model structure on dg-coalgebras

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#### Rational homotopy theory
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[[!include differential graded objects - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] structure on the [[category]] of  [[dg-coalgebras]].

## Definition

Let $k$ be a [[field]] of [[characteristic]] 0.

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

There is a pair of [[adjoint functors]]

$$
  (\mathcal{L} \dashv \mathcal{C})
  \;\colon\;
  dgLieAlg_k
   \underoverset
      {\underset{\mathcal{C}}{\longrightarrow}}
      {\overset{\mathcal{L}}{\longleftarrow}}
      {\bot}
  dgCoCAlg_k
$$

between the category of [[dg-Lie algebras]]and that of dg [[cocommutative coalgebras]], where the [[right adjoint]] sends a dg-Lie algebra $(\mathfrak{g}_\bullet, [-,-])$ to its [[Chevalley-Eilenberg algebras|Chevalley-Eilenberg coalgebra]], whose underlying coalgebra is the [[free graded co-commutative coalgebra]] on $\mathfrak{g}[1]$ and whose differential is given on the tensor product of two generators by the Lie bracket $[-,-]$.

=--

For (pointers to) the details, see at _[model structure on dg-Lie algebras -- Relation to dg-coalgebras](model%20structure%20on%20dg-Lie%20algebras#RelationToDgCoalgebras)_.

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

There exists a [[model category]] structure on $dgCoCAlg_k$ for which

* the cofibrations are the (degreewise) [[injections]];

* the weak equivalences are those morphisms that become [[quasi-isomorphisms]] under the functor $\mathcal{L}$ from prop. \ref{TheCETangentAdjunction}.

Moreover, this is naturally a [[simplicial model category]] structure.

=--

This is ([Hinich98, theorem, 3.1](#Hinich98)). More details on this are in the relevant sections at _[[model structure for L-infinity algebras]]_.

## Properties

### Relation to dg-Lie algebras
  {#RelationTodgLieAlgbras}

Throughout, let $k$ be of [[characteristic zero]].

+-- {: .num_defn #CEfunctor}
###### Definition
**(Chevalley-Eilenberg dg-coalgebra)**

Write

$$
  CE
    \;\colon\;
  dgLieAlg_{k}
    \longrightarrow
  dgCoCAlg_k
$$

for the [[Chevalley-Eilenberg algebra]] functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to the [[dg-coalgebra]]

$$
  CE(\mathfrak{g},\partial,[-,-])
   \;\coloneqq\;
  \left(
    \vee^\bullet \mathfrak{g}[1]
    ,\;
    D = \partial + [-,-]
  \right)
  \,,
$$

where on the right the extension of $\partial$ and $[-,-]$ to graded [[derivations]] is understood.

=--

For dg-Lie algebras concentrated in degrees $ \geq n \geq 1$ this is due to ([Quillen 69, appendix B, prop 6.2](#Quillen)). For unbounded dg-algebras, this is due to ([Hinich 98, 2.2.2](#Hinich98)).

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

For $(X,D) \in dgCoCAlg_k$  write

  $$
    \mathcal{L}(X,D)
     \coloneqq
    \left(
       F(\overline{X}[-1]),\;
       \partial \coloneqq D + (\Delta - 1 \otimes id - id \otimes 1)
    \right)
    \;\in dgLieAlg_k\;
  $$

  where

  1. $\overline{X} \coloneqq ker(\epsilon)$ is the [[kernel]] of the [[counit]], regarded as a [[chain complex]];

  1. $F$ is the [[free Lie algebra]] functor (as graded Lie algebras);

  1. on the right we are extending $(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X}$ as a Lie algebra [[derivation]]

=--

For dg-Lie algebras concentrated in degrees $ \geq n \geq 1$ this is due to ([Quillen 69, appendix B, prop 6.1](#Quillen)). For unbounded dg-algebras, this is due to ([Hinich 98, 2.2.1](#Hinich98)).

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

The [[functors]] from def. \ref{CEfunctor} and def. \ref{LeftAdjointToCEfunctor} are [[adjoint functor|adjoint]] to each other:

$$
  dgLieAlg_k
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {\bot}
  dgCoCAlg_k
  \,.
$$

Moreover, for $X \in dgCoCAlg_k$ and $\mathfrak{g} \in dgLieAlg_k$ then the adjoint [[hom sets]] are [[natural isomorphism|naturally isomorphic]]

$$
  Hom(\mathcal{L}(X), \mathfrak{g})
   \simeq
  Hom(X, CE(\mathfrak{g}))
    \simeq
  MC(Hom(\overline{X},\mathfrak{g}))
$$

to the [[Maurer-Cartan elements]] in the Hom-dgLie algebra from $\overline{X}$ to $\mathfrak{g}$.

=--

For dg-Lie algebras concentrated in degrees $ \geq n \geq 1$ this is due to ([Quillen 69, appendix B, somewhere](#Quillen)). For unbounded dg-algebras, this is due to ([Hinich 98, 2.2.5](#Hinich98)).

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

The adjunction $(\mathcal{L} \dashv CE)$ from prop. \ref{CEAdjunction} is a [[Quillen adjunction]] between then projective [[model structure on dg-Lie algebras]] as the [[model structure on dg-coalgebras]]

$$
  (dgLieAlg_k)_{proj}
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {\bot}
  (dgCoCAlg_k)_{Quillen}
  \,.
$$

=--

([Hinich 98, lemma 5.2.2, lemma 5.2.3](#Hinich98))

Moreover:

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

In non-negatively graded dg-coalgebras, both [[Quillen functors]] $(\mathcal{L} \dashv CE)$ from prop. \ref{QuillenAdjunctionCE} preserve all [[quasi-isomorphisms]], and both the [[adjunction unit]] and the [[adjunction counit]] are quasi-isomorphisms.

=--

For dg-algebras in degrees $\geq n \geq 1$ this is ([Quillen 76, theorem 7.5](#Quillen76)). In unbounded degrees this is ([Hinich 98, prop. 3.3.2](#Hinich98))

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

The Quillen adjunctin from prop. \ref{QuillenAdjunctionCE} is a [[Quillen equivalence]]:

$$
  (dgLieAlg_k)_{proj}
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {{}_{\phantom{qu}}\simeq_{Qu}}
  (dgCoCAlg_k)_{Quillen}
  \,.
$$

=--

([Hinich 98, theorem 3.2](#Hinich98)) using ([Quillen 76 II 1.4](#Quillen76))

## Related concepts

* [[model structure on dg-comodules]]

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

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

* [[model structure on coalgebras over a comonad]]

* [[rational homotopy theory]]

* [[relation between L-∞ algebras and dg-Lie algebras]]

## References

In [[characteristic zero]] and in positive degrees the model structure is due to

* {#Quillen69} [[Dan Quillen]], section II.5 and appendix B of _Rational homotopy theory_, Annals of Math., 90(1969), 205–295 ([JSTOR](http://www.jstor.org/stable/1970725), [pdf](http://www.math.northwestern.edu/~konter/gtrs/rational.pdf))

in non-negative degrees in

* [[Ezra Getzler]], [[Paul Goerss]], _A model category structure for differential graded coalgebras_ ([ps](http://www.math.northwestern.edu/~pgoerss/papers/model.ps), [[GetzlerGoerss99.pdf:file]])

and in unbounded degrees in

* {#Hinich98} [[Vladimir Hinich]], _DG coalgebras as formal stacks_, Journal of Pure and Applied Algebra Volume 162, Issues 2–3, 24 August 2001, Pages 209–250 [[arXiv:math/9812034](http://arxiv.org/abs/math/9812034), <a href="https://doi.org/10.1016/S0022-4049(00)00121-3">doi:10.1016/S0022-4049(00)00121-3</a]

* {#Pridham} [[Jonathan Pridham]], _Unifying derived deformation theories_, Adv. Math. 224 (2010), no.3, 772-826 ([arXiv:0705.0344](http://arxiv.org/abs/0705.0344))

See also

* {#Lurie} [[Jacob Lurie]], _[[Formal Moduli Problems]]_

Review with discussion of [[homotopy limits]] and [[homotopy colimits]] is in

* {#Walter06} [[Ben Walter]], section 5 of _Rational Homotopy Calculus of Functors_ ([arXiv:math/0603336](http://arxiv.org/abs/math/0603336))

[[!redirects model stucture on dg-coalgebras]]

[[!redirects model structure on dg cocommutative coalgebras]]


model structure on dg-comodules

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#### Model category theory
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#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] structure on the category of [[comodules]] in [[chain complexes]] over a [[dg-coalgebra]].

## Details

Let $C$ be a [[differential graded-cocommutative coalgebra]] over a [[field]].

**Model structure of the second kind**

There exists a [[model category]] structure on the category $C dgCoMod$ of [[dg-comodules]] over $C$ whose

* weak equivalences are the morphisms whose [[mapping cone]] is a co-acyclic comodule,

* fibrations are the [[surjections]] whose [[kernel]] is such that its underlying [[comodule]] over the underlying [[coalgebra]] of $C$ is an [[injective object]].

This is due to ([Positelski 11, 8.2 Theorem (a)](#Positelski11)).

**Model structure of the first kind**

There is another model structure where the fibrations in addition satisfy the condition that their kernel $K$ satisfies that for all acyclic $N$, then $\underline{Hom}_C(N,K)$ is acyclic.

This is due to ([Positelski 11, 8.2 Remark](#Positelski11)), there called the "model category structure of the first kind".  This is also reviewed as ([Pridham 13, prop. 2.2](#Pridham13)).

## Related concepts

* [[model structure on dg-coalgebras]]

* [[model structure on coalgebras over a comonad]]

## References

* {#Positelski11} [[Leonid Positselski]], _Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence_, Memoirs of the Amer. Math. Soc. 212 (2011), no.996, ([arXiv:0905.2621](https://arxiv.org/abs/0905.2621))

* {#Pridham13} [[Jonathan Pridham]], _Tannaka duality for enhanced triangulated categories_ ([arXiv:1309.0637](https://arxiv.org/abs/1309.0637))


model structure on dg-Lie algebras

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#### Model category theory
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[[!include model category theory - contents]]
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#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

[[dg-Lie algebras]] may be thought of (see [[relation between L-∞ algebras and dg-Lie algebras|here]]) as the "strict" [[strong homotopy Lie algebras]]. As such they support a [[homotopy theory]]. The _[[model category]]_ structure on dg-Lie algebras is one way to present this homotopy theory. This is used for instance in [[deformation theory]], see at _[[formal moduli problems]]_.

For dg-Lie algebras in positive degree and over the rational numbers this model structure, due to ([Quillen 69, theorem II](#Quillen69)) is one of the algebraic models for presenting [[rational homotopy theory]] (see there) of [[simply connected topological spaces]].

