongs to an [[adjunction]] on [[presheaves]]

$$
  (G \dashv \bar W)
  \colon
  [C^{op}, Grpd^\Delta]
   \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}}
  [C^{op}, sSet]_{inj}
  \,.
$$

The [[transferred model structure]] along $\bar W$ on $[C^{op}, Grpd^\Delta]$ of the  global injective [[model structure on simplicial presheaves]] exists on $[C^{op}, sSet, Grpd^\Delta]$: fibrations and weak equivalences are those that become global injective fibrations and weak equivalences, respectively, under $\bar W$.

=--

This appears as ([LBK, theorem 3.10](#LBK)).

## Related concepts

* [[model structure on simplicial sets]]

  * [[model structure on simplicial presheaves]]

* [[model structure on simplicial groupoids]]

## References

A model structure on [[sheaves]] with values in simplicial groupoids is due to

* [[Andre Joyal]] and [[Myles Tierney]], _Strong stacks and classifying spaces_, in: _Category theory_
(Como, 1990), volume 1488 of Lecture Notes in Math., pages 213–236. Springer, Berlin (1991) ([web](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2105376))

* [[Andre Joyal]] and [[Myles Tierney]], *On the homotopy theory of sheaves of simplicial groupoids*, Math. Proc. Cambridge Philos. Soc. **120** 2 (1996) 263-290

A model structure on [[presheaves]] with values in simplicial groupoids:

* {#LBK} Zhi-Ming Luo, Peter Bubenik, Peter Kim, _Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids_ ([arXiv](http://arxiv.org/abs/math.AT/0301045))

A model structure on simplicial sheaves of groupoids is discussed in

* [[Sjoerd Crans]], _Quillen closed model structure for sheaves_,  J. Pure Appl. Algebra  101 (1995), 35-57 ([web](http://home.tiscali.nl/secrans/papers/thcms.html))

* {#JardineLuo} [[John F. Jardine]], Luo, _Higher order principal bundles_ ([web](http://www.math.uiuc.edu/K-theory/0681/))

* {#Jardine15} [[John F. Jardine]], §9.4 in: *[[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)]

[[!redirects model structures on presheaves of simplicial groupoids]]


model structure on presheaves of spectra

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#### Stable Homotopy theory
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#Contents#
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## Idea

A [[model category]] structure on [[presheaves of spectra]]
[[presentable (infinity,1)-category|presents]] the [[stabilization]] of the corresponding [[(∞,1)-category]] presented by the corresponding [[model structure on simplicial presheaves]].

Over the trivial site it reproduces the [[model structure on spectra]] themselves.

## Related concepts

* [[model structure on simplicial presheaves]]

* [[model structure on parameterized spectra]]

## References

* [[Frederick Jardine]], _Stable homotopy theory of simplicial presheaves_, Canad. J. Math. 39(1987), 733-747 ([DOI](http://dx.doi.org/10.4153/CJM-1987-035-8))

[[!redirects model structures on presheaves of spectra]]


model structure on presheaves over a test category

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#Contents#
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## Idea

For $C$ a [[test category]], the canonical structure of a [[category with weak equivalences]] on the [[category of presheaves]] over $C$ lifts to the structure of a [[model category]]. All of these are models for the standard [[homotopy theory]] (the [[homotopy category]] of [[∞Grpd]]).

## Examples

* [[model structure on simplicial sets]]

* [[model structure on cubical sets]]

## References

The model structure is due to

* [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]_, Asterisque __308__, ([pdf](http://www.mathematik.uni-regensburg.de/cisinski/ast.pdf)).
  {#Cisinksi}

Further developments are in

* [[Rick Jardine]], _Categorical homotopy theory_, Homot. Homol. Appl. __8__ (1), 2006, pp.71–144, ([HHA](http://www.intlpress.com/HHA/v8/n1/a3/), [pdf](https://projecteuclid.org/download/pdf_1/euclid.hha/1140012467)).


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#Contents#
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## Idea

The _model structure on [[reduced simplicial sets]]_ is a [[presentable (infinity,1)-category|presentation]] of the full [[sub-(∞,1)-category]]

[[∞Grpd]]${}^{*/}_{\geq 1} \hookrightarrow$ [[∞Grpd]]${}^{*/}$ $\simeq$ [[Top]]${}^{*/}$

of [[pointed object|pointed]] [[∞-groupoids]] on those that are [[connected]].

By the [[looping and delooping]]-equivalence, this is [[equivalence of (∞,1)-categories|equivalent]] to the [[(∞,1)-category]] of [[∞-groups]] and this equivalence is presented by a [[Quillen equivalence]] to the [[model structure on simplicial groups]].

## Definition

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

A _[[reduced simplicial set]]_ is a [[simplicial set]] $S$ with a single vertex, hence with $X_0$ [[generalized the|the]] [[singleton set]]

$$
  S_0 \;=\; \ast
  \,.
$$

Write $sSet_0 \subset $ [[sSet]] for the [[full subcategory]] of the category of [[simplicial sets]] on those that are reduced.

=--

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

There is a [[model category]] structure on $sSet_0$ (def. \ref{ReducedSimplicialSet}) whose

* weak equivalences

* and cofibrations

are those whose [[underlying]] maps are such in the [[classical model structure on simplicial sets]] (i.e. the [[simplicial weak homotopy equivalences]] and the [[monomorphisms]], respectively).

=--

This appears as [Goerss & Jardine, Ch V, Prop. 6.2](#GoerssJardine).

## Properties

### Relation to ordinary simplicial sets
 {#RelationToOrdinarySimplicialSets}

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

Under the [[forgetful functor]] $U \colon sSet_0 \hookrightarrow sSet$

* a fibration $f \colon X \to Y$ with respect to the reduced model structure (prop. \ref{TheModelStructure}) maps to a fibration in the [[classical model structure on simplicial sets]] (a [[Kan fibration]]) precisely if it has the [[right lifting property]] against $* \to S^1 \coloneqq \Delta[1]/ \partial \Delta[1]$;

In particular:

* every [[fibrant object]] in $sSet_0$ is a fibrant object in $sSet$, hence a [[Kan complex]].

=--

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

The first statment is proven in [Goerss & Jardine, Ch. V, Lemma 6.6.](#GoerssJardine). The second is an immediate consequence, stated there as as [Goerss & Jardine, Ch. V, Corollary 6.8](#GoerssJardine).

=--

\begin{corollary}\label{KanFibrationBetweenReducedKanComplexesAsReducedFibration}
  Let $f \colon X \longrightarrow Y$ be a [[fibration]]
  in the model structure on reduced simplicial sets
  (Prop. \ref{TheModelStructure}) such that both $X$ and $Y$ are
  [[Kan complexes]].
  Then $f$ is a [[Kan fibration]]
  precisely if it induces a [[surjection]] on
  the first [[simplicial homotopy group]]
  $\pi_1(f) \colon \pi_1(X) \twoheadrightarrow \pi_1(Y)$.
\end{corollary}

([Goerss & Jardine, Ch. V, Cor. 6.9](#GoerssJardine))

As an example:

\begin{proposition}\label{DeloopedKanFibrationOfHomomorphismOfSimplicialGroups}
  Let $\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2$ be a [[homomorphism]] of [[simplicial groups]] which is a [[Kan fibration]].
Then the induced morphism of [[simplicial classifying spaces]]
$\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2$ is a [[Kan fibration]] if and only if $\phi$ is a [[surjection]] on [[connected components]]: $\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1)$.
\end{proposition}
([Goerss & Jardine, Ch. V, Cor. 6.9](#GoerssJardine))
\begin{proof}
  Since $\overline{W}(-)$ is a [[right Quillen functor]] to the model structure on reduced simplicial sets (Prop. \ref{QuillenAdjunctionWithSimplicialGroups}) it follows that $\overline{W}(\phi)$ is in any case a fibration in that model structure. Hence Cor. \ref{KanFibrationBetweenReducedKanComplexesAsReducedFibration} implies that $\overline{W}(\phi)$ is a Kan fibration precisely of $\pi_1 \circ \overline{W}(\phi)$ is surjective. But $\pi_1 \circ \overline{W} = \pi_0$, by [this Prop](simplicial+classifying+space#HomotopyGroupsOfBarWG).
\end{proof}

### Relation to pointed simplicial sets
 {#RelationToPointedSimplicialSets}

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

The [[coreflective embedding]]

$$
  sSet^{\ast/}
   \underoverset
     {\underset{cn}{\longrightarrow}}
     {\overset{i}{\longleftarrow}}
     {\bot}
  sSet_{0}
$$

into [[pointed simplicial sets]] (where $i$ the obvious inclusion, and $cn$ forms the 1st [[Eilenberg subcomplex]] over the basepoint) is a [[Quillen adjunction]]
(with respect to the reduced model structure from prop. \ref{TheModelStructure} and the [[coslice model structure]] under the point of the [[classical model structure on simplicial sets]]).

=--

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

By prop. \ref{TheModelStructure} the left adjoint preserves cofibrations and acyclic cofibrations (in fact all weak equivalences).

=--

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

The operation of [[reduced suspension]] $\Sigma_\ast$ ([[smash product]] with the simplicial circle $S^1 \coloneqq \Delta[1]/\partial \Delta[1]$) and forming [[loop space]] $\Omega_\ast$ ([[pointed mapping space]] out of the circle) constitute a [[Quillen adjunction]]

$$
  sSet_0
    \underoverset
      {\underset{\Omega_\bullet}{\longrightarrow}}
      {\overset{\Sigma_\ast}{\longleftarrow}}
      {\bot}
  sSet^{\ast/}
$$

=--

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

By the [[internal hom]] construction we have the [[adjunction]]

$$
  sSet^{\ast/}
    \underoverset
      {\underset{\Omega_\bullet}{\longrightarrow}}
      {\overset{\Sigma_\ast}{\longleftarrow}}
      {\bot}
  sSet^{\ast/}
$$

But by the formula at [[smash product]] the [[reduced suspension]] clearly lands in [[reduced simplicial sets]], so that we do have the restricted adjunction as claimed. Now by the fact that the [[classical model structure on simplicial sets]] is a [[simplicial model category]], the above is a [[Quillen adjunction]] and $\Sigma_\ast$ preserves cofibrations and acyclic cofibrations. Hence, by prop. \ref{TheModelStructure}, so does its factorization through the model structure on reduced simplicial sets.

=--

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

The operations of [[reduced suspension]] and [[loop space]] (co-)restrict to a [[Quillen adjunction]] on the reduced model structure (prop. \ref{TheModelStructure}) itself, by [[composition]] of the Quillen adjunctions from prop. \ref{CoreflectiveQuillenAdjunctionToPointedSimplicialSets} and prop. \ref{SuspensionQuillenAdjunctionWithReducedSimplicialSets}.

$$
  sSet_0
    \underoverset
      {\underset{\Omega_\bullet}{\longrightarrow}}
      {\overset{\Sigma_\ast}{\longleftarrow}}
      {\bot}
  sSet^{\ast/}
   \underoverset
     {\underset{cn}{\longrightarrow}}
     {\overset{i}{\longleftarrow}}
     {\bot}
  sSet_{0}
$$

=--

### Relation to simplicial groups
 {#RelationToSimplicialGroups}

+-- {: .num_prop #QuillenAdjunctionWithSimplicialGroups}
###### Proposition
**([[Quillen equivalence between simplicial groups and reduced simplicial sets]])**

The [[simplicial loop space]] functor $G$ and the [[simplicial classifying space]]-construction $\overline{W}(-)$ constitute a [[Quillen equivalence]]

$$
  (G \dashv \overline{W})
  \,\colon\,
    sGr
      \underoverset
        {\underset{\overline{W}}{\longrightarrow}}
        {\overset{G}{\longleftarrow}}
        {\;\;\;\;\;\;\;\;\bot\;\;\;\;\;\;\;\;}
    sSet_0
$$

between the model structure on reduced simplicial sets from prop. \ref{TheModelStructure} and [[model structure on simplicial groups]].