## Definition

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

There exists a [[model category]] structure $(dgLie_k)_{proj}$ on the category $dgLie_k$ of [[dg-Lie algebras]] over a commutative [[ring]] $k \supset \mathbb{Q}$ such that

* [[fibrations]] the [[surjective maps]]

* [[weak equivalences]] the [[quasi-isomorphisms]] on the underlying [[chain complexes]].

=--

For dg-Lie algebras in degrees $ \geq n \geq 1$, this is due to [Quillen 69](#Quillen69). For unbounded dg-Lie algebras this is due to ([Hinich 97](#Hinich97)).

This becomes a [[simplicial category]] with simplicial [[mapping spaces]] given by

$$
  dgLie(\mathfrak{g}, \mathfrak{h})
   \coloneqq
  ([k] \mapsto Hom_{dgLie}(\mathfrak{g} , \Omega^\bullet(\Delta^k) \otimes\mathfrak{h}))
  \,,
$$

where

* $\Omega^\bullet(\Delta^k)$ is the [[dg-algebra]] of [[polynomial differential forms]] on the $k$-[[simplex]];

* $\Omega^\bullet(\Delta^k)\otimes \mathfrak{h}$ is the canonical dg-Lie algebra structure on the [[tensor product]].

([Hinich 97, 4.8.2](#Hinich97), following [Bousfield & Gugenheim 1976](model+structure+on+dg-algebras#HomComplexes))

This enrichment satisfies together with the model structure some of the properties of a [[simplicial model category]] ([Hinich 97, 4.8.3, 4.8.4](#Hinich97)), but not all of them.

## Properties
 {#Properties}

### Relation to $L_\infty$-algebras
 {#RectificationResolution}

dg-Lie algebras with this model structure are a _rectification_ of [[L-∞ algebras]]: for $Lie$ the [[Lie operad]] and $\widehat Lie$ its standard [[cofibrant resolution]], [[algebras over an operad]] over $Lie$ in chain complexes are [[dg-Lie algebras]] and algebras over $\widehat Lie$ are [[L-∞ algebras]] and by the rectification result discussed at _[[model structure on dg-algebras over an operad]]_ there is an induced [[Quillen equivalence]]

$$
  Alg(\widehat Lie) \stackrel{\simeq}{\to} Alg(Lie)
$$

between the [[model structure for L-∞ algebras]] which is [[transferred model structure|transferred]] from the  [[model structure on chain complexes]] (unbounded propjective) to the above model structure on chain complexes.

There is also a [[Quillen equivalence]] from the model structure on dg-Lie algebras to the [[model structure on dg-coalgebras]]. This is part of a web of Quillen equivalences that identifies dg-Lie algebra/$L_\infty$-algebras with [[infinitesimal object|infinitesimal]] [[derived stack|derived]] [[∞-stacks]] ("[[formal moduli problems]]"). More on this is at _[[model structure for L-∞ algebras]]_.

Specifically, there is ([Quillen 69](#Quillen)) an [[adjunction]]

$$
  (\mathcal{R} \dashv i) \;\colon\;
  dgLie \stackrel{\overset{\mathcal{R}}{\leftarrow}}{\underset{i}{\to}} dgCoCAlg
$$

between [[dg-coalgebras]] and dg-Lie algebras, where the [[right adjoint]] is the (non-full) inclusion that regards a dg-Lie algebra as a [[differential graded coalgebra]] with co-binary differential, and where the [[left adjoint]] $\mathcal{R}$ ("rectification") sends a dg-coalgebra to a dg-Lie algebra whose underlying graded [[Lie algebra]] is the [[free Lie algebra]] on the underlying [[chain complex]]. Over a [[field]] of [[characteristic]] 0, this adjunction is a [[Quillen equivalence]] between the [[model structure for L-∞ algebras]] on $dgCoCAlg$ and the model structure on $dgLie$ ([Hinich 98, theorem 3.2](#Hinich98)).

In particular, therefore the composite $i \circ \mathcal{R}$ is a [[resolution]] functor for $L_\infty$-algebras.

For more see at *[[relation between L-∞ algebras and dg-Lie algebras]]*.

### Relation to dg-coalgebras
 {#RelationToDgCoalgebras}

Via the above relation to $L_\infty$-algebras, dg-Lie algebras are also connected by a composite adjunction to [[dg-coalgebras]]. We dicuss the direct adjunction.

Throughout, let $k$ be of [[characteristic zero]].

+-- {: .num_defn #CEfunctor}
###### Definition
**(Chevalley-Eilenberg dg-coalgebra)**

Write

$$
  CE
    \;\colon\;
  dgLieAlg_{k}
    \longrightarrow
  dgCocAlg_k
$$

for the [[Chevalley-Eilenberg algebra]] functor. It sends a dg-Lie algebra $(\mathfrak{g}, \partial, [-,-])$ to the [[dg-coalgebra]]

$$
  CE(\mathfrak{g},\partial,[-,-])
   \;\coloneqq\;
  \left(
    \vee^\bullet \mathfrak{g}[1]
    ,\;
    D = \partial + [-,-]
  \right)
  \,,
$$

where on the right the extension of $\partial$ and $[-,-]$ to graded [[derivations]] is understood.

=--

For dg-Lie algebras concentrated in degrees $ \geq n \geq 1$ this is due to ([Quillen 69, appendix B, prop 6.2](#Quillen)). For unbounded dg-algebras, this is due to ([Hinich 98, 2.2.2](#Hinich98)).

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

For $(X,D) \in dgCocalg_k$  write

  $$
    \mathcal{L}(X,D)
     \coloneqq
    \left(
       F(\overline{X}[-1]),\;
       \partial \coloneqq D + (\Delta - 1 \otimes id - id \otimes 1)
    \right)
    \;\in dgLieAlg_k\;
  $$

  where

  1. $\overline{X} \coloneqq ker(\epsilon)$ is the [[kernel]] of the [[counit]], regarded as a [[chain complex]];

  1. $F$ is the [[free Lie algebra]] functor (as graded Lie algebras);

  1. on the right we are extending $(\Delta - 1 \otimes id - id \otimes 1) \colon \overline{X} \to \overline{X} \otimes \overline{X}$ as a Lie algebra [[derivation]]

=--

For dg-Lie algebras concentrated in degrees $ \geq n \geq 1$ this is due to ([Quillen 69, appendix B, prop 6.1](#Quillen)). For unbounded dg-algebras, this is due to ([Hinich 98, 2.2.1](#Hinich98)).

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

The [[functors]] from def. \ref{CEfunctor} and def. \ref{LeftAdjointToCEfunctor} are [[adjoint functor|adjoint]] to each other:

$$
  dgLieAlg_k
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {\bot}
  dgCocAlg_k
  \,.
$$

Moreover, for $X \in dgCocAlg_k$ and $\mathfrak{g} \in dgLieAlg_k$ then the adjoint [[hom sets]] are [[natural isomorphism|naturally isomorphic]]

$$
  Hom(\mathcal{L}(X), \mathfrak{g})
   \simeq
  Hom(X, CE(\mathfrak{g}))
    \simeq
  MC(Hom(\overline{X},\mathfrak{g}))
$$

to the [[Maurer-Cartan elements]] in the Hom-dgLie algebra from $\overline{X}$ to $\mathfrak{g}$.

=--

For dg-Lie algebras concentrated in degrees $ \geq n \geq 1$ this is due to ([Quillen 69, appendix B, somewhere](#Quillen)). For unbounded dg-algebras, this is due to ([Hinich 98, 2.2.5](#Hinich98)).

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

The adjunction $(\mathcal{L} \dashv CE)$ from prop. \ref{CEAdjunction} is a [[Quillen adjunction]] between then projective [[model structure on dg-Lie algebras]] as the [[model structure on dg-coalgebras]]

$$
  (dgLieAlg_k)_{proj}
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {\bot}
  (dgCocAlg_k)_{Quillen}
  \,.
$$

=--

([Hinich 98, lemma 5.2.2, lemma 5.2.3](#Hinich98))

Moreover:

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

In non-negatively graded dg-coalgebras, both [[Quillen functors]] $(\mathcal{L} \dashv CE)$ from prop. \ref{QuillenAdjunctionCE} preserve all [[quasi-isomorphisms]], and both the [[adjunction unit]] and the [[adjunction counit]] are quasi-isomorphisms.

=--

For dg-algebras in degrees $\geq n \geq 1$ this is ([Quillen 76, theorem 7.5](#Quillen76)). In unbounded degrees this is ([Hinich 98, prop. 3.3.2](#Hinich98))

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

The Quillen adjunctin from prop. \ref{QuillenAdjunctionCE} is a [[Quillen equivalence]]:

$$
  (dgLieAlg_k)_{proj}
   \underoverset
     {\underset{CE}{\longrightarrow}}
     {\overset{\mathcal{L}}{\longleftarrow}}
     {{}_{\phantom{qu}}\simeq_{Qu}}
  (dgCocAlg_k)_{Quillen}
  \,.
$$

=--

([Hinich 98, theorem 3.2](#Hinich98)) using ([Quillen 76 II 1.4](#Quillen76))

### Relation to simplicial Lie algebras

The [[normalized chains complex]] functor from [[simplicial Lie algebras]] constitutes a [[Quillen adjunction]] from the projective [[model structure on simplicial Lie algebras]], see [there](model+structure+on+simplicial+Lie+algebras#QuillenAdjunctionTodgLieAlgebras).

## Related concepts

* [[model structure on dg-coalgebras]]

* [[model structure on simplicial Lie algebras]]

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

* [[relation between L-∞ algebras and dg-Lie algebras]]

## References

The model structure on dg-Lie algebras in [[characteristic zero]] and in  degrees $\geq n \geq 1$ goes back to

* {#Quillen69} [[Dan Quillen]], section II.5 and appendix B of _Rational homotopy theory_, Annals of Math., 90(1969), 205–295 ([JSTOR](http://www.jstor.org/stable/1970725), [pdf](http://www.math.northwestern.edu/~konter/gtrs/rational.pdf))

This is extended to a model structure on dg-Lie algebras in unbounded degrees in

* {#Hinich97} [[Vladimir Hinich]], _Homological algebra of homotopy algebras_, Comm. in algebra, 25(10)(1997), 3291–3323 ([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), _Erratum_ ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453)))

and the corresponding [[Quillen adjunction]] to the [[model structure on dg-coalgebras]] in unbounded degrees is discussed in

* {#Hinich98} [[Vladimir Hinich]], _DG coalgebras as formal stacks_, Journal of Pure and Applied Algebra Volume 162, Issues 2–3, 24 August 2001, Pages 209–250 ([arXiv:math/9812034](http://arxiv.org/abs/math/9812034))

See also section 2.1 of

* [[Jacob Lurie]], _[[Formal moduli problems]]_

Review with discussion of [[homotopy limits]] and [[homotopy colimits]] is in

* {#Walter06} [[Ben Walter]], section 4 of _Rational Homotopy Calculus of Functors_ ([arXiv:math/0603336](http://arxiv.org/abs/math/0603336))

[[!redirects model structures on dg-Lie algebras]]


model structure on dg-modules

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[[!include model category theory - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The category of [[dg-modules]] over a [[dg-algebra]], or more generally a [[dg-category]], admits [[dg-model structures]] which present the [[derived dg-category]].

In the case of [[dg-algebras]] in an [[abelian category]] $\mathcal{A}$, [[dg-modules]] are the same as [[module over an algebra over an operad|modules]] over [[algebra over an operad|algebras]] over the [[associative operad]] in $Ch(\mathcal{A})$. These admit model structures as described in [[model structure on modules over an algebra over an operad]], [[transferred model structure|transferred]] from the [[model structures on chain complexes]] in $\mathcal{A}$.

More generally there are [[model structures]] on [[dg-modules]] over a [[dg-category]], analogous to [[model structures on simplicial presheaves]].

## Definitions
 {#Definitions}

+-- {: .num_theorem #ProjectiveModelStructure}
###### Theorem
**(projective model structure)**

There is a [[cofibrantly generated model structure]] on the category of [[dg-modules]], called the **projective model structure**, where

1. the [[weak equivalences]] are object-wise [[quasi-isomorphisms]] of [[chain complexes]],

1. the [[fibrations]] are object-wise [[epimorphisms]].

Moreover, this is a [[dg-model structure]].

=--

+-- {: .num_theorem #InjectveModelStructure}
###### Theorem
**(injective model structure)

There is a [[model structure]] on the category of [[dg-modules]], called the **injective model structure**, where

1. the [[weak equivalences]] are object-wise [[quasi-isomorphisms]] of [[chain complexes]],

1. and the [[cofibrations]] are object-wise [[monomorphisms]].

=--

See ([Toën 04](#Toen), section 3) and ([Keller 06](#Keller06), theorem 3.2).

These model structures present the [[derived dg-category]].

## Related concepts

* [[model structure on chain complexes]]

* [[module structure on modules in a monoidal model category]]

* [[model structure on dg-algebras]]

* [[model structure on dg-comodules]]

## References

Section 3 of

* {#Toen2004} [[Bertrand Toën]], _The homotopy theory of dg-categories and derived Morita theory_, [arXiv:math/0408337](http://arxiv.org/abs/math/0408337).

Paragraph 3.2 of

* {#Keller06} [[Bernhard Keller]], _On differential graded categories_, [arXiv:math/0601185](http://arxiv.org/abs/math/0601185).

### For dg-algebras

For the case of [[dg-algebras]], see the references below.

A general account is around section 11.2.5 of

* [[Benoit Fresse]], _Modules over operads and functors_ Springer Lecture Notes in Mathematics, (2009) ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.5833&rep=rep1&type=pdf))
{#Fresse}

and in section 3 of

* {#Hinich}[[Vladimir Hinich]],  _Homological algebra of homotopy algebras_ Communications in algebra, 25(10). 3291-3323 (1997)([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), _Erratum_ ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453)))

The [[homotopy category]] and [[triangulated category]] of dg-modules is discussed for instance also in

* {#Bernstein} Joseph Bernstein, _DG-modules and equivariant cohomology_ ([pdf](http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/luntz_ESF/4-dg-modules-and-equivariant-cohomology.pdf)).

See also

* {#BarthelMayRiehl14} [[Tobias Barthel]], [[Peter May]], [[Emily Riehl]], _Six model structures for DG-modules over DGAs: model category theory in homological action_, New York J. Math. 20 (2014) 1077–1159 ([pdf](http://www.math.harvard.edu/~eriehl/six.pdf))

Discussion as a model for [[rational parameterized stable homotopy theory]] is due to

* [[Vincent Braunack-Mayer]], _[[schreiber:thesis Braunack-Mayer|Rational parameterized stable homotopy theory]]_, Zurich 2018

[[!redirects model structures on dg-modules]]

[[!redirects dg-model structure on dg-modules]]
[[!redirects dg-model structures on dg-modules]]

[[!redirects category of dg-modules]]
[[!redirects categories of dg-modules]]


model structure on dg-operads

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[[!include model category theory - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Many general results discussed at [[model structure on operads]] are based on the existence of a _cocommutative Hopf interval object_ . In the [[category of chain complexes]] -- the case for [[dg-operad]]s -- the Hopf interval is not cocommutative, so this case requires special discussion.

## Related concepts

* **model structure on dg-operads**

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

  * [[model structure on dg-algebras]]

* [[model structure on dg-categories]]

## References

* [[Vladimir Hinich]],  _Homological algebra of homotopy algebras_ Communications in algebra, 25(10). 3291-3323 (1997)([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), _Erratum_ ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453)))
{#Hinich}


model structure on diffeological spaces

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###Context###
#### Model category theory
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#### Differential geometry
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#### Homotopy theory
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[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There may be different [[model category]]-[[mathematical structure|structures]] on the [[category]] of [[diffeological spaces]].

Of practical interest would be a model structure whose [[weak equivalences]] are the [[isomorphisms]] on standard _smooth_ [[homotopy groups]] of [[diffeological spaces]], i.e. the [[weak equivalences]] between the [[cohesive shapes]] of [[diffeological spaces]] regarded as the [[concrete objects]] in the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]]. By the discussion at _[[shape via cohesive path ∞-groupoid]]_ these are detected by smooth functions out of [[simplices]] or [[cubes]] with their canonical diffeological structure (as discussed in [Christensen-Wu 14](#ChristensenWu14)).

A model category with this property has been claimed in [Haraguchi-Shimakawa 13](#HaraguchiShimakawa13),, [Haraguchi-Shimakawa 20](#HaraguchiShimakawa20)

Another model category structure is discussed in [Kihara 16](#Kihara16), but this uses a non-standard diffeology on simplices (to enforce that all smooth [[singular simplicial complexes]] are [[fibrant objects]]).

## Details

The approach of [Haraguchi-Shimakawa 13](#HaraguchiShimakawa13), [Haraguchi-Shimakawa 20](#HaraguchiShimakawa20)
proceeds as follows:

[[!include adjunction between topological spaces and diffeological spaces]]

## References
 {#References}

A proof of a model structure on diffeological spaces, with weak equivalences detected on standard smooth homotopy groups, is claimed in

* {#HaraguchiShimakawa13} [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _A model structure on the category of diffeological spaces_ ([arXiv:1311.5668](https://arxiv.org/abs/1311.5668))

though [Kihara 16](#Kihara16) writes ([p. 2](https://arxiv.org/pdf/1605.06794.pdf#page=2)) that

> there exists a gap in the proof of [Haraguchi-Shimakawa 13](#HaraguchiShimakawa13), Theorem 5.6

This gap is addressed in

* {#Haraguchi18} [[Tadayuki Haraguchi]], _Homotopy structures of smooth CW complexes_ ([arXiv:1811.06175](https://arxiv.org/abs/1811.06175))

* {#HaraguchiShimakawa20} [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _A model structure on the category of diffeological spaces, I_ ([arXiv:2011.12842](https://arxiv.org/abs/2011.12842))

A different model structure, however not using the standard smooth homotopy groups(!), is claimed (and published) in

* {#Kihara16} [[Hiroshi Kihara]], _Model category of diffeological spaces_, Journal of Homotopy and Related Structures, (2018), 1-40 ([arXiv:1605.06794](https://arxiv.org/abs/1605.06794))

See also

* {#ChristensenWu14} [[J. Daniel Christensen]],  [[Enxin Wu]], _The homotopy theory of diffeological spaces, I. Fibrant and cofibrant objects_, New York J. Math. 20 (2014), 1269-1303 ([arXiv:1311.6394](http://arxiv.org/abs/1311.6394))

* [[Hiroshi Kihara]], *Quillen equivalences between the model categories of smooth spaces, simplicial sets, and arc-gengerated spaces* ([arXiv:1702.04070](https://arxiv.org/abs/1702.04070))

* [[Tadayuki Haraguchi]], [[Kazuhisa Shimakawa]], _On homotopy types of diffeological cell complexes_ ([arXiv:1912.05359](https://arxiv.org/abs/1912.05359))

[[!redirects model category of diffeological spaces]]


model structure on differential graded-commutative superalgebras

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#### Model category theory
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#Contents#
* table of contents
{:toc}

## Idea

The [[category]] of [[differential graded-commutative superalgebras]] over a [[field]] of [[characteristic zero]] carries a _projective_ [[model category]] [[structure]] whose [[weak equivalences]] are the underlying [[quasi-isomorphisms]] and whose [[fibrations]] are the degreewise [[surjections]] (all either in unbounded degree, in non-negative degree or in non-positive degree).

This is the [[transferred model structure]] of the projective [[model structure on chain complexes of super vector spaces]], transferred along the [[forgetful functor]] to underlying chain complexes.

The model structure is hence the direct generalization of the projective [[model structure on differential graded-commutative algebras]], to which it reduces on the objects concentrated in even super-degree.

## Related concepts

* [[super L-infinity algebra]]

## References

A unified treatmeant generalizing to arbitary super [[Fermat theories]] is in

* {#CarchediRoytenberg12} [[David Carchedi]], [[Dmitry Roytenberg]],  theorem 1, theorem 6.5 of _Homological Algebra for Superalgebras of Differentiable Functions_ ([arXiv:1212.3745](https://arxiv.org/abs/1212.3745))

[[!redirects model structures on differential graded-commutative superalgebras]]

[[!redirects model structure on dg-commutative superalgebras]]
[[!redirects model structures on dg-commutative superalgebras]]

[[!redirects model structure on dg-supercommutative superalgebras]]
[[!redirects model structures on dg-supercommutative superalgebras]]

[[!redirects model structure on dgc-superalgebras]]
[[!redirects model structures on dgc-superalgebras]]

[[!redirects model structure on dg-superalgebras]]
[[!redirects model structures on dg-superalgebras]]


model structure on enriched categories

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# Model structures on enriched categories
* automatic table of contents goes here
{:toc}

## Idea

If $V$ is a [[monoidal model category]], then in many cases there is a [[model category]] of $V$-[[enriched categories]].  This includes the [[model structure on simplicial categories]] and the [[model structure on dg-categories]], for instance.