=--

This appears as ([Goerss-Jardine, ch. V prop. 6.3](#GoerssJardine)).

## Related concepts

* [[model structure on simplicial groups]]

* **model structure on reduced simplicial sets**

* [[groupoid object in an (∞,1)-category]], [[∞-group]], [[looping and delooping]]

## References

Textbook account for the Kan model structure on reduced simplicial sets:

* {#GoerssJardine} [[Paul Goerss]], [[J. F. Jardine]], Chapter V in: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/))

The analog (in fact [[transferred model structure|transfer]]) of the [[Joyal model structure]] for reduced simplicial sets, modelling [[quasi-categories]] with a single object:

* {#Burke21} Nigel Burke, §2 of: *Homotopy Theory of Monoids and Group Completion*, PhD thesis, Cambridge (2021) [[doi](https://doi.org/10.17863/CAM.72815), [cam:1810/325358](https://www.repository.cam.ac.uk/handle/1810/325358), [pdf](https://www.repository.cam.ac.uk/bitstreams/1f6603cb-e442-491c-8478-f86c4e03f05d/download)]


model structure on relative categories

[[!redirects model structure on categories with weak equivalences]]

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{:toc}

## Idea

The [[model category]] structure on the [[category]] of [[category with weak equivalences|categories with weak equivalences]] is a model for the [[(∞,1)-category of (∞,1)-categories]].

Every [[category with weak equivalences]] $C$ presents under Dwyer-Kan [[simplicial localization]] a [[simplicially enriched category]] or alternatively under [[Charles Rezk]]'s _simplicial nerve_ a [[complete Segal space|Segal space]], both of which are incarnations of a corresponding [[(∞,1)-category]] $\mathbf{C}$ with the same objects of $C$, at least the 1-morphisms of $C$ and such that every weak equivalence in $C$ becomes a true equivalence ([[homotopy equivalence]]) in $\mathbf{C}$.

By the work of Barwick and Kan, there is a model structure on relative categories presenting the (∞,1)-category of simplicial spaces.

## Details

For the purposes of the present entry, a [[category with weak equivalences]] means the bare minimum of what may reasonably go by that name:

**Definition** A [[relative category]] $(C,W)$ is a [[category]] $C$ equipped with a choice of [[wide subcategory]] $W$.

A morphism in $W$ is called a **weak equivalence** in $C$. Notice that we do not require here that these weak equivalence satisfy [[category with weak equivalences|2-out-of-3]], nor even that they contain all [[isomorphisms]].

A morphism $(C_1,W_1) \to (C_2,W_2)$ of relative catgeories is a [[functor]] $C_1 \to C_2$ that preserves weak equivalences.

Write $RelCat$ for the [[category]] of relative categories and such morphisms between them.

### Model category structure

+-- {: .num_theorem}
###### Theorem
The subdivided nerve of Barwick-Kan induces a right [[transferred model structure]]
on $RelCat$ such that there is a Quillen equivalence
$$
  K_\xi : sSet^{\Delta^{\op}}_{Reedy} \leftrightarrows (RelCat, BK) : N_\xi
$$
Both functors preserve and reflect weak equivalences, and the adjunction unit and counit are natural weak equivalences. The simplicial nerve $N$ is naturally weakly equivalent to $N_\xi$.

Furthermore, this all remains true if the Reedy model structure on $sSet^{\Delta^{\op}}$ is replaced with any of its Bousfield localizations.

The model structure on RelCat is left proper. It is also right proper when the model structure on $ sSet^{\Delta^{\op}}$ is right proper.
=--
+-- {: .proof}
###### Proof
The existence and properness of the model structure and the Quillen equivalence is Theorem 6.1 of [Barwick-Kan](#BarKan), as is the fact $N_\xi$ preserves and reflects weak equivalences.

Proposition 10.3 shows the adjunction unit is a natural weak equivalence.
That the counit is a natural weak equivalence follows by considering the triangle identity, 3-for-2, and the fact $N_\xi$ reflects weak equivalences
$$
N_\xi C \xrightarrow{\eta N_\xi C} N_\xi K_\xi N_\xi C
\xrightarrow{N_\xi \varepsilon C} N_\xi C
$$
Finally, by considering the diagram
$$
 \array{
  X & \stackrel{\eta}{\to} & N_\xi K_\xi X
  \\
  \downarrow f && \downarrow
  \\
  Y & \stackrel{\eta}{\to} & N_\xi K_\xi Y
 }
$$
we see $f$ is a weak equivalence iff $N_\xi K_\xi f$ is,
and iff $K_\xi f$ is.
=--

In particular, Rezk's complete Segal space structure on bisimplicial sets is a Bousfield localization of the Reedy model structure, so we can define:

+-- {: .num_defn}
###### Definition
Let $(RelCat, Rezk)$ denote the model structure corresponding to the complete
Segal spaces.
=--

+-- {: .num_theorem}
###### Theorem
The localization $L(RelCat, Rezk) \simeq (\infty,1)Cat$ is the ∞-localization functor $(C,W) \mapsto L(C, W)$ inverting the weak equivalences
=--

+-- {: .proof}
###### Proof
We will see below that we can compute this through the map $(C,W) \to (NC, NW)$ to marked simplicial sets. Suppose we're given a fibrant replcaement $(NC, NW) \to Y^\natural$. Since $NC$ is a quasi-category, composition induces an equivalence for every quasi-category $Z$
$$
  Fun(Y, Z) \simeq Map^\flat(Y^\natural, Z^\natural)
  \to Map^\flat((NC, NW), Z^\natural) \simeq Fun_{NW}(NC, Z)
$$
thus $Y$ satisfies the universal property of $L(NC, NW)$.
=--

It is shown in [Meier](#Meier) that [[categories of fibrant objects]] are fibrant in this model structure.

### Compatibility with other models for (∞,1)-categories

One often uses the [[simplicial localization|hammock localization]] $L^H : RelCat \to sSetCat$, where $sSetCat$ is given the Bergner model structure whose weak equivalences are the Dwyer-Kan equivalences: i.e. the local weak homotopy equivalences.

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

A functor $f$ in $RelCat$ is a Rezk equivalence iff $L^H(f)$ is a Dwyer-Kan equivalence.
=--
This is main theorem 1.4 of [Barwick-Kan](#RezkIsDK).

The idea underlying marked simplicial is directly analogous to the idea underlying relative categories. In fact, the functor $RelCat \to sSet^+ : (C,W) \to (NC, NW)$ preserves and reflects equivalences, since

+-- {: .num_prop}
###### Proposition
Let $R$ be a fibrant replacement in $sSetCat$. Then the natural transformation
$(NC, NW) \mapsto NRL^H(C,W)^\natural$ of marked simplicial sets is a natural weak equivalence.
=--
This is theorem 1.2.1 of [Hinich 13](#Hinich13)

### Nerve functors

The compatibility of the various nerve and simplicial localization functors is in section 1.11 of

* [[Clark Barwick]] and [[Dan Kan]], _A characterization of simplicial localization functors_ ([arXiv:math/1012.1540](http://arxiv.org/abs/1012.1540))

## References

* [[Clark Barwick]] and [[Dan Kan]],

  * {#BarKan} _Relative categories: another model for the homotopy theory of homotopy theories_ ([arXiv:math/1011.1691](http://arxiv.org/abs/1011.1691))

  * _A characterization of simplicial localization functors_ ([arXiv:math/1012.1540](http://arxiv.org/abs/1012.1540))

  * {#RezkIsDK} _In the category of relative categories the Rezk equivalences are exactly the DK-equivalences_ ([arXiv:math/1012.1541](http://arxiv.org/abs/1012.1541))

  * _A Thomason-like Quillen equivalence between quasi-categories and relative categories_ ([arXiv:math/1101.0772](http://arxiv.org/abs/1101.0772))

  * _Partial model categories and their simplicial nerves_ ([arXiv:math/1102.2512](http://arxiv.org/abs/1102.2512))

* {#Hinich13} [[Vladimir Hinich]] *Dwyer-Kan Localization Revisited*,
Homology, Homotopy and Applications Volume 18 (2016) Number 1 ([arXiv:1311.4128](https://arxiv.org/abs/1311.4128), [doi:10.4310/HHA.2016.v18.n1.a3](https://dx.doi.org/10.4310/HHA.2016.v18.n1.a3))

* Lennart Meier, *Fibration Categories are Fibrant Relative Categories*, [arxiv](https://arxiv.org/abs/1503.02036)
 {#Meier}

[[!redirects Barwick-Kan equivalence]]
[[!redirects Barwick–Kan equivalence]]


model structure on S-modules

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[model structure on spectra]] for [[S-modules]].

The [[category]] of _S-modules_ ([EKMM 97](#EKMM97)) is a [[presentable (∞,1)-category|presentation]] of the [[symmetric monoidal (∞,1)-category]] [[stable (infinity,1)-category of spectra|of spectra]], with the special property that it implements the [[smash product of spectra]] such as to yield itself a [[symmetric monoidal category|symmetric]] [[monoidal model category|monoidal]] [[model category of spectra]]: the _[[model structure on symmetric spectra]]_. This implies in particular that with respect to this [[symmetric smash product of spectra]] an [[E-∞ ring]] is presented simply as a plain [[commutative monoid]] [[internalization|in]] S-modules.

## Properties

### Relation to the model structure on symmetric spectra

There is a [[Quillen equivalence]] to the [[model structure on symmetric spectra]] ([Schwede 01](#Schwede01)).

### Relation to model structure on orthogonal structure

Comparison to the [[model structure on orthogonal spectra]] is due to ([Mandell 04](#Mandell04)).

## Related concepts

[[model structure on spectra]]

* [[sequential spectrum]], [[model structure on sequential spectra]]

with [[symmetric monoidal smash product of spectra]]

* [[symmetric spectrum]], [[model structure on symmetric spectra]]

* [[orthogonal spectrum]], [[model structure on orthogonal spectra]]

* [[S-module]], **model structure on S-modules**

* [[excisive functor]], [[model structure on excisive functors]]

* [[Model categories of diagram spectra]]

## References

The construction originates in

* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], _Rings, modules and algebras in stable homotopy theory_, AMS 1997, 2014

Review includes

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], sections 2 and 3 of _[[Modern foundations for stable homotopy theory]]_ ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

Comparison to the [[model structure on symmetric spectra]] is due to

* {#Schwede01} [[Stefan Schwede]], _S-modules and symmetric spectra_, Math. Ann. 319, 517–532 (2001) ([pdf](http://www.math.uni-bonn.de/people/schwede/comparison.pdf))

Comparison to the [[model structure on orthogonal spectra]] is due to

* {#Mandell04} [[Michael Mandell]], _Equivariant orthogonal spectra and S-modules_ ([pdf](http://www.math.uiuc.edu/K-theory/0408/MMM.pdf))


model structure on sections

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

# Model structure on sections

* table of contents
{: toc}

## Definitions

### Sections

Let $K$ be a [[small category]] and $F:K\to ModelCat$ a [[pseudofunctor]], where $ModelCat$ is the 2-category of [[model categories]], [[Quillen adjunctions]] pointing in the direction of their left adjoints, and [[mate]]-pairs of natural isomorphisms.  This is sometimes called a **Quillen presheaf** (with "right" or "left" attached according to whether one is thinking about the left adjoints or the right adjoints in the Quillen adjunctions); we denote the Quillen adjunction $F(f)$ by $f_! \dashv f^*$.