## Definition

Let $V$ be a monoidal model category.  The localization functor $\gamma: V \to Ho(V)$ is then a [[lax monoidal functor]], and hence any $V$-category $C$ induces a $Ho(V)$-category $\gamma_\bullet C$.  The *homotopy category* of a $V$-category $C$ is the underlying ordinary category $(\gamma_\bullet C)_o$.  We say a $V$-functor $F:C\to D$ is *locally X* if each morphism $F:C(x,y) \to D(F x, F y)$ is X.

Define a $V$-functor $F:C\to D$ to be:

* A **weak equivalence** if $\gamma_\bullet F :\gamma_\bullet C \to \gamma_\bullet D$ is an equivalence of $Ho(V)$-categories (that is, an internal equivalence in the [[2-category]] of $Ho(V)$-categories).  This is equivalent to asking that (1) $F$ is locally a weak equivalence, and (2) the ordinary functor $(\gamma_\bullet F)_o$ is essentially surjective.

* A **naive fibration** if (1) $F$ is locally a fibration, and (2) $\gamma_\bullet F$ is an [[isofibration]].

Define a $V$-category $C$ to be

* **fibrant** if the functor $C\to 1$ is a naive fibration.  This is equivalent to $C$ being locally fibrant, i.e. each $C(x,y)$ is fibrant.

By a theorem of Joyal, these weak equivalences and fibrant objects determine at most one model structure on the category $V Cat$.  When it exists, it is called the **(canonical, categorical) model structure on $V$-categories**.

Usually, the fibrations between fibrant objects in this model structure are precisely the naive fibrations (although between non-fibrant objects, the two classes are distinct).  Usually also, the trivial fibrations are precisely the weak equivalences that are also naive fibrations, which is to say the $V$-functors that are (1) locally trivial fibrations and (2) surjective on objects.

See the references for general conditions under which this model structure exists.

## Existence

\begin{theorem}
(Muro, \cite{Muro}, Theorem 1.1.)
For any combinatorial closed symmetric monoidal model category $V$
satisfying the [[monoid axiom]],
the category $Cat(V)$ admits the Dwyer–Kan model structure.
Moreover, this model structure is combinatorial.
\end{theorem}

## Examples

* The [[model structure on simplicial categories]] which presents [[(∞,1)-categories]] is induced from the Quillen [[model structure on simplicial sets]].

* The [[model structure on simplicial categories]] which presents [[(∞,2)-categories]] is induced from the [[Joyal model structure]] on simplicial sets

* The [[model structure on dg-categories]] is induced from the [[projective model structure on chain complexes]].

* The [[canonical model structure]] on [[Cat]] is induced from the [[trivial model structure]] on [[Set]].

* The canonical (Lack) model structure on $2 Cat$ is induced from the canonical model structure on $Cat$.

* The canonical (Lack) model structure on [[Gray-categories]] is induced from the canonical model structure on $2 Cat$.

## Related concepts

* [[model structure on internal categories]]

* [[model structure on operads]]

## References

* [[Julia E. Bergner]], _A model category structure on the category of simplicial categories_, Transactions of the American Mathematical Society 359:5 (2006), 2043-2058.  [arXiv](https://arxiv.org/abs/math/0406507), [doi](http://dx.doi.org/10.1090/s0002-9947-06-03987-0).

* [[Jacob Lurie]], [[Higher Topos Theory]] section A.3.2

* [[Alexandru Stanculescu]], *Constructing model categories with prescribed fibrant objects*, Theory and Applications of Categories, **29** 23 (2014) 635-653  [[arXiv:1208.6005](http://arxiv.org/abs/1208.6005), [tac:29-23](http://www.tac.mta.ca/tac/volumes/29/23/29-23abs.html)]

* {#BergerMoerdijk12} [[Clemens Berger]], [[Ieke Moerdijk]], _On the homotopy theory of enriched categories_ ([arXiv:1201.2134](http://arxiv.org/abs/1201.2134))

* {#Muro} [[Fernando Muro]], _Dwyer–Kan homotopy theory of enriched categories_.  [arXiv:1201.1575](https://arxiv.org/abs/1201.1575).

[[!redirects model structures on enriched categories]]

[[!redirects Dwyer-Kan model structure on enriched categories]]
[[!redirects Dwyer–Kan model structure on enriched categories]]


model structure on equivariant chain complexes

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#### Representation theory
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#### Model category theory
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#### Rational homotopy theory
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[[!include differential graded objects - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

Let $G$ be a [[finite group]].

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

There is a [[model category]]-[[mathematical structure|structure]] on the [[category]]

$$
  Functors
  \big(
    G Orbits
    \,,\,
    CochainComplexes^{\geq 0}_{\mathbb{Q}}
  \big)
$$

of connective $G$-[[equivariant cochain complexes]] (i.e. with [[differential]] of degree +1) over the [[rational numbers]], whose

$\mathrm{W}$ -- [[weak equivalences]] are the [[quasi-isomorphisms]] over each $G/H \in G Orbits$;

$Cof$ -- [[cofibrations]] are the [[positive number|positive]]-degree wise [[injections]] over each $G/H \in G Orbits$;

$Fib$ -- [[fibrations]] are the morphisms which over each $G/H \in G Orbits$ are degree-wise [[surjections]] whose degreewise [[kernels]] are [[injective objects]] (in the [[category]] of [[vector G-spaces]]).

=--

([Scull 08, Theorem 3.1](#Scull08))

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

For $G = 1$ the [[trivial group]], this reduces to the [[injective model structure on connective cochain complexes]].

=--

## Related concepts

* [[model structure on equivariant dgc-algebras]]

## References

* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))

[[!redirects model structures on equivariant chain complexes]]

[[!redirects model structure on equivariant connective cochain complexes]]
[[!redirects model structures on equivariant connective cochain complexes]]


model structure on equivariant dgc-algebras

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#### Rational homotopy theory
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#Contents#
* table of contents
{:toc}

## Idea

Where the [[projective model structure on connective dgc-algebras]] models the [[rational homotopy theory]] of rationally [[finite type]] [[nilpotent topological spaces]] (by the [[fundamental theorem of dg-algebraic rational homotopy theory]]), its $G$-[[equivariant cohomology|equivariant]] enhancement ([Scull 08](#Scull08)) models $G$-[[equivariant rational homotopy theory]] of [[topological G-spaces]]:

Using [[Elmendorf's theorem]], the underlying [[category]] is that of [[diagrams]] of connective [[dgc-algebras]] parametrized over the [[orbit category]] of $G$: _$G$-[[equivariant dgc-algebras]]_. A key technical aspect of this generalization is that not all [[objects]] are [[injective object|injective]] anymore, but otherwise the definitions and properties of the model structure proceed in analogy to [Bousfield-Gugenheim](model+structure+on+dg-algebras#BousfieldGugenheim76)'s [[projective model structure on connective dgc-algebras]]. Notably, the [[minimal cofibrations]] coincide with the equivariant [[minimal Sullivan models]] earlier considered by [Triantafillou 82](#Triantafillou82).

## Definition
 {#Definition}

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

There is a [[model category]]-[[mathematical structure|structure]] on the [[category]]

$$
  Functors
  \big(
    G Orbits
    \,,\,
    dgcAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)
$$

of connective $G$-[[equivariant dgc-algebras]] (i.e. with [[differential]] of degree +1) over the [[rational numbers]], whose [[weak equivalences]] and [[fibrations]] are those of the underlying [[model structure on equivariant connective cochain complexes]], hence:

$\mathrm{W}$ -- [[weak equivalences]] are the [[quasi-isomorphisms]] over each $G/H \in G Orbits$;

$Fib$ -- [[fibrations]] are the morphisms which over each $G/H \in G Orbits$ are degree-wise [[surjections]] whose degreewise [[kernels]] are [[injective objects]] (in the [[category]] of [[vector G-spaces]]).

=--

([Scull 08, Theorem 3.2](#Scull08))

## Properties

+-- {: .num_prop}
###### Proposition
**([[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]])**

Let $G$ be a [[finite group]].

The $G$-[[equivariant PL de Rham complex]]-construction is the [[left adjoint]] in a [[Quillen adjunction]] between

* the [[opposite model structure|opposite]] of the [[projective model structure on equivariant connective dgc-algebras]]

* the [[model structure on equivariant simplicial sets]]

  (i.e.: the global projective [[model structure on functors]] from the [[opposite category|opposite]] of the [[orbit category]] to the [[classical model structure on simplicial sets]])

$$
  \big(
    G dgcAlgebras^{\geq 0}_{k}
  \big)^{op}_{proj}
  \underoverset
    {
      \underset
        {\;\;\; exp \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\;\Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot_{\mathrlap{Qu}}}
  G SimplicialSets_{Qu}
$$

=--

([Scull 08, Prop. 5.1](#Scull08))

## Related concepts

* [[vector G-space]]

* [[equivariant chain complex]], [[model structure on equivariant chain complexes]]

* [[equivariant dgc-algebra]]

* [[equivariant PL de Rham complex]]

* [[equivariant Sullivan model]]

* [[equivariant cohomology]]

* [[equivariant rational homotopy theory]]

* [[equivariant rational stable homotopy theory]]

* [[model structure on topological G-spaces]]

## References

* {#Triantafillou82} [[Georgia Triantafillou]], _Equivariant minimal models_, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) ([jstor:1999119](http://www.jstor.org/stable/1999119))

* [[Marek Golasiński]], _Equivariant rational homotopy theory as a closed model category_, Journal of Pure and Applied Algebra Volume 133, Issue 3, 30 December 1998, Pages 271-287 (doi:10.1016/S0022-4049(97)00127-8)

  (for Hamiltonian groups)

* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))

[[!redirects model structures on equivariant dgc-algebras]]

[[!redirects model structure on equivariant connective dgc-algebras]]
[[!redirects model structures on equivariant connective dgc-algebras]]

[[!redirects projective model structure on equivariant connective dgc-algebras]]
[[!redirects projective model structures on equivariant connective dgc-algebras]]


model structure on functors

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#Contents#
* table of contents
{:toc}

## Idea

The [[model category]] structures on [[functor category|functor categories]] are models for [[(∞,1)-category of (∞,1)-functors|(∞,1)-categories of (∞,1)-functors]].

For $C$ a [[model category]] and $D$ any [[small category]] there are two "obvious" ways to put a [[model category]] structure on the [[functor category]] $[D,C]$, called the _projective_ and the _injective_ model structures.  For completely general $C$, neither one need exist, but there are rather general conditions that ensure their existence.  In particular, the projective model structure exists as long as $C$ is [[cofibrantly generated model category|cofibrantly generated]], while both model structures exist if $C$ is [[accessible model category|accessible]] (and in particular if it is [[combinatorial model category|combinatorial]]).  In the case of [[enriched category|enriched]] diagrams, additional cofibrancy-type conditions are required on $D$.