The category of **sections** of $F$ is the category of [[sections]] $K \to \int F$ of its [[Grothendieck construction]].  Explictly, a section of $F$ consists of, for each $k\in K$ an object $X_k\in F_k$, and for each $f:\ell\to k$ in $K$ a morphism $X_\ell \to f^* X_k$ (or equivalently $f_! X_\ell \to X_k$) satisfying a functoriality condition.  The category $Sect(F)$ is the [[lax limit]] of the diagram $K\to Cat$ consisting of the left Quillen functors $f_!$, and also the [[colax limit]] of the analogous diagram consisting of the right Quillen functors.  Accordingly, it is [[locally presentable]] if each category $F_k$ is.

### Projective and injective model structures

\begin{theorem}
  If each $F_k$ is a [[combinatorial model category]], [[accessible model category]], or [[tractable model category]], then $Sect(F)$ has two model structures that are also combinatorial, accessible, or tractable respectively:

1. A **projective model structure** in which a morphism $X\to Y$ is a fibration or weak equivalence if each $f_k : X_k\to Y_k$ is such in $F_k$.
1. An **injective model structure** in which a morphism $X\to Y$ is a cofibration or weak equivalence if each $f_k : X_k\to Y_k$ is such in $F_k$.

\end{theorem}
\begin{proof}
  The combinatorial and tractable cases are [Barwick, 2.28 and 2.30](#Barwick10).  They are a straightforward application of [[transferred model structures]], generalizing the usual constructions of [[model structures on functors]].  The accessible case follows similarly using the accessible version of transfer.
\end{proof}

These model structures are a presentation of the $(\infty,1)$-categorical *lax limit* of $F$.  To present the *pseudo (homotopy) limit* of $F$ we have to localize them:

\begin{theorem}
  The above projective model structure has a [[left Bousfield localization]], called the **(projective) homotopy limit model structure**, in which the fibrant objects are the projectively fibrant ones that are **homotopy cartesian**, i.e. for each $f:\ell\to k$ in $K$ the induced map $X_\ell \to f^*(X_k) \to f^*(R X_k)$ is a weak equivalence, where $R$ denotes a fibrant replacement.
\end{theorem}
\begin{proof}
  This is [Barwick, 4.38](#Barwick10).
\end{proof}

### Reedy model structures

If each $F_k$ is a model category (not even necessarily cofibrantly generated) and $K$ is a [[Reedy category]], then there is also a **Reedy model structure** on $Sect(F)$ defined analogously to the ordinary case.  This is in [Balzin18](#Balzin18), together with a generalization to any "admissible model semifibration" over a Reedy category.

## Properties

* These model structures are presentations of the $(\infty,1)$-category of sections (i.e. (co)lax limit) of the induced functor of $(\infty,1)$-categories.  For the projective model structure this is shown in [Harpaz18](#Harpaz18) (hence also for the injective model structure, since they have the same weak equivalences), which considers more generally sections defined on a [[simplicial category]].  For the Reedy model structure it is shown in [Balzin18](#Balzin18), with the special case of an [[inverse category]] being in [Spitzweck10](#Spitzweck10).

* A subcategory of the injective model structure analogous to the homotopy limit model structure is shown in [Bergner10](#Bergner10) to present the homotopy limit of the corresponding diagram of [[complete Segal spaces]].

## Related pages

* The dual construction is the [[Grothendieck construction for model categories]] (a lax colimit).

## References

* {#Barwick10} [[Clark Barwick]], _On left and right model categories and left and right Bousfield localizations_.  Homology, Homotopy and Applications, 12.2, 2010, p. 245-320.  [pdf](https://core.ac.uk/download/pdf/16520637.pdf), [related arxiv preprint](https://arxiv.org/abs/0708.2067)

* {#Bergner10} [[Julia E. Bergner]], _Homotopy limits of model categories and more general homotopy theories_, [arXiv:1010.0717](https://arxiv.org/abs/1010.0717)

* {#Spitzweck10} [[Markus Spitzweck]], _Homotopy limits of model categories over inverse index categories_, 2010, [doi](https://doi.org/10.1016/j.jpaa.2009.08.001)

* Johnson, Mark W. _On modified Reedy and modified projective model structures_. Theory Appl. Categ. 24 (2010), No. 8, 179–208.

* {#Harpaz18} [[Yonatan Harpaz]], _Lax limits of model categories_, [pdf](https://www.math.univ-paris13.fr/~harpaz/lax_limits.pdf)

* {#Balzin18} Edouard Balzin, _Reedy Model Structures in Families_, [arXiv:1803.00681](https://arxiv.org/abs/1803.00681)

[[!redirects model structures on sections]]
[[!redirects model structure for sections]]
[[!redirects model structures for sections]]
[[!redirects lax limit model structure]]
[[!redirects lax limit model structures]]
[[!redirects lax limit of model categories]]
[[!redirects lax limits of model categories]]
[[!redirects colax limit model structure]]
[[!redirects colax limit model structures]]
[[!redirects colax limit of model categories]]
[[!redirects colax limits of model categories]]
[[!redirects oplax limit model structure]]
[[!redirects oplax limit model structures]]
[[!redirects oplax limit of model categories]]
[[!redirects oplax limits of model categories]]
[[!redirects homotopy limit model structure]]
[[!redirects homotopy limit model structures]]
[[!redirects homotopy limit of model categories]]
[[!redirects homotopy limits of model categories]]


model structure on semi-simplicial sets

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

#Contents#
* table of contents
{:toc}

## Idea

There exists the [[model category]] structure on the [[category]] of [[semi-simplicial sets]] which is [[transferred model structure|transferred]] along the [[right adjoint]] to the [[forgetful functor]] from the [[classical model structure on simplicial sets]] ([van den Berg 13](#vandenBerg)). See [this discussion](https://nforum.ncatlab.org/discussion/4861/model-structure-on-semisimplicial-sets/?Focus=56290#Comment_56290), which seems to conclude that this is [[Quillen equivalence|Quillen equivalent]] to the [[classical model structure on simplicial sets]].

There is also a [[weak model category]] structure ([Henry 18](#Henry18)), for which the Quillen equivalence to simplicial sets is proven as [Henry 18, Thm 5.5.6 (iv)](#Henry18).

Also there is the structure of a [[semimodel category]] ([Rooduijn 2018](#Rooduijn2018)) and of a [[fibration category]] on semisimplicial sets ([Sattler 18, Th, 3.18](#Sattler18)) (and [[cofibration category]] on fibrant-cofibrant objects).

## References
 {#References}

As a [[model category]]-structure:

* {#vandenBerg} [[Benno van den Berg]], _A note on semisimplicial sets_, 2013 ([[vandenBerg_SemisimplicialSets.pdf:file]])

As a [[weak model category]]:

* {#Henry18} [[Simon Henry]], Theorem 5.5.6 of: *Weak model categories in classical and constructive mathematics*, Theory and Applications of Categories, Vol. 35, 2020, No. 24, pp 875-958.  ([arXiv:1807.02650](https://arxiv.org/abs/1807.02650), [tac:35-24](http://www.tac.mta.ca/tac/volumes/35/24/35-24abs.html))

As a right [[semimodel category]]:

* {#Rooduijn2018} Jan Rooduijn, *A right semimodel structure on semisimplicial sets*, Amsterdam 2018 ([pdf](https://eprints.illc.uva.nl/id/eprint/1663/1/MoL-2018-34.text.pdf), [mol:4787](https://msclogic.illc.uva.nl/theses/archive/publication/4787/A-right-semimodel-structure-on-semisimplicial-sets))

As a [[fibration category]]:

* {#Sattler18} [[Christian Sattler]], *Constructive homotopy theory of marked semisimplicial sets* ([arXiv:1809.11168](https://arxiv.org/abs/1809.11168))

The above note contains a mistake in Theorem 3.43: semisimplicial sets do not form a [[cofibration category]] as considered there: (cofibration, weak equivalence)-factorizations do not generally exist (cf. the paragraph at the end of Subsection 3.2). Instead, the notion of cofibration category has to be weakened using a notion of pseudofactorizations (similar to the notion of weak model category).

[[!redirects model structure on semisimplicial sets]]

[[!redirects weak model structure on semi-simplicial sets]]
[[!redirects weak model structure on semisimplicial sets]]

[[!redirects semimodel structure on semisimplicial sets]]
[[!redirects semi-model structure on semi-simplicial sets]]

[[!redirects fibration category of semi-simplicial sets]]
[[!redirects fibration category of semisimplicial sets]]


model structure on sequential spectra

## Idea

A [[model category]] structure on [[sequential spectra]].

For sequential spectra in the [[classical model structure on simplicial sets]]:

* [[Bousfield-Friedlander model structure]]

For sequential spectral in the [[classical model structure on topological spaces]]

* [[model structure on topological sequential spectra]]

## Related concepts

[[model structure on spectra]]

* [[sequential spectrum]], **model structure on sequential spectra**

with [[symmetric monoidal smash product of spectra]]

* [[symmetric spectrum]], [[model structure on symmetric spectra]]

* [[orthogonal spectrum]], [[model structure on orthogonal spectra]]

* [[S-module]], [[model structure on S-modules]]

* [[excisive functor]], [[model structure on excisive functors]]


model structure on simplicial abelian groups

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

#Contents#
* table of contents
{:toc}

## Definition

The [[category]] of [[simplicial abelian groups]] carries the [[mathematical structure|structure]] of a [[model category]] $sAbGrp_{proj}$ whose [[weak equivalences]] and [[fibrations]] are those of the [[forgetful functor|underlying]] morphisms in the [[classical model structure on simplicial sets]], hence whose

* [[weak equivalences]] are the [[simplicial weak homotopy equivalences]],

* [[fibrations]] are the [[Kan fibrations]]

of [[forgetful functor|underlying]] [[simplicial sets]].

Hence the [[free functor|free]]/[[forgetful functor|underlying]]-[[adjoint functors]] (where the [[free functor]] produces [[free simplicial abelian groups]] $\mathbb{Z}[-]$) is a [[Quillen adjunction]] with the [[classical model structure on simplicial sets]] $sSet_{Qu}$:

\[
  \label{QuillenAdjunctionWithClassicalModelStructureOnSimplicialSets}
  sAbGrp_{proj}
  \underoverset
    {\underset{undrlng}{\longrightarrow}}
    {\overset{\mathbb{Z}[-]}{\longleftarrow}}
    {\bot_{\mathrm{Qu}}}
  sSet_{Qu}
\]

## Properties

### Monoidal structure
 {#MonoidalStructure}

With respect to the degreewise [[tensor product of abelian groups]] this is a [[monoidal model category]] ([Schwede & Shipley 2003, p. 312 (26 of 48)](#SchwedeShipley03)).

### Dold-Kan correspondence

The [[Dold-Kan correspondence]] yields (see [there](Dold-Kan+correspondence#ModelCatVersion) for more) a [[Quillen equivalence]] to the projective [[model structure on connective chain complexes]].

([Quillen 67, Section II.4 item 5](#Quillen67), see also [Schwede-Shipley 03, section 4.1](#SchwedeShipley03), [p.17](http://www.math.uic.edu/~bshipley/monoidalequi.final.pdf#page=17), [Jardine 03, Lemma 1.5](#Jardine03)).

In fact, much more is true: all five classes of maps
in a [[model category]] ([[weak equivalences]], (acyclic) [[cofibrations]],
and (acyclic) [[fibrations]]) are [[preserved]] and [[reflected]]
by both of these equivalences.
That is to say, each [[model structure]] is obtained from the other
one by transferring it along the corresponding equivalence of categories.

## Related concepts

* [[Dold-Kan correspondence]]

* [[model structure on cosimplicial abelian groups]]

* [[model structure on simplicial algebras]]

## References

Due to:

* {#Quillen67} [[Daniel Quillen]], Section II.4 in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

Further discussion:

* {#SchwedeShipley03} [[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), [euclid:euclid.agt/1513882376](https://projecteuclid.org/euclid.agt/1513882376))

* {#Jardine03} [[J. F. Jardine]], Lemma 1.5 in: _Presheaves of chain complexes_, K-theory 30.4 (2003): 365-420 ([pdf](https://pdfs.semanticscholar.org/360c/fd4eb76da956e0c8dd897377f6d6f342e44e.pdf), [[JardinePresheavesOfChainComplexes.pdf:file]])


model structure on simplicial algebras

+-- {: .rightHandSide}
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###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $T$ a [[Lawvere theory]] and $T Alg$ the [[category]] of [[algebra over a Lawvere theory]], there is a [[model category]] structure on the category $T Alg^{\Delta^{op}}$ of [[simplicial object|simplicial]] $T$-algebras which models the $\infty$-algebras for $T$ regarded as an [[(∞,1)-algebraic theory]].

## Details

First we consider the case of [[simplicial objects]] in algebras over an ordinary Lawvere theory:

* _[For algebras over ordinary Lawvere theories](#OverOrdinaryLawvereTheories)_

Then we generalize to the case that the Lawvere theory itself is simplicial:

* _[For algebras over simplicial Lawvere theories](#ForAlgebrasOverSimplicialLawvereTheories)_

### For algebras over ordinary Lawvere theories
  {#OverOrdinaryLawvereTheories}

Recall that the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C^{op})$ itself is modeled by the [[model structure on simplicial presheaves]]

$$
  PSh_{(\infty,1)}(C^{op})
  \simeq
  [C, sSet]^\circ
  \,,
$$

where we regard $C$ as a [[Kan complex]]-[[enriched category]] and have on the right the [[sSet]]-[[enriched functor category]] with the projective or injective model structure, and $(-)^\circ$ denoting the full enriched subcategory on fibrant-cofibrant objects.

This says in particular that every weak $(\infty,1)$-functor $f : C \to \infty \mathrm{Grp}$ is equivalent to a _rectified_ one $F : C \to KanCplx$. And $f \in PSh_{(\infty,1)}(C^{op})$ belongs to $Alg_{(\infty,1)}(C)$ if $F$ preserves finite products _weakly_  in that for $\{c_i \in C\}$ a finite collection of objects, the canonical natural morphism

$$
  F(c_1 \times \cdots, \c_n)
  \to
  F(c_1) \times \cdots \times F(c_n)
$$

is a [[homotopy equivalence]] of [[Kan complexes]].

We now look at model category structure on _strictly_ product preserving functors $C \to sSet$, which gives an equivalent model for $Alg_{(\infty,1)}(C)$. See [[model structure on homotopy T-algebras]].

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

Let $T$ be the [[syntactic category]] of a [[Lawvere theory]], and let $T Alg^{\Delta^{op}} \subset Func(T,sSet)$ be the [[full subcategory]] of the [[functor category]] from $T$ to [[sSet]] on those functors that preserve these [[finite products]].

Then $T Alg^{\Delta^{op}}$ carries the structure of a [[model category]] $(T Alg^{\Delta^{op}})_{proj}$ where the weak equivalences and the fibrations are those maps underlying which are weak equivalences or fibrations, respectively, in the [[classical model structure on simplicial sets]].

=--

This is due to ([Quillen 67, II.4 theorem 4](#Quillen67)).

The inclusion $i : sAlg(C) \hookrightarrow sPSh(C^{op})_{proj}$ into the projective [[model structure on simplicial presheaves]] evidently preserves fibrations and acyclic fibrations and gives a [[Quillen adjunction]]

$$
  sAlg(C)_{proj}
  \stackrel{\leftarrow}{\underset{i}{\hookrightarrow}}
  sPSh(C^{op})
  \,.
$$

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

The total right [[derived functor]]

$$
  \mathbb{R}i \;\colon\; Ho(sAlg(C)_{proj}) \to Ho(sPSh(C^{op})_{proj})
$$

is a [[full and faithful functor]] and an object $F \in sPSh(C^{op})$ belongs to the [[essential image]] of $\mathbb{R}i$ precisely if it preserves products up to [[weak homotopy equivalence]].

=--

This is due to ([Badzioch 02, def. 5.2, prop. 5.4, cor. 5.7](#Badzioch02)).

It follows that the natural $(\infty,1)$-functor

$$
  (sAlg(C)_{proj})^\circ
    \stackrel{}{\longrightarrow}
  PSh_{(\infty,1)}(C^{op})
$$

is an [[equivalence of quasi-categories|equivalence]].

### For algebras over simplicial Lawvere theories
 {#ForAlgebrasOverSimplicialLawvereTheories}

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

Let $\Gamma = (Skel(FinSet^{\ast/}))$ be [[Segal's category]], the [[opposite category]] of a [[skeleton]] of [[finite set|finite]] [[pointed sets]].

A _[[simplicial Lawvere theory]]_ is a a pointed [[simplicial category]] $T$ equipped with a [[functor]] $i \;\colon\;\Gamma \to T$ such that

1. $T$ has the same [[set]] of [[objects]] as $\Gamma$;

1. $i$ is the identity on objects

1.  $i$ preserves [[finite products]]

Given a simplicial theory $T$, then a _simplicial $T$-algebra_ is a product preserving [[simplicial functor]] $X$ to the [[simplicial category]] of [[pointed simplicial sets]]. The simplicial set

$$
  X(1_+) \in sSet
$$

(the value on the pointed 2-element set) is called the _underlying simplicial set_ of the $T$-algebra.

A [[homomorphism]] of $T$-algebras is a simplicial [[natural transformation]] between such functors. Write

$$
  T Alg \in sSet Cat
$$

for the resulting [[simplicial category]].

A homomorphism is called a _weak equivalence_ or a _fibration_ if on underlying simplicial sets it is a weak equivalence or fibration, respectively, in the [[classical model structure on simplicial sets]].
Write

$$
  (T Alg)_{proj}
$$

for the category equipped with these [[classes]] of morphisms.

=--

([Schwede 01, def. 2.1 and def. 2.2 and beginning of section 3](#Schwede01))

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

In the case that the [[simplicial Lawvere theory]] $T$ happens to be simplicially discrete, i.e. an ordinary [[category]], then the category $(T Alg)_{proj}$ of simplicial $T$-algebras from def. \ref{SimplicialLawvereTheory} with that from prop. \ref{ModelStructureOnSimplicialAlgebrasOverOrdinaryLawveretheory}.

=--

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

For $T$ a [[simplicial Lawvere theory]] (def. \ref{SimplicialLawvereTheory}) the category $(T Alg)_{proj}$ from def. \ref{SimplicialLawvereTheory} is a [[simplicial model category]].

=--

This is due to ([Reedy 74, theorem I](#Reedy74)), reviewed in ([Schwede 01](#Schwede01)).

The analogous statement with the [[classical model structure on simplicial sets]] replaced by the [[classical model structure on topological spaces]] is due to ([Schwänzl-Vogt 91](#SchwänzlVogt91))

A comprehensive statement of these facts is in [[Higher Topos Theory|HTT, section 5.5.9]].

### Other cases
 {#OtherCases}

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

Let $\mathcal{C}$ be a [[category]] with [[enough projectives]]. Say that a morphism $f$ in the category $\mathcal{C}^{\Delta^{\mathrm{op}}}$ of [[simplicial objects]] in $\mathcal{C}$ is a _weak equivalence_ or _fibration_ if for every [[projective object]] $P \in \mathcal{C}$ then $Hom(P,f)$ is a weak equivalence or fibration, respectively, in the [[classical model structure on simplicial sets]] (i.e. a [[Kan fibration]] or [[weak homotopy equivalence]], respectively). If now at least one of the following conditions is satisfied,

* every object of $\mathcal{C}^{\Delta^{op}}$ is fibrant;

* $\mathcal{C}$ is closed under [[colimits]] and has a [[small set]] of [[projective object|projective]] [[generators]];

then this makes $\mathcal{C}^{\Delta^{op}}$ a [[simplicial model category]].

=--

([Quillen 67, II.4, theorem 4](#Quillen67))

## Properties

### Relation to homotopy $T$-algebras

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

There is a [[Quillen equivalence]] between the model structure on simplicial $T$-algebras and the model structure for [[homotopy T-algebra]]s. (See there).

=--

This is ([Badzioch 02, theorem 1.3](#Badzioch02)).

### Homotopy groups

Let $T$ be an _abelian Lawvere theory_,  a theory that contains the theory of [[abelian group]], $Ab \to T$. Then every simplicial $T$-algebra has an underlying [[simplicial abelian group]] and is necessarily a [[Kan complex]].

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

The [[homotopy groups]] $\pi_*$ of a simplicial abelian $T$-agebra form an $\mathbb{N}$-graded $T$-algebra $\pi_*(A)$

=--

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

The inclusion of the full [[subcategory]] $i : T Alg \hookrightarrow T Alg^{\Delta^{op}}$ of ordinary $T$-algebra as the simplicially constant ones constitutes a [[Quillen adjunction]]

$$
  (\pi_0  \dashv i) : T Alg \stackrel{\overset{\pi_0}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  T Alg^{\Delta^{op}}
$$

from the [[trivial model structure]] on $T Alg$.

The [[derived functor]] $\mathbb{R} i : T Alg \to Ho(T Alg^{\Delta^{op}})$ is a  [[full and faithful functor]].

=--

This allows us to think of ordinary $T$-algebras a sitting inside $\infty$-$T$-algebras.

> all this is certainly true for ordinary $k$-algebras. Need to spell out general proof.
> I believe this is the Badzioch paper cited above - JB

## Examples

### Simplicial associative $k$-algebras

A [[simplicial ring]]s is a simplicial $T$-algebras for $T$ the Lawvere theory of rings.