A related kind of model structure is the [[Reedy model structure]]/[[generalized Reedy model structure]] on functor categories, which applies for *any* model category $C$, but requires $D$ to be a very special sort of category, namely a [[Reedy category]]/[[generalized Reedy category]].

In the special case that $C = $ [[sSet]] is the [[classical model structure on simplicial sets]] the projective and injective model structure on the functor categories $[D,SSet]$ are described in more detail at [[global model structure on simplicial presheaves]] and [[model structure on sSet-enriched presheaves]].

## Definition

Let $\mathbf{S}$ be a [[symmetric monoidal category]], let $C$ be an $\mathbf{S}$-[[model category]] that is an $\mathbf{S}$-[[enriched category]], and let $D$ be a [[small category|small]] $\mathbf{S}$-enriched category. Usually we have either $\mathbf{S}=Set$ or else $\mathbf{S}$ is a [[monoidal model category]] and $C$ an $\mathbf{S}$-[[enriched model category]].

Let $[D,C]$ denote the enriched [[functor category]], whose objects are $\mathbf{S}$-enriched functors $D\longrightarrow C$.

+-- {: .num_defn #ProjectiveAndInjectiveStructure}
###### Definition
We define the following classes of maps in $[D,C]$:

* the **projective weak equivalences** and **projective fibrations** are the [[natural transformations]] that are objectwise such morphisms in $C$.
* the **injective weak equivalences** and **injective cofibrations** are the [[natural transformations]] that are objectwise such morphisms in $C$.

If either of these choices defines a model structure on $[D,C]$, we call it the **projective model structure** $[D,C]_{proj}$ or **injective model structure** $[D,C]_{inj}$ respectively.  Of course, the projective cofibrations and injective fibrations can then be characterized by lifting properties.
=--

## Existence

### Projective case

The projective model structure can be regarded as a right-[[transferred model structure]].  This yields the following basic result on its existence.

\\begin{theorem}\label{ExistenceOfProjectiveStructureOnEnrichedFunctors}
**(existence and cofibrant generation of projective structure on enriched fuinctors)**
\linebreak
Given

* $\mathbf{S}$ a [[cosmos for enrichment]],

* $D$, $C$ a pair of $\mathbf{S}$-enriched categories

where $C$ carries the [[structure]] of

1. a [[cofibrantly generated model category]],

2. which admits [[copowers]] by the [[hom-objects]] $D(x,y)\in \mathbf{S}$, which preserve [[acyclic cofibrations]].

   (This is the case for instance if $\mathbf{S}=$ [[Set]], or if $\mathbf{S}$ is a [[monoidal model category]], $C$ is an [[enriched model category|$\mathbf{S}$-model category]], and the [[hom-objects]] $D(x,y)$ are cofibrant in $\mathbf{S}$.)

Then the projective model structure $[D,C]_{proj}$ on the [[enriched functor category]] exists, and is again cofibrantly generated.
\end{theorem}
(For $\mathbf{S} = $ [[Sets]] this is [Hirschhorn 2002, Thm. 11.6.1](#Hirschhorn02).).
\begin{proof}
Assuming the existence of such copowers, for any $x\in ob(D)$ the "[[evaluation]] at $x$" functor $ev_x \colon [D,C]\longrightarrow C$ has a [[left adjoint]] $F_x$ sending $A\in C$ to the functor $y\mapsto D(x,y)\odot A$, where $\odot$ denotes the [[copower]].  Now if $I$ and $J$ are generating sets of cofibrations and trivial cofibrations for $C$, let $I^D$ be the set of maps $F_x(i)$ in $[D,C]$, for all $i\in I$ and $x\in ob(D)$, and similarly for $J$.  Then the projective fibrations and trivial fibrations are characterized by having the right lifting property with respect to $J^D$ and $I^D$ respectively, while both $I^D$ and $J^D$ permit the [[small object argument]] since $I$ and $J$ do and colimits in $[D,C]$ are pointwise.  Since the trivial fibrations in $[D,C]$ clearly coincide with the fibrations that are weak equivalences, it remains only to show that all $J^D$-cell complexes are weak equivalences.  But a $J^D$-cell complex is objectwise a cell complex built from cells $D(x,y)\odot j$ for maps $j\in J$, and the assumption ensures that these are trivial cofibrations in $C$, hence so is any cell complex built from them.
\end{proof}

There do exist projective model structures that do not fall under this theorem, however, such as the following.

+-- {: .un_theorem}
###### Theorem
If $C$ is a [[locally presentable category|locally presentable]] [[2-category]] with its [[2-trivial model structure]] and $D$ is a small 2-category, then the projective model structure on $[D,C]$ exists.
=--
+-- {: .proof}
###### Proof
This follows from the result of [Lack](#Lack06) on [[transferred model structures]] for algebras over [[2-monads]], since $[D,C]$ is the category of algebras for an accessible 2-monad on $C^{ob(D)}$.
=--

Note that $C$ need not be cofibrantly generated (and the 2-trivial model structure often fails to be cofibrantly generated), so the generality of this result is not entirely included in the previous one.

### Accessible case

In the case when $C$ is an [[accessible model category]], i.e. it is a [[locally presentable category]] and its constituent [[weak factorization systems]] have [[accessible functor|accessible]] realizations as [[functorial factorizations]], we have the following general result from [Moser](#Moser) (the unenriched case appears in [HKRS15](#HKRS15) and [GKR18](#GKR18)).

\begin{theorem}\label{Accessible}
Let

* $\mathbf{S}$ be a [[locally presentable category|locally presentable]] [[cosmos]]

* $C$ an $\mathbf{S}$-cocomplete locally $\mathbf{S}$-presentable $\mathbf{S}$-[[enriched category]] that is an [[accessible model category]],

* $D$ a [[small category|small]] $\mathbf{S}$-category.

Then:

1. If [[copowers]] by the [[hom-objects]] $D(x,y)$ preserve [[acyclic cofibrations]], then the projective model structure on $[D,C]$ exists and is accessible.

1. If [[copowers]] by the hom-objects $D(x,y)$ preserve [[cofibrations]], then the injective model structure on $[D,C]$ exists and is accessible

\end{theorem}

### Combinatorial case
 {#CombinatorialCase}

Every [[combinatorial model category]] (i.e. locally presentable and cofibrantly generated) is accessible, so Theorem \ref{Accessible} shows that both model structures exist, and Theorem \ref{CofGenProj} shows that the projective model structure is cofibrantly generated, hence (by [this Prop.](locally+presentable+category#PresheavesWithValuesInLocPresAreLocPres)) also [[combinatorial model category|combinatorial]].

In fact the injective model structure is also combinatorial, although the proof is much more involved, because there is no explicit description of the generating cofibrations and acyclic cofibrations; they have to be produced by a cardinality argument.  This was first proven by in [[Higher Topos Theory|HTT, prop. A.2.8.2 and A.3.3.2]] under strong assumptions on the enriching category (in particular that all objects are cofibrant), and later generalized by [Makkai & Rosický 2014](#MakkaiRosický14) to essentially the following:

\begin{theorem}\label{Combinatorial}
Let $\mathbf{S}$ be a [[locally presentable category|locally presentable]] [[cosmos]], $C$ an $\mathbf{S}$-cocomplete locally $\mathbf{S}$-presentable $\mathbf{S}$-[[enriched category]] that is a [[combinatorial model category]], and $D$ a small $\mathbf{S}$-category.  Then:

1. If [[copowers]] by the hom-objects $D(x,y)$ preserve trivial cofibrations, then the projective model structure on $[D,C]$ exists and is combinatorial.

1. If [[copowers]] by the hom-objects $D(x,y)$ preserve cofibrations, then the injective model structure on $[D,C]$ exists and is combinatorial.

\end{theorem}
\begin{proof}
It suffices to construct the factorizations, which follows from [Makkai & Rosický 2014, Remark 3.8](#MakkaiRosický14) about left-lifting combinatorial weak factorization systems.
\end{proof}

## Properties

### General

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

The projective and injective structures $[D,C]_{proj}$ and $[D,C]_{inj}$, def. \ref{ProjectiveAndInjectiveStructure}, are (insofar as they exist):

* right or left [[proper model categories]] if $C$ is right or left proper, respectively.

* $\mathbf{S}$-[[enriched model categories]] if $C$ is an $\mathbf{S}$-model category.

=--

The statement about properness appears as [[Higher Topos Theory|HTT, remark A.2.8.4]].

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

For $C$ a [[combinatorial simplicial model category]], the [[(∞,1)-category]] [[presentable (∞,1)-category|presented by]] $[D,C]_{proj}$ and $[D,C]_{inj}$ under the above assumptions is the [[(∞,1)-category of (∞,1)-functors]] $Func(D,C^\circ)$ from the ordinary category $D$ to the $(\infty,1)$-category presented by $C$.

=--

See at *[[(∞,1)-category of (∞,1)-functors]]* for more.

### Relation to other model structures

+-- {: .num_prop}
###### Proposition
If [[copowers]] by the hom-objects of $D$ preserve trivial cofibrations, then every every fibration in $[D,C]_{inj}$ is in particular a fibration in $[D,C]_{proj}$.  Similarly, if [[powers]] by the hom-objects of $D$ preserve trivial fibrations, then every cofibration in $[D,C]_{proj}$ is in particular a cofibration in $[D,C]_{inj}$.  The hypotheses are satisfied if $D$ is unenriched, or in the monoidal model category case if the hom-objects of $D$ are cofibrant.
=--

This is argued in the beginning of the proof of [[Higher Topos Theory|HTT, lemma A.2.8.3]].  For $Top$-enriched functors, this is ([Piacenza 91, section 5](#Piacenza91)). For details see at _[classical model structure on topological spaces -- Model structure on enriched functors](classical+model+structure+on+topological+spaces#ModelStructureOnTopEnrichedFunctors)_.

+-- {: .proof}
###### Proof
If $i:A\longrightarrow B$ is a trivial cofibration in $C$ and $x\in ob(D)$, then the first assumption implies that $F_x(i) : F_x(A) \longrightarrow F_x(B)$, for $F_x(A) (y) = D(x,y) \odot A$ the left adjoint of $ev_x : [D,C] \longrightarrow C$, is a trivial cofibration in $[D,C]_{inj}$.  Thus, any fibration $p$ in $[D,C]_{inj}$ has the right lifting property with respect to it, which is to say that $ev_x(p)$ has the right lifting property with respect to $i$.  Since this is true for any $i$, each $ev_x(p)$ is a fibration, i.e. $p$ is a fibration in $[D,C]_{inj}$.  The other half is dual.
=--

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

The [[identity functors]]

$$
  [D,C]_{inj}
    \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}}
  [D,C]_{proj}
$$

form a [[Quillen equivalence]] (with $Id : [D,C]_{proj} \longrightarrow [D,C]_{inj}$ being the left Quillen functor).

If $D$ is a [[Reedy category]] this factors through the [[Reedy model structure]]

$$
  [D,C]_{inj}
    \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}}
  [D,C]_{Reedy}
    \stackrel{\overset{Id}{\longleftarrow}}{\underset{Id}{\longrightarrow}}
  [D,C]_{proj}
$$

=--

### Functoriality in domain and codomain

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

The functor model structures depend [[Quillen adjunction|Quillen-functorially]] on their [[codomain]], in that for

$$
  D_1
  \stackrel
    {\overset{L}{\longleftarrow}}
    {\underset{R}{\longrightarrow}}
  D_2
$$

an $\mathbf{S}$-[[enriched Quillen adjunction]] between [[combinatorial model categories|combinatorial]] $\mathbf{S}$-[[enriched model categories]], postcomposition induces $\mathbf{S}$-[[enriched Quillen adjunctions]]

$$
  [C,D_1]_{proj}
    \stackrel
      {\overset{[C,L]}{\longleftarrow}}
      {\underset{[C,R]}{\longrightarrow}}
  [C,D_2]_{proj}
$$

and

$$
  [C,D_1]_{inj}
    \stackrel
      {\overset{[C,L]}{\longleftarrow}}
      {\underset{[C,R]}{\longrightarrow}}
  [C,D_2]_{inj}
  \,.
$$

Moreover, if $(L \dashv R)$ is a [[Quillen equivalence]], then so is $([C,L] \dashv [C,R])$.

=--

For the case that $C$ is a small category this is [[Lurie, remark A.2.8.6](#Lurie)], for the enriched case this is [[Lurie, prop. A.3.3.6](#Lurie)].

The Quillen-functoriality on the [[domain]] is more asymmetric.

\begin{proposition}
\label{QuillenFunctorialityInDomain}
**(Quillen functoriality in the domain category)**
\linebreak
For $p \colon C_1 \longrightarrow C_2$ a functor between small categories or an $\mathbf{S}$-[[enriched functor]] between $\mathbf{S}$-[[enriched categories]], let

$$
  (p_! \dashv p^* \dashv p_*)
  \;\colon\;
  [C_2,D]
  \stackrel{\overset{p_!}{\longleftarrow}}{\stackrel{\overset{p^*}{\longrightarrow}}{\underset{p_*}{\longleftarrow}}}
 [C_1,D]
$$

be the [[adjoint triple]] where $p^*$ is precomposition with $p$ and where $p_!$ and $p_*$ are left and right [[Kan extension]] along $p$, respectively.

Then we have [[Quillen adjunctions]]

$$
  (p_! \dashv p^*)
  \;\colon\;
  [C_1,D]_{proj} \stackrel{\overset{p_!}{\longrightarrow}}{\underset{p^*}{\longleftarrow}}
  [C_2,D]_{proj}
$$

and

$$
  (p^* \dashv p_*)
  \;\colon\;
  [C_1,D]_{inj} \stackrel{\overset{p^*}{\longleftarrow}}{\underset{p_*}{\longrightarrow}}
  [C_2,D]_{inj}
  \,.
$$

\end{proposition}

For $C$ not enriched this appears as [[Lurie, prop. A.2.8.7](#Lurie)], for the enriched case it appears as [[Lurie, prop. A.3.3.7](#Lurie)].

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

In the $sSet$-enriched case, if $p : D_1 \longrightarrow D_2$ is a [[weak equivalence]] in the [[model structure on sSet-categories]], then these two Quillen adjunctions are both [[Quillen equivalence]]s.

=--

### Relation to homotopy Kan extensions/limits/colimits

Often functors $D \longrightarrow C$ are thought of as [[diagram]]s in the model category $C$, and one is interested in obtaining their [[homotopy limit]] or [[homotopy colimit]] or, generally, for $p : D \longrightarrow D'$ any functor, their left and right [[homotopy Kan extension]].

These are the left and right [[derived functor]]s $HoLan := \mathbb{L} p_1$ and $HoRan := \mathbb{R} p_*$ of

$$
  [D,C]_{proj} \stackrel{p_!}{\longrightarrow}  [D',C]_{proj}
$$

and

$$
  [D,C]_{inj} \stackrel{p_*}{\longrightarrow} [D',C]_{inj}
$$

respectively.

For more on this see [[homotopy Kan extension]]. For the case that $D' = *$ this reduces to [[homotopy limit]] and [[homotopy colimit]].

### Relation to $\infty$-functor categories
 {#RelationToInfinityFunctorCategories}

\begin{proposition}
  For

  * $\mathbf{C}$ a [[combinatorial simplicial model category]]

  * $\mathbf{D}$ a [[small category|small]] [[sSet-enriched category]]

then both the projective and the injective model structure on the [[sSet]]-[[enriched functor category]] $sFunc(\mathbf{D}, \mathbf{C})$ (which exist by the above discussion) [[presentable (infinity,1)-category|present]] the corresponding [[(infinity,1)-category of (infinity,1)-functors|$\infty$-category of $\infty$-functors]]. Concretely:

The [[homotopy coherent nerve]] of the [[full subcategory|full]] [[sSet-enriched category|sSet-enriched]] [[subcategory]] of the functor model category on its [[bifibrant objects]], $\big(\mathbf{sFunc}(\mathbf{D}, \mathbf{C})\big)^\circ$, is [[equivalence of quasi-categories|equivalent]] to the [[(infinity,1)-functor (infinity,1)-category|$\infty$-functor]] [[quasi-category]] between the [[homotopy coherent nerves]]
$$
  N \big( \mathbf{sFunc}(\mathbf{D}, \mathbf{C})^\circ \big)
  \;\;
  \underset{qCat}{\simeq}
  \;\;
  Func_\infty
  \Big(
    N(\mathbf{D})
    ,\,
    N\big( \mathbf{C}^\circ \big)
  \Big)
  \,.
$$
\end{proposition}
This is due to [Lurie (2009), Prop. 4.2.4.4](#Lurie). See also the discussion *[here](infinity1-category+of+infinity1-functors#ModelCategoryPresentation)* at *[[(infinity,1)-category of (infinity,1)-functors|$\infty$-category of $\infty$-functors]]*.

## Examples

Examples of cofibrant objects in the projective model structure are discussed at *[[projectively cofibrant diagram]]*.

\begin{example}
**([[model structure on simplicial presheaves]])**
\linebreak
  Model structures on [[sSet]]-valued [[functors]], hence on [[simplicial presheaves]], play a central role in the theory of [[combinatorial model categories]] and specifically in [[model topos]]-theory presenting [[(infinity,1)-topos|$\infty$-toposes]]. See at *[[model structure on simplicial presheaves]]* for more.
\end{example}

Specifically:

\begin{example}
\label{BorelModelStructureOnSimplicialGroupActions}
**([[Borel model structure]] on [[simplicial group actions]])**
\linebreak
For $\mathcal{G} \,\in\, Grp(sSet)$ a [[simplicial group]] with [[sSet]]-[[enriched groupoid|enriched]] [[delooping groupoid]] denoted $\mathbf{B}\mathcal{G} \in sSet\text{-}Grpd$, an [[sSet]]-[[enriched functor]] $\mathbf{B}\mathcal{G} \longrightarrow sSet$ is equivalently a [[simplicial group action]] of $\mathcal{G}$.

Under this identification, the projective model strcuture on [[simplicial functors]] (Prop. \ref{ExistenceOfProjectiveStructureOnEnrichedFunctors}) is equivalently the *[[Borel model structure]]* on [[simplicial group actions]], a context of Borel-[[equivariant homotopy theory]]:

$$
  sFunc\big(
    \mathbf{B}\mathcal{G}
    ,\,
    sSet
  \big)_{proj}
  \;\;
  =
  \;\;
  \mathcal{G}Act(sSet)_{Borel}
  \,.
$$

More generally, for $\mathcal{X} \in sSet\text{-}Grpd$ an [[sSet]]-[[enriched groupoid]] (Dwyer-Kan [[simplicial groupoid]]) with a single [[connected component]] $\pi_0(\mathcal{X}) \simeq \{[x]\}$, so that the inclusion
$$
  \iota
    \,\colon\,
  \mathbf{B}(\mathcal{X}(x,x))
  \xhookrightarrow{\phantom{---}}
  \mathcal{X}
$$
is an [[sSet]]-[[enriched adjoint equivalence]] (see discussion [there](simplicial+groupoid#RelationToSimplicialGroups)) the projective model structure on [[simplicial functors]] from Prop. \ref{ExistenceOfProjectiveStructureOnEnrichedFunctors} is [[transferred model structure|transferred]] under the induced [[adjoint equivalence]] of [[sSet]]-[[enriched functor categories]]
$$
  sFunc\big(
    \mathcal{X}
    ,\,
    sSet
  \big)_{proj}
  \underoverset
    {\underset{\iota^\ast}{\longrightarrow}}
    {\overset{\iota_!}{\longleftarrow}}
    {\;\; \bot_{\simeq} \;\;}
  sFunc\Big(
    \mathbf{B}\big(\mathcal{X}(x,x)\big)
    ,\,
    sSet
  \Big)_{proj}
  \;=\;
  \big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}
$$
By [this example](transferred+model+structure#RightTransferAlongAdjointEquivalence) it follows that morphisms in all three classes $(\mathrm{W}, Fib, Cof)$ in $sFunc(\mathcal{X}, \, sSet)_{proj}$ are those which restrict on $x \in Obj(\mathcal{X})$ to the respective class in $\big(\mathcal{X}(x,x)\big) Act(sSet)_{Borel}$.