Let $k$ be an ordinary commutative ring and $T$ the theory of commutative [[associative algebras]] over $k$. We write $T Alg$ as $sCAlg_k$ or $CAlg_k^{op}$.

Such simplicial $k$-algebras are discussed for instance in ([Toën-Vezzosi, section 2.2.1](#ToenVezzosi)).
According to ([Schwede 97](#Schwede97), Lemma 3.1.3), this model structure is [[proper model category|proper]].

### Simplicial commutative $k$-algebras

The following is a variant of the model structure on simplicial commutative algebras that is implied by the above general theorem.

+-- {: .num_prop}
###### Proposition
**(second model structure)**

Let $T$ be the [[Lawvere theory]] for  ([[associative algebra|associative]] and) [[commutative algebras]] over a [[ring]] $k$. Then $(cAlg_k)^{\Delta^{op}}$ becomes a [[simplicial model category]] with

* weak equivalences the morphisms whose underlying morphism of simplicial sets are weak equivalences in the [[classical model structure on simplicial sets]];

* fibrations the morphisms $X \to Y$ such that $X \to \pi_0 X \times_{\pi_0 Y} Y$ is a degreewise surjection.

=--

This appears as ([Goerss-Schemmerhorn, theorem 4.17](#GoerssSchemmerhorn)).

### Simplicial Lie algebras

See at _[[model structure on simplicial Lie algebras]]_.

### Simplicial complete Hopf algebras
 {#SimplicialCommutativeHopfAlgebras}

Complete rational [[Hopf algebras]] are not the models of a [[Lawvere theory]], even though in some sense they are close ([Quillen 67, bottom of p. 265 (61 of 92)](#Quillen67)).

But they do satisfy the assumptions of proposition \ref{QuillenCriterion} ([Quillen 69, appendix B, prop. 2.24](#Quillen69)). Hence simplicial rational complete Hopf algebras form a [[simplicial model category]].

## Related concepts

* [[model structure on simplicial abelian groups]]

* [[model structure on cosimplicial algebras]]

* [[model structure on monoids]]

* [[algebra over a monad]]

  [[∞-algebra over an (∞,1)-monad]]

  * [[model structure on algebras over a monad]]

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * [[homotopy T-algebra]]

* [[algebra over an operad]]

   [[∞-algebra over an (∞,1)-operad]]

   * [[model structure on algebras over an operad]]

## References

The classical reference for the [[transferred model structure]] on simplicial $T$-algebras is

* {#Quillen67} [[Daniel Quillen]], Section II.4 item 5 in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

with some further remarks in

* {#Quillen69} [[Dan Quillen]], Section II.3 of: _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))

The generalization to the case that the theory $T$ itself is allowed to be simplicial is due to

* {#Reedy74} [[Christopher Reedy]], _Homology of algebraic theories_, Ph.D. Thesis, University of California, San Diego, 1974

and the topological version is due to

* {#SchwänzlVogt91} [[Roland Schwänzl]], [[Rainer Vogt]], _The categories of $A_\infty$- and $E_\infty$-monoids and ring spaces as closed simplicial and topological model categories_, Archives of Mathematics 56 (1991) 405-411 ([doi:10.1007/BF01198229](https://doi.org/10.1007/BF01198229))

More is in

* {#Schwede01} [[Stefan Schwede]], _Stable homotopy of algebraic theories_, Topology 40 (2001) 1-41 ([pdf](http://www.math.uni-bonn.de/people/schwede/stable.pdf))

In

* {#Rezk} [[Charles Rezk]], _Every homotopy theory of simplicial algebras admits a proper model_ ([math/0003065](http://arxiv.org/abs/math/0003065)) ,

it is discussed that every model category of simplicial $T$-algebras is [[Quillen equivalence|Quillen equivalent]] to a left [[proper model category]].

The fact that the model structure on simplicial $T$-algebras serves to model $\infty$-algebras is in

* {#Badzioch02} [[Bernard Badzioch]], _Algebraic theories in homotopy theory_, Annals of Mathematics, 155 (2002), 895-913 ([JSTOR](http://www.jstor.org/stable/3062135))

and the multi-sorted version is in

* {#Bergner07} [[Julie Bergner]], _Rigidification of algebras over multi-sorted theories_, Algebraic and Geometric Topoogy 7 (2007) ([arXiv:0508152](https://arxiv.org/abs/math/0508152))

The simplicial model structure on ordinary simplicial algebras (i.e. simplicial [[associative algebras]]) is discussed in

* [[Paul Goerss]], [[Kirsten Schemmerhorn]], _Model categories and simplicial methods_ ([pdf](http://www.math.northwestern.edu/~pgoerss/papers/ucnotes.pdf))

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

Discussion of simplicial commutative associative algbras over a ring in the context of [[derived geometry]] is in

* {#ToenVezzosi} [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry II: geometric stacks and applications _ ([arXiv](http://arxiv.org/abs/math/0404373))

Discussion of divided [[power operations]] on simplicial algebras is in

* [[Benoit Fresse]], _On the homotopy of simplicial algebras over an operad_, Transactions of the AMS, volume 352, number 9 ([jstor](http://www.jstor.org/stable/118177))

[[!redirects model structure on simplicial T-algebras]]

[[!redirects simplicial T-algebra]]

[[!redirects simplicial algebra]]
[[!redirects simplicial algebras]]


model structure on simplicial groupoids

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

Before we start, beware the usual terminology issue with "[[simplicial groupoids]]":

\begin{remark}
**(terminology)**
This entry is concernd with  _[[simplicial groupoids]]_ as traditionally understood (following [Dwyer & Kan (1984)](#DwyerKan84)), referring to [[simplicial objects]] in the [[category]] [[Grpd]] of [[groupoids]] with the special property that their [[simplicial set]] of objects is simplicially constant. Any such "Dwyer-Kan simplicial groupoid" is equivalently an [[sSet-enriched category]] that is a [[groupoid]] in the [[enriched category theory|enriched]] sense: an *[[sSet]]-[[enriched groupoid]]*. Therefore, and since in applications it is often this [[sSet]]-[[enriched category|enriched]] [[structure]] which matters, a more accurate term would be *simplicially enriched groupoids*, but this terminology is not at all standard. See the corresponding discussion at *[[simplicial groupoid]]* ([here](simplicial+groupoid#Definition)).
\end{remark}

## Definition

Write $sGrpd_{DK}$ for the [[category]] of [[simplicial groupoids]] (whose [[simplicial sets]] of [[objects]] are understood to be constant, see the discussion [there](simplicial+groupoid#Definition)).

\begin{definition}\label{FreeMorphismsOfSimplicialGroupoids}
**(free morphisms of simplicial groupoids)**
\linebreak
  Say that a [[morphism]] $\mathbf{f} \colon \mathbf{X} \to \mathbf{Y}$ of [[simplicial groupoids]] is *free* iff:

1. it is degreewise [[injective]] (i.e. on the [[sets]] of [[objects]] and on the sets of [[morphisms]] in each degree);

1. there is a [[subset]]

   \[
     \label{SetOfFreeGenerators}
     \Gamma
       \;\subset\;
     \mathbf{Y}
   \]

   of [[morphisms]] in $\mathbf{Y}$ (of any degree) with the following properties:

   1. $\Gamma$ contains no [[identity morphisms]];

   1. $\Gamma$ is closed under forming degenerate cells;

   1. every non-[[identity morphism]] in $\mathbf{Y}$ is *uniquely* the [[composition]] of a *reduced* sequence of morphisms

      1. in $\Gamma$,

      1. in the [[image]] under $\mathbf{f}$ of non-identity morphisms in $\mathbf{X}$,

      1. their [[inverse morphisms|inverses]],

      where *reduced* means that:

      1. no morphism in the sequences is consecutive with its [[inverse morphisms|inverse]],

      1. no two non-[[identity morphisms]] in the [[image]] of $\mathbf{f}$ are consecutive.
\end{definition}
[[Dwyer & Kan 1984, §2.3](#DwyerKan84)]

\begin{proposition}\label{TheModelStructure}
**(Dwyer-Kan model structure on simplicial groupoids)**
\linebreak
There is a [[model category]] structure on $sGrpd_{DK}$ whose

* [[weak equivalences]] are the [Dwyer-Kan equivalences](model+structure+on+sSet-categories#DwyerKanEquivalences), hence those morphisms $\mathbf{f} \colon \mathbf{H} \to \mathbf{K}$ such that

  1. $\mathbf{f}$ induces in [[isomorphism]] on [[connected components]] $\pi_0 \mathbf{f} \colon \pi_0 \mathbf{H} \to \pi_0 \mathbf{K}$;

  1. for each object $x$ of $\mathbf{H}$ the induced morphism $\mathbf{H}(x,x) \to \mathbf{K}\big(\mathbf{f}(x), \mathbf{f}(x)\big)$ is a weak equivalence in the [[model structure on simplicial groups]], which in turn equivalently means that it is a weak equivalence of [[underlying]] simplicial sets in the [[classical model structure on simplicial sets]] (a [[simplicial weak homotopy equivalence]]).

* [[fibrations]] are the [[Kan fibration|Kan]]-[[isofibrations]], namely those morphisms $\mathbf{f} \colon \mathbf{H} \to \mathbf{K}$ such that

  1. for every [[object]] $x$ of $\mathbf{H}$ and every [[morphism]] $\omega \colon \mathbf{f}(x) \to y$ in $\mathbf{K}_0$ there is a morphism $\hat \omega \colon x \to z$ of $H_0$ such that $\mathbf{f}(\hat \omega) = \omega$;

  1. for every object $x$ in $\mathbf{H}$ the induced morphism $\mathbf{f} \colon \mathbf{H}(x,x) \to \mathbf{K}\big(\mathbf{f}(x), \mathbf{f}(x)\big)$ is a [[Kan fibration]].

* {#CofIsRetractOfFree} [[cofibrations]] are the [[retracts]] (in the [[arrow category]]) of the free maps from Def. \ref{FreeMorphismsOfSimplicialGroupoids}.