It follows that for $\mathcal{X} \,\in\, sSet\text{-}Grpd$ a [[simplicial groupoid]] with any set $\pi_0(\mathcal{X})$ of [[connected components]], the projective model structure of simplicial functors over it is the [product model structure](model+category#ProductModelStructure) of the [[Borel model structures]] of [[simplicial group actions]], one for each connected component:
$$
  sFunc\big(
    \mathcal{X}
    ,\,
    sSet
  \big)_{proj}
  \underoverset
    {\underset{\iota^\ast}{\longrightarrow}}
    {\overset{\iota_!}{\longleftarrow}}
    {\;\; \bot_{\simeq} \;\;}
  \underset{i \in \pi_0(\mathcal{X})}{\prod}
  sFunc\Big(
    \mathbf{B}\big(\mathcal{X}(x_i,x_i)\big)
    ,\,
    sSet
  \Big)_{proj}
  \;=\;
  \underset{i \in \pi_0(\mathcal{X})}{\prod}
  \big(\mathcal{X}(x_i,x_i)\big) Act(sSet)_{Borel}
  \,.
$$
\end{example}

## Related concepts

* [[Reedy model structure]], [[generalized Reedy model structure]],

* [[model structure on sections]]

* [[lim^1 and Milnor sequences]]

* [[model structure on homotopical presheaves]], [[model structure on simplicial presheaves]]

* [[model structure for n-excisive functors]]

## References

The projective model structure on $Top_{Quillen}$-enriched functors is discussed in

* {#Piacenza91} [[Robert Piacenza]]  section 5 of _Homotopy theory of diagrams and CW-complexes over a category_, Can. J. Math. Vol 43 (4), 1991 ([[Piazenza91.pdf:file]])

  also chapter VI of [[Peter May]] et al., _Equivariant homotopy and cohomology theory_, 1996 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf))

See also

* [[Alex Heller]], _Homotopy in functor categories_, Transactions of the AMS, vol 272, Number 1, July 1982 ([JSTOR](http://www.jstor.org/stable/1998955))

Textbook account of the projective model structure

* {#Hirschhorn02} [[Philip Hirschhorn]], §6.11 in: _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs **99** (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

The injective model structure for unenriched diagrams of simplicial sets was first constructed by

* [[Alex Heller]], _Homotopy theories_

Probably the first general construction of injective model structures for enriched diagrams in combinatorial model categories was in

* {#Lurie} [[Jacob Lurie]],  sections A.2.8 (unenriched) and section A.3.3 (enriched) of _[[Higher Topos Theory]]_, 2009

The projective model structure for functors to [[sSet]] on a _[[large category|large]]_ domain is discussed in

* [[Boris Chorny]], [[William Dwyer]], _Homotopy theory of small diagrams over large categories_, [arXiv:math/0607117](http://arxiv.org/abs/math/0607117)

The case of diagrams in a 2-category is a special case of

* {#Lack06} [[Steve Lack]], *Homotopy-theoretic aspects of 2-monads*, [arXiv](http://arxiv.org/abs/math.CT/0607646)

The use of accessible model structures to construct both projective and injective model structures on unenriched diagrams was introduced in

* Marzieh Bayeh, [[Kathryn Hess]], [[Varvara Karpova]], Magdalena Kędziorek, [[Emily Riehl]], [[Brooke Shipley]], _Left-induced model structures and diagram categories_ ([arXiv:1401.3651](http://arxiv.org/abs/1401.3651))

* {#HKRS15} [[Kathryn Hess]], Magdalena Kędziorek, [[Emily Riehl]], [[Brooke Shipley]], _A necessary and sufficient condition for induced model structures_ ([arXiv:1509.08154](http://arxiv.org/abs/1509.08154)).  This paper contains an error, corrected by:

* {#GKR18} [[Richard Garner]], Magdalena Kedziorek, [[Emily Riehl]], _Lifting accessible model structures_, [arXiv:1802.09889](https://arxiv.org/abs/1802.09889)

It was generalized to enriched diagrams in

* {#Moser} Lyne Moser, *Injective and Projective Model Structures on Enriched Diagram Categories*, [arXiv:1710.11388](https://arxiv.org/abs/1710.11388)

The more general result above on combinatoriality of injective model structures follows from Remark 3.8 of

* {#MakkaiRosický14} [[M. Makkai]], [[J. Rosický]], _Cellular categories_, J. Pure Appl. Alg. **218** (2014) 1652-1664 [[arXiv:1304.7572](https://arxiv.org/abs/1304.7572), [doi:10.1016/j.jpaa.2014.01.005](https://doi.org/10.1016/j.jpaa.2014.01.005)]

See also

* [[David White]], _Modified projective model structure_ ([MO comment](http://mathoverflow.net/questions/76160/acyclic-models-via-model-categories/104423#104423))

[[!redirects model structures on functors]]

[[!redirects injective model structure]]
[[!redirects injective model structures]]
[[!redirects injective model structure on functors]]
[[!redirects projective model structure]]
[[!redirects projective model structures]]
[[!redirects projective model structure on functors]]

[[!redirects global model structure on functor categories]]
[[!redirects global model structures on functor categories]]
[[!redirects global model structure on functors]]
[[!redirects global model structures on functors]]

[[!redirects model structure on enriched functors]]
[[!redirects model structures on enriched functors]]

[[!redirects projective model structure on enriched functors]]
[[!redirects projective model structures on enriched functors]]
[[!redirects injective model structure on enriched functors]]
[[!redirects injective model structures on enriched functors]]

[[!redirects model structure on simplicial functors]]
[[!redirects projective model structure on simplicial functors]]
[[!redirects injective model structure on simplicial functors]]


model structure on homotopical presheaves

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

#Contents#
* automatic table of contents goes here
{:toc}

## Idea ##

For $V$ a sufficiently nice ([[monoidal model category|monoidal]]) [[model category]] and $C$ a [[small category]] equipped with a [[Grothendieck topology]] $\tau$, there are left [[Bousfield localization]]s of the [[global model structure on functors]] $[C^{op}, V]$ whose fibrant objects satisfy [[descent]] with respect to [[?ech cover]]s or even [[hypercover]]s with respect to $\tau$.

These model structures are expected to model $V$-valued [[∞-stack]]s on $C$. This is well understood for the case $V = $ [[SSet]] equipped with the standard [[model structure on simplicial sets]] modelling [[∞-groupoid]]s. In this case the resulting [[local model structure on simplicial presheaves]] is known to be one of the  [[models for ∞-stack (∞,1)-toposes]].

But the general localization procedure works for choices of $V$ different from and more general than [[SSet]] with its standard model structure. In particular it should work for

* $V = $ [[SSet]] equipped with its [[model structure on simplicial sets|Joyal model structure]], modelling [[(∞,1)-category|(∞,1)-categories]]

* $V = (n,r)\Theta Spaces$, the model structure on [[Theta space]]s modelling weak [[(n,r)-category|(n,r)-categories]];

* $V = dSet$, the [[model structure on dendroidal sets]] modelling [[(∞,1)-operad]]s.

For these cases the local model structure on $V$-valued presheaves should model, respectively, $(n,r)$-category valued sheaves/stacks and $(\infty,1)$-operad valued sheaves/stacks.

## References ##

The general localization result is apparently due to

* [[Clark Barwick]], _On left and right model categories and left and right Bousfield localization_ Homology, Homotopy and Applications, vol. 12(2), 2010, pp.245–320 ([pdf](http://www.intlpress.com/HHA/v12/n2/a9/v12n2a9.pdf))

which considers the [[?ech cover]]-localization assuming $V$ to be [[monoidal category|monoidal]] and

* [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique_ ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/THESE.PDF))

which apparently does the [[hypercover]] [[descent]] and without assuming $V$ to be monoidal.

Much of this was kindly pointed out by [[Denis-Charles Cisinski]] in discussion [here](http://mathoverflow.net/questions/5808/local-joyal-simplicial-presheaves).

## Related concepts

* [[enriched Reedy category]]

* [[model structure on sSet-presheaves]]

[[!redirects model structures on homotopical presheaves]]


model structure on internal categories

\tableofcontents

## Idea

[…]

## Related concepts

* [[canonical model structure]]

* [[model structure on enriched categories]]

## References

[[internal category|Categories internal to]]  [[toposes]]:

* {#JoyalTierney91} [[André Joyal]], [[Myles Tierney]], Thm. 4 (p. 232) of: *Strong stacks and classifying spaces*, in: *Category Theory* Lecture Notes in Mathematics **1488**, Springer (1991) 213-236 [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]

Categories internal to a [[site]] with [[finite limits]]:

* [[Tomas Everaert]], [[Rudger W. Kieboom]], [[Tim van der Linden]], _Model structures for homotopy of internal categories_, Theory and Applications of Categories 15:3 (2005) 66-94.  [tac:15-03](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html), [EuDML:125875](https://eudml.org/doc/125875).

Categories internal to [[simplicial sets]] (see [[simplicial object in Cat]]):

* [[Geoffroy Horel]], _A model structure on internal categories in simplicial sets_, Theory and Applications of Categories 30:20 (2015), 704-750.  [arXiv](https://arxiv.org/abs/1403.6873), [TAC](http://www.tac.mta.ca/tac/volumes/30/20/30-20abs.html).


model structure on modules in a monoidal model category

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

#Contents#
* table of contents
{:toc}

## Idea

For $\mathcal{C}$ a [[monoidal model category]] and $A \in Mon(\mathcal{C})$ a [[monoid in a monoidal category|monoid in]] $\mathcal{C}$, there is under mild conditions a natural [[model category]] structure on its [[category of modules|category of]] [[module over a monoid|modules over]] $A$. ([Schwede & Shipley 2000, Thm. 4.1 (3.1 in the preprint)](#SchwedeShipley00)).

## Statement

\begin{proposition}\label{SchwedeShipleyExistenceStatement}
  Let $(\mathcal{C}, \otimes, \mathbb{1})$ be a [[cofibrantly generated model category|cofibrantly generated]] [[monoidal model category]] and let $R \,\in\, Mon(\mathcal{C}, \otimes, \mathbb{1})$ be a [[monoid object]] whose [[underlying]] object is [[cofibrant object|cofibrant]] in $\mathcal{C}$. Then:

1. The category $R Mod(\mathcal{C}, \otimes, \mathbb{1})$ of [[internalization|internal]] $R$-[[module objects]] carries a [[cofibrantly generated model category|cofibrantly generated model structure]] whose [[weak equivalences]] and [[fibrations]] are those whose [[underlying]] maps are so in $\mathcal{C}$, hence which is [[transferred model structure|right transferred]] along the [[forgetful functor]] $U$:

   $$
     R Mod(\mathcal{C})
     \underoverset
       {\underset{ U }{\longrightarrow}}
       {\overset{F}{\longleftarrow}}
       {\;\;\;\;\;\; \bot_{\mathrlap{Qu}} \;\;\;\;\;\;}
     \mathcal{C}
     \,.
   $$

1. If $R$ is in addition a [[commutative monoid object]] then the [[tensor product of modules]] makes $R Mod(\mathcal{C})$ itself into a [[monoidal model category]].