\end{proposition}
This is due to [Dwyer & Kan (1984), §1.1, §1.2, Thm. 2.5](#DwyerKan84), reviewed in [Goerss & Jardine (2009), p. 316 and after cor V.7.3](#GoerssJardine09).

\begin{remark}\label{RelationToModelStructureOnSimplicialGroups}
**(relation to model structure on simplicial groups)**
\linebreak
  An $X \in sGrpd_{DK}$ for which $X_0 = \ast$ is the [[terminal groupoid]] is, when regarded as a [[pointed object]], equivalently the ([[delooping]] of the) [[simplicial group]] which is its unique [[hom-object]]. Restricted to such "simplicial [[delooping groupoids]]" of simplicial groups and under this identitification, the fibrations and weak equivalences in Prop. \ref{FreeMorphismsOfSimplicialGroupoids} are those of the [[model structure on simplicial groups]].
\end{remark}

## Properties

### Extra model category properties
 {#ExtraModelCategoryProperties}

We write $sSet\text{-}Grpd$ for the category of Dwyer-Kan simplicial groupoids.

\begin{definition}\label{SimplicialIntervalGroupoid}
**(simplicial interval groupoid)**
\linebreak
 Write
$$
  \mathcal{F}
  \;\colon\;
  sSet
   \overset{\mathcal{I}}{\longrightarrow}
  sSet\text{-}Cat
    \longrightarrow
  sSet\text{-}Grpd
$$
for the [[functor]] which sends a [[simplicial set]] $S$ to the [[sSet]]-[[enriched groupoid]] $\mathcal{F}(S)$ which has precisely two objects $0,1$, no non-trivial [[endomorphisms]] and isomorphisms between $0$ and $1$ freely generated from the cells of $X$.
\end{definition}
This is [Dwyer & Kan (1984), §2.8](#DwyerKan84), related to the [[Milnor construction]] in [Goerss & Jardine (2009), pp. 314](#GoerssJardine09). (Beware that in [Bergner (2008), p. 4](#Bergner08) the statement of free generation is missing.)

\begin{example}
  The construction of Def. \ref{SimplicialIntervalGroupoid} applied to the [[terminal object|terminal]]  [[simplicial set]] $\Delta[0]$ ([[0-simplex]], "[[singleton]]") is the [[interval groupoid]]:
$$
  \mathcal{F}(\ast) \,=\,
  \big\{
    0 \overset{\sim}{\leftrightarrows} 1
  \big\}
  \,.
$$
\end{example}

\begin{proposition}
  The model structure on simplicial groupoids from Prop. \ref{TheModelStructure} is [[cofibrantly generated model category|cofibrantly generated]] with generating (acyclic) cofibrations the images under the simplicial interval functor (Def. \ref{SimplicialIntervalGroupoid}) of the generating (acyclic) cofibrations in the [[classical model structure on simplicial sets]] (see [there](classical+model+structure+on+simplicial+sets#AsACofibrantlyGeneratedModelCategory)), hence of the [[boundary]]- and [[horn]]-inclusions of [[simplices]], respectively:

$$
  \begin{array}{l}
  I
    \,\coloneqq\,
  \Big\{
    \mathcal{F}(i_n)
      \,\colon\,
    \mathcal{F}\big(\partial \Delta[n]\big)
     \hookrightarrow
    \mathcal{F}\big(\Delta[n]\big)
  \Big\}_{n \in \mathbb{N}}
  \\
  J
    \,\coloneqq\,
  \Big\{
    \mathcal{F}(j^k_n)
      \,\colon\,
    \mathcal{F}\big(\Lambda_k[n]\big)
     \hookrightarrow
    \mathcal{F}\big(\Lambda_k[n]\big)
  \Big\}_{n \in \mathbb{N}_+, 0 \leq k \leq n}
  \end{array}
$$

\end{proposition}
This is essentially [Dwyer & Kan (1984), Prop. 2.9, 2.10](#DwyerKan84), made more explicit in [Bergner (2008), Thm. 2.2](#Bergner08).

\begin{proposition}
  The model structure on simplicial groupoids from Prop. \ref{TheModelStructure} is [[right proper model category|right proper]].
\end{proposition}
This is [Bergner (2008), Prop. 2.5](#Bergner08).

### Relation to simplicial groups
 {#RelationToSimplicialGroups}

\begin{remark}
Forming simplicial [[delooping groupoids]] constitutes a [[fully faithful functor]] of [[1-categories]] from [[simplicial groups]] to [[sSet]]-[[enriched groupoids]] (DK-[[simplicial groupoids]]):

\[
  \label{SimplicialDeloopingGroupoid}
  \array{
    sGrp &\hookrightarrow& sSet\text{-}Grpd
    \\
    \mathcal{G} &\mapsto& \mathbf{B}\mathcal{G}
    \mathrlap{\,.}
  }
\]

With respect to the above model structure (Prop. \ref{TheModelStructure}) this functor clearly preserves the [[weak equivalences]] and [[fibrations]] of the [[model structure on simplicial groups]]. However, it does not have a [[left adjoint]] and thus fails to be a [[right Quillen functor]].
\end{remark}

### Relation to skeletal simplicial groupoids

The following proposition assumes the [[axiom of choice]] in the underlying category of [[Sets]].

\begin{proposition}
\label{BifibrantResolutionBySkeleta}
  Every $\mathbf{X} \,\in\, sSet\text{-}Grpd$ has a [[bifibrant object|bifibrant]] [[resolution]] by a [[skeletal groupoid]].
\end{proposition}
Here we mean a skeletal [[sSet]]-[[enriched groupoid]], hence a [[disjoint union]] of simplicial [[delooping groupoids]] (eq:SimplicialDeloopingGroupoid).
\begin{proof}
  By the existence of the model structure (Prop. \ref{TheModelStructure}) all objects admit a [[cofibrant resolution]]. Moreover every $sSet$-groupoid has a [[deformation retraction]] onto a skeleton (by [this Prop.](simplicial+groupoid#SimplicialGroupoidsAdjointEquivalentToSkeleta), assuming the [[axiom of choice]]), and since cofibrancy is [preserved](model+category#ClosureOfMorphisms) by [[retraction]] it follows that every $\mathbf{X}$ has a [[cofibrant resolution]] by a skeleton. To conclude we just need to see that skeletal simplicial groupoids are fibrant, which is the case by [Moore's theorem](simplicial+group#EverySimplicialGroupIsAKanComplex).
\end{proof}

### Relation to simplicial sets

\begin{proposition}
\label{QuillenEquivalenceWithSimplicialSets}

The

* [[Dwyer-Kan loop groupoid]]-construction $\mathcal{G}$ ([[left adjoint]])

* [[simplicial classifying space]]-construction  $\overline{W}$ ([[right adjoint]], see [here](simplicial+classifying+space#BarWForSimplicialGroupoid))

constitute a [[Quillen equivalence]]

$$
  sGrpd_{DK}
  \underoverset
    {\underset{\overline{W}}{\longrightarrow}}
    {\overset{\mathcal{G}}{\longleftarrow}}
    {\;\;\;\; \simeq_{_{\mathrlap{Qu}}} \;\;\;\;}
  sSet_{Qu}
$$

between the model structure on $sGrpd_{DK}$ from Prop. \ref{TheModelStructure} and the [[classical model structure on simplicial sets]].

In addition both $\mathcal{G}$ and $\overline W$ preserve all [[weak equivalences]].

\end{proposition}

This is due to [Dwyer & Kan (1984), Thm. 3.3](#DwyerKan84), reviewed in [Goerss & Jardine (2009), Thm. 7.8](#GoerssJardine).

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

When restricted to simplicial groupoids of the form $(\mathbf{B} G)_\bullet$ for $G_\bullet$ a [[simplicial group]] and $\mathbf{B} G_n$ its [[delooping groupoid]] this produces a standard presentation of [[looping and delooping]] for [[infinity-groups]]. See there and at *[[model structure on simplicial groups]]* for more.

=--

\begin{lemma}\label{AcyclicFibrationsSurjectiveOnObjects}
Any [[acyclic fibration]] $f \in Fib \cap \mathrm{W}$ of simplicial groupoids is [[surjective]] on objects.
\end{lemma}
One lazy way to see this:
\begin{proof}
For $f_\bullet \in Fib \cap \mathrm{W}$ an acyclic Kan fibration, notice that:

1. Since the [[simplicial classifying space]] functor on simplicial groupoids ([here](simplicial+classifying+space#BarWForSimplicialGroupoid)) is a [[right Quillen functor]] by Prop. \ref{QuillenEquivalenceWithSimplicialSets} it follows that $\overline{W}(f) \in Fib \cap \mathrm{W}$ is an [[acyclic Kan fibration]]

1. All acylic fibrations in the [[classical model structure on simplicial sets]] are degreewise surjective (see [this Prop.](acyclic+Kan+fibration#AcyclicKanFibrationsAreSurjective)).

But since $\overline{W}$ is the identity on the sets of objects/vertices (by its definition [here](simplicial+classifying+space#BarWForSimplicialGroupoid)), the claim follows.
\end{proof}

Also useful to notice is:

\begin{proposition}
\label{sGrpdCofibrationIsObjectwiseSSetCofibration}
  A (acyclic) cofibration of simplicial groupoids is [[hom-object]]-wise an (acyclic) injection, hence a [[classical model structure on simplicial sets|Kan-Quillen]] (acyclic) cofibration, of simplicial [[hom-sets]].
\end{proposition}
\begin{proof}
  By Def. \ref{FreeMorphismsOfSimplicialGroupoids} and Prop. \ref{TheModelStructure} these cofibrations of simplicial groupoids are, in particular, retracts of monomorphisms of simplicial sets. Since [[sSet]] is a [[topos]], its [[epi-mono factorization system]] implies that monomorphisms are preserved under retracts. And of course also the Kan-Quillen weak equivalences are preserved under retracts.
\end{proof}

### Relation to simplicial categories
 {#RelationToSimplicialCategories}

\begin{proposition}
The canonical inclusion functor
$$
  \iota
    \,\colon\,
  sSet\text{-}Grpd
   \xhookrightarrow{\phantom{--}}
  sSet\text{-}Cat
$$
of the category of [[sSet]]-[[enriched 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 model structures on simplicial groupoids and the Bergner-[[model structure on sSet-categories]] (see [there](model+structure+on+sSet-categories#ModelForInfinityCategories)),

hence we have a [[Quillen adjunction]]:
\[
  \label{SimplicialGroupoidificationAdjunction}
  sSet\text{-}Grpd_{DK}
  \underoverset
     {\underset{\iota}{\hookrightarrow}}
     {\overset{F}{\longleftarrow}}
     {\;\;\; \bot_{\mathrm{Qu}} \;\;\;}
  sSet\text{-}Cat_{Berg}
  \,.
\]
\end{proposition}
(see also [Minichiello, Rivera & Zeinalian (2023), Prop. 2.8](#MinichielloRiveraZeinalian23))

Consider the [[pair]] of [[adjoint functors]] given by [[homotopy coherent nerve]] $N$ and [[rigidification of quasi-categories]] $\mathfrak{C}$:
\[
  \label{HoCoherentNerveAndRigidificationAdjunction}
  sSet\text{-}Cat_{Berg}
  \underoverset
     {\underset{N}{\longrightarrow}}
     {\overset{\mathfrak{C}}{\longleftarrow}}
     {\;\;\; \bot \;\;\;}
  sSet
\]
This is a [[Quillen adjunction]] with respect to the [[Joyal model structure]] for [[quasi-categories]] on the right, but next we are after a slightly different role of this adjunction:
\begin{proposition}
 \label{ComparisonFromLocalizationOfRigidificationAndToDKGroupoid}
 For $S \in sSet$ there is a [[natural transformation]]
 $$
    F \circ \mathfrak{C}(S)
    \longrightarrow
    \mathcal{G}(S)
 $$
 which is a [[Dwyer-Kan equivalence]] (from the localization (eq:SimplicialGroupoidificationAdjunction) of the rigidification (eq:HoCoherentNerveAndRigidificationAdjunction) to the [[Dwyer-Kan fundamental simplicial groupoid]]).
\end{proposition}
[[Minichiello, Rivera & Zeinalian (2023), Thm. 1.1 (Cor. 4.2)](#MinichielloRiveraZeinalian23)]

As a corollary:
\begin{proposition}
\label{LocalizationOfRegidificationAsLeftQuillenFunctor}
The composite of the adjunctions (eq:SimplicialGroupoidificationAdjunction) and (eq:HoCoherentNerveAndRigidificationAdjunction)
$$
  sSet\text{-}Grpd_{DK}
  \underoverset
     {\underset{\iota}{\hookrightarrow}}
     {\overset{F}{\longleftarrow}}
     {\;\;\; \bot \;\;\;}
  sSet\text{-}Cat
  \underoverset
     {\underset{N}{\longrightarrow}}
     {\overset{\mathfrak{C}}{\longleftarrow}}
     {\;\;\; \bot \;\;\;}
  sSet_{KQ}
$$
is a [[Quillen adjunction]] between the Dwyer-Kan model structure from Prop. \ref{TheModelStructure} and the [[Kan-Quillen model structure]] on simplicial sets.
\end{proposition}
\begin{proof}
 It is sufficient to see that $F \circ \mathfrak{C}$ is a [[left Quillen functor]]. This is immediate for the [[Joyal model structure]] on $sSet$, where $\mathfrak{C} \dashv N$ is a [[Quillen adjunction]] ([this Prop.](relation+between+quasi-categories+and+simplicial+categories#TheQuillenEquivalence)) so that $F \circ \mathfrak{C}$, being the composite of the left Quillen functor $\mathfrak{C}$ with the left Quillen functor $F$ (eq:SimplicialGroupoidificationAdjunction) is itself left Quillen. But the [[cofibrations]] of the [[Joyal model structure]] coincide with those of the [[Kan-Quillen model structure]] on [[sSet]] (both being  [[monomorphisms]]), so that $F \circ \mathfrak{C}$ also preserves the Kan-Quillen cofibrations.
To conclude it is therefore sufficient to see that $F \circ \mathfrak{C}$ sends all [[simplicial weak homotopy equivalences]] to [[Dwyer-Kan equivalences]]. But by Prop. \ref{ComparisonFromLocalizationOfRigidificationAndToDKGroupoid} every map $f \colon S \longrightarrow S'$ of [[simplicial sets]] gives rise to a [[commuting diagram]] in [[sSet Cat]] of the form
$$
  \array{
    F \circ \mathfrak{C}(S)
    &\overset{F \circ \mathfrak{C}(f)}{\longrightarrow}&
    F \circ \mathfrak{C}(S')
    \\
    \Big\downarrow
    &&
    \Big\downarrow
    \\
    \mathcal{G}(S)
    &\underset{\;\; \mathcal{G}(f) \;\;}{\longrightarrow}&
    \mathcal{G}(S')
  }
$$
where the vertical morphisms are [[Dwyer-Kan equivalences]]. Now observe that the [[Dwyer-Kan fundamental simplicial groupoid]]-functor $\mathcal{G}$ preserves all weak equivalences by [[Ken Brown's lemma]]: because it is a [[left Quillen functor]]  on a model category all whose objects are cofibrant. Therefore also $F \circ \mathfrak{C}$ preserves all weak equivalences, by [[2-out-of-3]].
\end{proof}

## Examples
 {#Examples}

### Fiber products with action 1-groupoids
 {#FiberProductsWithActionOneGroupoids}

While the underlying category of [[sSet-enriched groupoids]] is [[cartesian closed monoidal category|cartesian closed]] (see [there](simplicial+groupoid#CartesianClosedStructure)), the model category is not [[cartesian monoidal model category|cartesian monoidal as a model structure]].

But here is a (very) simple special case at least of an [[internal hom]] Quillen adjunction for [[action groupoid|action]] [[1-groupoids]].

{#AssumptionForFiberProductWithActionOneGroupoid} Consider:

* a [[group]] $G \,\in\, Grp(Set)$,

* with [[delooping groupoid]] denoted $\mathbf{B}G \,\in\, Grpd(Set) \subset sSet\text{-}Grpd$,

* an [[inhabited set]] $W \,\in\, Set$,

* equipped with a [[group action]] $G \curvearrowright W \,\in\, G Act(Set)$,

* with [[action groupoid]] denoted $W \sslash G \,\in\, Grpd \hookrightarrow sSet\text{-}Grpd$,

  canonically regarded in the [[slice category|slice]] $sSet\text{-}Grpd_{/\mathbf{B}G}$,

* [[sSet-enriched groupoids]]$\;\mathbf{X}, \mathbf{Y} \,\in\, sSet\text{-}Grpd$ lifted to the [[slice category|slice]] over $\mathbf{B}G$ via $p_\mathbf{X},\, p_\mathbf{Y} \,\in\, sSet\text{-}Grpd_{/\mathbf{B}G}$.

\begin{lemma}
\label{FiberProductWithActionOneGroupoidPreservesCofibrations}
With the [above assumptions](#AssumptionForFiberProductWithActionOneGroupoid),
for
$\mathbf{f} \colon \mathbf{X} \to \mathbf{Y}$ a morphism in the slice over $\mathbf{B}G$ which is a cofibration in the [[slice model category]] in that its [[underlying]] morphism in $sSet\text{-}Grpd$ is a Dwyer-Kan cofibration (Def. \ref{TheModelStructure}), then also the fiber product morphism
$$
  (W \sslash G) \times_{\mathbf{B}G} \mathbf{f}
  \;\colon\;
  (W \sslash G) \times_{\mathbf{B}G} \mathbf{X}
  \longrightarrow
  (W \sslash G) \times_{\mathbf{B}G} \mathbf{Y}
$$
is a DK-(slice-)cofibration.
\end{lemma}
\begin{proof}
By [definition](#CofIsRetractOfFree), the cofibration $\mathbf{f}$ is a retraction (in the [[arrow category]] of $sSet\text{-}Grpd$) of a free map $\widehat {f}$ (Def. \ref{FreeMorphismsOfSimplicialGroupoids}). But the right part of such a retraction diagram exhibits also $\widehat{f}$ as morphism in the slice over $\mathbf{B}G$ such that also the left part lifts to the slice:
\begin{tikzcd}[sep=15pt]
  \mathbf{X}
  \ar[rr]
  \ar[
    dd,
    "{
      \mathbf{f}
    }"{description}
  ]
  \ar[
    rrrr,
    equals,
    rounded corners,
    to path={
         ([yshift=+0pt]\tikztostart.north)
      -- ([yshift=+8pt]\tikztostart.north)
      -- ([yshift=+8pt]\tikztotarget.north)
      -- ([yshift=+0pt]\tikztotarget.north)
    }
  ]
  &&
  \widehat{\mathbf{X}}
  \ar[rr]
  \ar[
    dd,
    "{
      \widehat{\mathbf{f}}
    }"{description}
  ]
  \ar[drrr]
  &&
  \mathbf{X}
  \ar[
    dr,
    "{ p_{\mathbf{X}} }"
  ]
  \\
  && && & \mathbf{B}G
  \\
  \mathbf{Y}
  \ar[rr]
  \ar[
    rrrr,
    equals,
    rounded corners,
    to path={
         ([yshift=+0pt]\tikztostart.south)
      -- ([yshift=-8pt]\tikztostart.south)
      -- ([yshift=-8pt]\tikztotarget.south)
      -- ([yshift=+0pt]\tikztotarget.south)
    }
  ]
  &&
  \widehat{\mathbf{Y}}
  \ar[rr]
  \ar[urrr]
  &&
  \mathbf{Y}
  \ar[
    from=uu,
    crossing over,
    "{ \mathbf{f} }"{description}
  ]
  \ar[
    ur,
    "{ p_{\mathbf{Y}} }"{swap}
  ]
\end{tikzcd}
Regarded this way as a diagram in the slice, the functor $(W \sslash G) \times_{\mathbf{B}G} (-)$ preserves its retraction property.

Therefore it is sufficient to prove the claim under the assumption that $\mathbf{f}$ is actually a free map.

So let $\Gamma \subset Mor(\mathbf{Y})$ denote a [set of generators](#SetOfFreeGenerators) which exhibits $\mathbf{f}$ as a free map. We claim that then the set
$$
  (W \sslash G)
    \times_{\mathbf{B}G}
  \Gamma
  \;\;
  \subset
  \;\;
  Mor\big(
    (W \sslash G)
      \times_{\mathbf{B}G}
    \mathbf{Y}
  \big)
$$
serves as a set of generators exhibiting $(W \sslash G) \times_{\mathbf{B}G} f$ as a free map, and hence as a cofibration.

Namely, by the nature of the [[fiber product]], every morphism in $(W \sslash G) \times_{\mathbf{B}G} \mathbf{Y}$ is of the form
$$
  \big( p_{\mathbf{Y}}(\phi), \phi \big)
  \;\colon\;
  (w, y) \longrightarrow (w', y')
  \,,
$$
where $w' \,=\, p_{\mathbf{Y}}(\phi) \cdot w$ is determined by the pair consisting of $\phi \colon y \to y'$ and of $w \in Obj(\mathbf{Y})$. By assumption on $\Gamma$, $\phi$ has a unique reduced decomposition into $\Gamma$-components and this uniquely lifts to a reduced decomposition in $(W \sslash G)\times_{\mathbf{B}G} \Gamma$, determined by $w$.
\end{proof}

\begin{proposition}
  With the [above assumptions](#AssumptionForFiberProductWithActionOneGroupoid), we have a [[Quillen adjunction]] on the [[slice model category]] of the DK-model structure:
$$
  sSet\text{-}Grpd_{/\mathbf{B}G}
  \underoverset
    {
      \underset{
         Map\big( W \sslash G, - \big)_{\mathbf{B}G}
      }{\longleftarrow}
    }
    {
      \overset{
        (W \sslash G)
          \times_{\mathbf{B}G}
        (-)
      }{\longrightarrow}
    }
    { \bot_{\mathrlap{Qu}} }
  sSet\text{-}Grpd_{/\mathbf{B}G}
  \,.
$$
\end{proposition}
\begin{proof}
  With Lem. \ref{FiberProductWithActionOneGroupoidPreservesCofibrations}
 it is now sufficient to show that $(W \sslash G) \times_{\mathbf{B}G}(-)$ preserves all weak equivalences.

So assume that
$$
  \mathbf{f}_{x,x'}
    \,\colon\,
  \mathbf{X}(x,x')
  \longrightarrow
  \mathbf{Y}\big(\mathbf{f}(x),\mathbf{f}(x')\big)
$$
is a weak equivalence for all $w,w' \,\in\, Obj(\mathbf{X})$.

By the fact that $\mathbf{f}$ is a map in the slice over $\mathbf{B}G$, these component maps decompose into [[disjoint unions]] indexed by $g \in G$:
$$
  \mathbf{f}_{x,x'}
  \;=\;
  \underset{g \in G}{\coprod}
  \Big(
    \mathbf{f}^g_{x,x'}
      \,\colon\,
    \big((p_{\mathbf{X}})_{x,x'}\big)^{-1}
    \big(\{g\}\big)
    \longrightarrow
    \big(
      (p_{\mathbf{Y}})_{
        \mathbf{f}(x),
        \mathbf{f}(x')
      }\big)^{-1}
      \big(\{g\}\big)
  \Big)
  \,.
$$
Accordingly, all the components $\mathbf{f}^g_{x,x'}$ must be weak equivalences.
But then
$$
  \big(
    (W \sslash G)
      \times_{\mathbf{B}G}
    \mathbf{f}
  \big)_{(w,x), (w',x') }
  \;=\;
    \underset{
      {g \in G}
      \atop
      g \cdot w = w'
    }{\coprod}
      \mathbf{f}^g_{x,x'}
$$
is a coproduct of weak equivalences and hence a weak equivalence.
\end{proof}

## Related concepts

* [[Dwyer-Kan loop groupoid]]

* [[model structure on simplicial groups]]

* [[model structure on simplicial categories]]

* [[model structure on simplicial sets]]

* [[model structure on presheaves of simplicial groupoids]]

## References

The original article:

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]],  *Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [<a href="https://doi.org/10.1016/1385-7258(84)90038-6">doi:10.1016/1385-7258(84)90038-6</a>]

Textbook accounts:

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], after corollary 7.3 in chapter V of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/)]

* {#Jardine15} [[John F. Jardine]], §9.3 in: *[[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)]

A proof of the model structure closer to that establishing the [[model structure on simplicial categories]] and making explicit the [[cofibrantly generated model category|cofibrant generation]]:

* {#Bergner08} [[Julia E. Bergner]], Section 2 of: *Adding inverses to diagrams II: Invertible homotopy theories are spaces*, Homology, Homotopy Appl. **10** 2 (2008) 175-193 [[doi:10.4310/HHA.2008.v10.n2.a9](https://dx.doi.org/10.4310/HHA.2008.v10.n2.a9), [doi:0710.2254](https://arxiv.org/abs/0710.2254), erratum:[doi:10.4310/HHA.2012.v14.n1.a15](https://dx.doi.org/10.4310/HHA.2012.v14.n1.a15)]

See also:

* A. R. Garzon, J. G. Miranda and R. Osorio, *A simplicial description of the homotopy category of simplicial groupoids*, Theory and Applications of Categories **7**  14  (2000)263-283 [[tac:7-14](http://www.tac.mta.ca/tac/volumes/7/n14/7-14abs.html)]

* {#MinichielloRiveraZeinalian23} [[Emilio Minichiello]], [[Manuel Rivera]], [[Mahmoud Zeinalian]], *Categorical models for path spaces*, Advances in Mathematics **415** (2023) 108898 [[arXiv:2201.03046](https://arxiv.org/abs/2201.03046), [doi:10.1016/j.aim.2023.108898](https://doi.org/10.1016/j.aim.2023.108898)]

[[!redirects model category of simplicial groupoids]]


model structure on simplicial groups

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _model structure on simplicial groups_ is a presentation of the
[[(∞,1)-category]] of [[∞-groups]] in [[∞Grpd]] $\simeq$ [[Top]]. See [[group object in an (∞,1)-category]].

## Definition

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

There is a "projective" [[model category]] structure on the [[category]] $sGrp$ of [[simplicial groups]] where a morphism is

* a [[weak equivalence]] if the [[underlying]] morphism is a weak equivalence in the [[classical model structure on simplicial sets]], hence a [[simplicial weak homotopy equivalence]];

* a [[fibration]] if the underlying morphism is a fibration in the [[classical model structure on simplicial sets]], hence a  [[Kan fibration]];

* a [[cofibration]] if it has the [[left lifting property]] with respect to all [[acyclic fibrations]].

=--

([Quillen 67, II 3.7](#Quillen67), see also [Goerss-Jardine 99, V](#GoerssJardine99))

## Properties

### Further model category properties
 {#FurtherModelCategoryProperties}

#### Fibrations

\begin{proposition}\label{EveryObjectIsFibrant}
  Every object in the projective model structure (Prop. \ref{ModelStructureOnSimplicialGroups}) is [[fibrant object|fibrant]].
\end{proposition}
\begin{proof}
  This statement amounts to saying that the underlying simplicial set of any simplicial group is a [[Kan complex]], which is the case by Moore's theorem ([here](simplicial+group#EverySimplicialGroupIsAKanComplex)).
\end{proof}

#### Cofibrations
 {#Cofibrations}

\begin{proposition}\label{sGrpIsCofibrantlyGenerated}
  The model structure on [[sGrp]] from Prop. \ref{ModelStructureOnSimplicialGroups} is [[cofibrantly generated model category|cofibrantly generated]], with generating (acyclic) cofibrations the images under the degreewise [[free group]]-functor $F$ of the [[boundary of a simplex|boundary]] and [[horn]]-inclusions into simplicial [[simplices]]:
$$
  I \;=\;
  \big\{
    F(i_n) \,\colon\, F(\partial \Delta[n]) \hookrightarrow F(\Delta[n])
  \big\}_{n \in \mathbb{N}}
$$
$$
  J
    \;=\;
  \big\{
    F(j_n^k) \,\colon\, F(\Lambda_k[n]) \hookrightarrow F(\Delta[n])
  \big\}_{
    {n \in \mathbb{N}_+,}
    \atop
    {0 \leq k \leq n}
  }
$$
\end{proposition}
\begin{proof}
  By definition, the model structure on simplicial groups is the right [[transferred model structure]] from the [[classical model structure on simplicial sets]], along the [[forgetful functor]] $sGrp \to sSet_{Qu}$ with [[left adjoint]] the degreewise [[free group]]-functor $F \,\colon\,sSet \to sGrp$. Now since $sSet_{Qu}$ is [[cofibrantly generated model category|cofibrantly generated]] with generating (acyclic) cofibrations $i_n$ ($j_n^k$)
([this Prop.](classical+model+structure+on+simplicial+sets#AsACofibrantlyGeneratedModelCategory)) it follows (by [this Prop.](transferred+model+structure#SufficientConditions)) that so is [[sGrp]], with generating (acyclic) cofibrations their images under $F$, as claimed. \end{proof}

\begin{definition}\label{AlmostFreeMaps}
**(almost free maps)** \linebreak
A homomorphims of simplicial groups is called *almost free* if it is degreewise the inclusion into the [[amalgamation of groups|amalgamation]] with a [[free group]]

$$
  f_n
    \,\colon\,
  \mathcal{G}_n
    \xhookrightarrow{\phantom{---}}
  \mathcal{G}_n \star F(S_n)
$$
in a compatible way.
\end{definition}
([Goerss & Jardine (1999), p. 270](#GoerssJardine99); [Speirs (2015), §A.4](#Speirs15))

\begin{proposition}
\label{ClosurePropertiesOfAlmostFreeMaps}
 The [[class]] of almost free maps (Prop. \ref{AlmostFreeMaps}) is closed under

1. [[isomorphism]]

1. [[pushout]]

1. [[transfinite composition]]

\end{proposition}
[Speirs (2015), Prop. A.4.1](#Speirs15)

\begin{proposition}\label{CharacterizationOfCofibrations}
In the model structure on simplicial groups (Prop. \ref{ModelStructureOnSimplicialGroups}):

1. Every *almost free* homomorphism (Def. \ref{AlmostFreeMaps}) is a cofibration.

1. Every cofibration is a [[retract]] of an almost free map

\end{proposition}
\begin{proof}
The first statement is proven in [Goerss & Jardine (1999) Cor. 1.10](#GoerssJardine99).
For the second statement, use with Prop. \ref{sGrpIsCofibrantlyGenerated}, that the model category is cofibrantly generated by $I = \{F(i_n)\}_{n \in \mathbb{N}}$, which [means](cofibrantly+generated+model+category#CellsAndInjectives) that the cofibrations of $sGrp$ are the [[retracts]] of the [[relative cell complexes]] (i.e. of the [[transfinite compositions]] of [[pushouts]]) of the $F(i_n)$. Therefore it is now sufficient to see that
$$
  Cell\big(
    \{ F(i_n) \}
  \big)
  \;\;
  \supset
  \;\;
  AlmostFreeMaps
  \,
  \;\;\;\;\;\;\;\;
  \text{and}
  \;\;\;\;\;\;\;\;
  Cell\big(
    \{ F(i_n) \}
  \big)
  \;\;
  \subset
  \;\;
  AlmostFreeMaps
  \,.
$$
The first statement is [Goerss & Jardine (1999), V Prop. 1.9](#GoerssJardine99). The second statement follows by Prop. \ref{ClosurePropertiesOfAlmostFreeMaps} after observing that $F(i_n)$ itself is almost free.
\end{proof}

\begin{corollary}\label{AllCofibrationsOfSimplicialGroupsAreMonomorphisms}
  All cofibrations in [[sGrp]] are [[monomorphisms]], hence the [[underlying]] maps are [[cofibrations]] in [[classical model structure on simplicial sets|$sSet_{Qu}$]].
\end{corollary}
\begin{proof}
  Since the almost free maps (Def. \ref{AlmostFreeMaps}) are manifestly monomorphisms and since all cofibrations are retracts of almost free maps (Prop. \ref{CharacterizationOfCofibrations}) this follows form the fact that injections (cofibrations in $sSet_{Qu}$) are closed under retracts.
\end{proof}

### Relation to reduced simplicial sets

+-- {: .num_prop #QuillenEquivalenceWithReducedSimplicialSets}
###### Proposition
**([[Quillen equivalence between simplicial groups and reduced simplicial sets]])**

Forming [[simplicial loop space]] objects and [[simplicial classifying spaces]] gives a [[Quillen equivalence]]

$$
  \big(
    \Omega \dashv \overline{W}
  \big)
  \;\colon\;
  sGrp
   \stackrel{\overset{}{\longleftarrow}}{\longrightarrow}
  sSet_0
$$

with the [[model structure on reduced simplicial sets]].

=--

([Goerss-Jardine 99, Chapter V, Prop. 6.3](#GoerssJardine99))

### Relation to simplicial groupoids

\begin{remark}
Forming simplicial [[delooping groupoids]] constitutes a [[fully faithful functor]] of [[1-categories]] from [[simplicial groups]] to [[sSet]]-[[enriched groupoids]] (DK-[[simplicial groupoids]]):

$$
  \array{
    sGrp &\hookrightarrow& sSet\text{-}Grpd
    \\
    \mathcal{G} &\mapsto& \mathbf{B}\mathcal{G}
    \mathrlap{\,.}
  }
$$

With respect to the above model structure (Prop. \ref{ModelStructureOnSimplicialGroups}) and the [[model structure on simplicial groupoids]], this functor clearly preserves the [[weak equivalences]] and [[fibrations]]. However, it does not have a [[left adjoint]] and thus fails to be a [[right Quillen functor]].
\end{remark}

## Related concepts

* [[model structure on simplicial groupoids]]

* [[Borel model structure]] (presenting [[infinity-actions]] of simplicial groups)

## References

The model structure on simplicial groups is due to:

* {#Quillen67} [[Daniel Quillen]], II §3.7 of: *[[Homotopical Algebra]]*, Lecture Notes in Mathematics **43**, Springer (1967) [[doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)]

Further detailed discussion:

* {#GoerssJardine99} [[Paul Goerss]], [[J. F. Jardine]], chapter V of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/))

Review:

* {#Speirs15} [[Martin Speirs]], *Model Categories with a view towards rational homotopy theory*, MSc thesis, Copenhagen (2015) [[pdf](https://speirs.sites.ku.dk/files/2017/10/Master.pdf), [[Speirs-ModelCategories.pdf:file]]]

See also:

* {#Baues99} [[Hans-Joachim Baues]], *Quadratic Homology*, Transactions of the American Mathematical Society **351** 2 (1999) [[jstor:117809](https://www.jstor.org/stable/117809), [pdf](https://www.ams.org/journals/tran/1999-351-02/S0002-9947-99-02335-1/S0002-9947-99-02335-1.pdf)]