\end{proposition}

([Schwede & Shipley 2000, Thm. 3.1 with Rem. 3.2](#SchwedeShipley00))

## Related concepts

* **[[model structure on monoids in a monoidal model category]]**

  * [[model structure for ring spectra]]

* [[model structure on commutative monoids in a symmetric monoidal model category]]

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

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

* [[model structure on simplicial T-algebras]]

## References

General discussion:

* {#SchwedeShipley00} [[Stefan Schwede]], [[Brooke Shipley]], *Algebras and modules in monoidal model categories*, Proc. London Math. Soc. **80** 2  (2000)  491-511  [[arXiv:math/9801082](https://arxiv.org/abs/math/9801082), [doi:10.1112/S002461150001220X](https://doi.org/10.1112/S002461150001220X)]

Examples:

* [[Jordan Williamson]], §2.1 in: *Algebraic models of change of groups functors in (co)free rational equivariant spectra*, J. Pure Appl. Algebra **226** 11 (2022) 107108 [[doi:10.1016/j.jpaa.2022.107108](https://doi.org/10.1016/j.jpaa.2022.107108), [arXiv:2003.12412](https://arxiv.org/abs/2003.12412)]

{#TheSpecialCaseInFunctorCategory} The special case of the model structure on modules in a [[functor category]] with values in a closed symmetric [[monoidal model category]] is (re-)derived (see the discussion [here](https://nforum.ncatlab.org/discussion/14400/model-structure-on-modules-in-a-monoidal-model-category/?Focus=99231#Comment_99231)) in:

* Angelos Anastopoulos, [[Marco Benini]], Sec. 2.4 of *Homotopy theory of net representations*, Rev. Math. Phys. (2023) [[arXiv:2201.06464](https://arxiv.org/abs/2201.06464), [doi:10.1142/S0129055X23500083](https://doi.org/10.1142/S0129055X23500083)]

(there for the purpose of desribing representations of [[nets of observables]] in [[homotopical algebraic quantum field theory]]).

[[!redirects model structures on modules in a monoidal model category]]


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

#Contents#
* table of contents
{:toc}

## Definition

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

Let

* $\mathcal{E}$ be a [[cofibrantly generated model category|cofibrantly generated]] [[symmetric monoidal category|symmetric]] [[monoidal model category]];

* $P$ an admissible $\Sigma$-cofibrant [[operad]] in $\mathcal{E}$ (see [[model structure on operads]]);

* $A$ a cofibrant $P$-algebra (see [[model structure on algebras over an operad]]).

Then then category $Mod_P(A)$ of [[modules over an algebra over an operad]] carries the [[transferred model structure]] along the [[forgetful functor]]
$U : Mod_P(A) \to \mathcal{E}$.

Every [[morphism]] of cofibrant $P$-algebras $f : A \to B$ induced a [[Quillen adjunction]]

$$
  (f_! \dashv f^*) : Mod_P(B) \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} Mod_P(A)
$$

which is a [[Quillen equivalence]] if $f$ is a weak equivalence.
=--

This is ([BergerMoerdijk, theorem 2.6](#BergerMoerdijk)).

## Examples

* [[model structure on dg-modules]]

## Related concepts

* [[(∞,1)-operad]], [[model structure on operads]]

  * [[algebra over an (∞,1)-operad]], [[model structure on algebras over an operad]]

    * [[module over an algebra over an (∞,1)-operad]], **model structure on modules over an algebra over an operad**

* [[operad for modules over an algebra]]

## References

* [[Clemens Berger]], [[Ieke Moerdijk]], _On the derived category of an algebra over an operad_ ([arXiv](http://arxiv.org/abs/0801.2664))
{#BergerMoerdijk}

* [[Benoit Fresse]], _Modules over operads and functors_ Springer Lecture Notes in Mathematics, (2009)  ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.159.5833&rep=rep1&type=pdf))
{#Fresse}


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

#Contents#
* table of contents
{:toc}

## Idea

For $C$ a [[monoidal model category]] there is under mild conditions a natural [[model category]] structure on its [[category of monoids]].

## Definition

For $C$ a [[monoidal category]] with all [[colimits]], its [[category of monoids]] comes equipped (as discussed there) with a [[free functor]]/[[forgetful functor]] [[adjunction]]

$$
  (F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
   C
  \,.
$$

Typically one uses on $Mon(C)$ the [[transferred model structure]] along this adjunction, if it exists.

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

If $C$ is [[monoidal model category]] that

* satisfies the [[monoid axiom in a monoidal model category]];

* is a [[cofibrantly generated model category]];

* all objects are [[small objects]],

then the [[transferred model structure]] along the [[free functor]]/[[forgetful functor]] [[adjunction]] $(F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C$ exists on its [[category of monoids]].
=--

This is part of ([SchwedeShipley, theorem 4.1](#SchwedeShipley)).

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

If the symmetric monoidal model category $C$

* is a [[cofibrantly generated model category]]

* its unit is cofibrant

* it has a suitable [[interval object]]

then the transferred model structure on monoids exists.

=--

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

Regard monoids a [[algebras over an operad]] for the [[associative operad]]. Then apply the existence results discussed at _[[model structure on algebras over an operad]]_. See there for more details.

=--

## Properties

### Homotopy pushouts

Suppose the transferred model structure exists on $Mon(C)$. By the discussion of free monoids at [[category of monoids]] we have that then [[pushout]]s of the form

$$
  \array{
     F(A) &\stackrel{F(f)}{\to}& F(B)
     \\
     \downarrow && \downarrow
     \\
     X &\to& P
  }
$$

exist in $Mon(C)$, for all $f : A \to B$ in $C$

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

Let the monoidal model category $C$ be

* [[symmetric monoidal category|symmetric monoidal]];

* and satisfy the [[monoid axiom in a monoidal model category]].

If $f : A\to B$ is an acyclic cofibration in the model structure on $C$, then the pushout $X \to P$ as above is a weak equivalence in $Mon(C)$.

=--

This is [SchwedeShipley, lemma 6.2](#SchwedeShipley).

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

Use the description of the pushout as a [[transfinite composition|transfinite composite]] of pushouts as described at [[category of monoids]] in the section free and relative free monoids.

One sees that the [[pushout product axiom]] implies that all the intermediate pushouts produce acyclic cofibrations and the [[monoid axiom in a monoidal model category]] implies then that each $P_{n-1} \to P_n$ is a weak equivalence. Moreover, all these moprhisms are of the kind used in the monoid axioms, so also their [[transfinite composition]] is a weak equivalence.

=--

### $A_\infty$-Algebras

Under mild conditions on $C$ the model structure on monoids in $C$ is [[Quillen equivalence|Quillen equivalent]] to that of [[A-infinity algebras]] in $C$. See [[model structure on algebras over an operad]] for details.

## Related concepts

* **model structure on monoids in a monoidal model category**

  * [[model structure for ring spectra]]

* [[model structure on modules in a monoidal model category]]

* [[model structure on commutative monoids in a symmetric monoidal model category]]

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

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

* [[model structure on simplicial T-algebras]]

## References

* {#SchwedeShipley} [[Stefan Schwede]], [[Brooke Shipley]], _Algebras and modules in monoidal model categories_ Proc. London Math. Soc. (2000) 80(2): 491-511  ([pdf](http://www.math.uic.edu/~bshipley/monoidal.pdf))

* {#White14} [[David White]], _Model Structures on Commutative Monoids in General Model Categories_ ([arXiv:1403.6759](http://arxiv.org/abs/1403.6759))

[[!redirects model structure on monoids]]

[[!redirects model structures on monoids]]


model structure on operads

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

#Contents#
* table of contents
{:toc}

## Idea

If $V$ is a [[symmetric monoidal category]] that is also a [[monoidal model category]], then under suitable conditions there is also the structure of a [[model category]] on the category of [[symmetric operad|symmetric]] $V$-[[operads]].

This is important for the notion of _homotopy_ [[algebra over an operad]], such as [[A-∞ algebras]] and [[E-∞ algebras]].

## Definition

Throughout, let $V$ be a [[symmetric monoidal category|symmetric]] [[closed monoidal category|closed]] [[monoidal model category]] with all small [[colimit]]s and finite [[limit]]s.

### Symmetric collections

We first consider the _collections_ of operations underlying a [[symmetric operad]] (with no notion of composition of operations yet).

For $G$ a [[discrete group]] write $\mathbf{B}G$ for the [[delooping]] [[groupoid]]: the category with a single object and $G$ as its set of [[morphisms]]. Then for $V$ any other category, write $V^{\mathbf{B}G}$ for the [[functor category]], consisting of [[functors]] $\mathbf{B}G \to V$. This is the category of [[actions]] of $G$ on [[objects]] in $V$ (the [[category of representations]]).

For $G$ a [[finite group]] also $V^{\mathbf{B}G}$ inherits the structure of a [[closed monoidal category|closed]] [[symmetric monoidal category]] with small [[colimits]] and [[finite limits]]. There is a [[forgetful functor]]/[[free functor]] [[adjunction]]

$$
  V^{\mathbf{B}G} \stackrel{(-)[G]}{\underset{U}{\to}}
  V
  \,.
$$

Write $\Sigma_n$ for the [[symmetric group]] on $n \in \mathbb{N}$ elements. Take $\Sigma_0$ and $\Sigma_1$ both to be the trivial group.

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

The category of **collections** (of potential operations) in $V$  is the [[product]]

$$
  Coll(V) := \prod_{n \geq 0} V^{\mathbf{B}\Sigma_n}
  \,.
$$

=--

A collection $P$ is a tuple of objects

$$
  P = (P(n))_{n \in \mathbb{N}}
$$

each equipped with an [[action]] by the respective $\Sigma_n$.

### Hopf interval object

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

An object $H \in V$ is a [[Hopf algebra]] object if it is equipped with the structure of a [[monoid]], that of a [[comonoid]] such that product and coproduct preserve each other.

If $V$ is equipped with a compatible structure of a [[monoidal model category]] we say that a a Hopf algebra object is an **Hopf [[interval object]]** if it is equipped with morphisms

$$
  I \coprod I \hookrightarrow H \stackrel{\simeq}{\to} I
$$

that factor the [[codiagonal]] on $I$ by a cofibration followed by a weak equivalence.

=--

+-- {: .num_example}
###### Examples

Such cocommutative coalgebra intervals exist in

* the [[model structure on topological spaces|model structure on compactly generated topological spaces]];

* the [[model structure on simplicial sets]];

* the [[model structure on symmetric spectra]].

In

* the [[model structure on chain complexes]]

there is a coalgebra interval.

=--

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

Since the coalgebra interval in the [[category of chain complexes]] is not cocommutative, this case requires special discussion, as some of the statements below will not apply to it. For more on this case see [[model structure on dg-operads]].

=--

### Model category structure

Assume now that $V$ is moreover equipped with a compatible structure of a [[monoidal model category]].

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

If $V$ is a [[cofibrantly generated model category]], then for each [[finite group]] $G$ the [[transferred model structure]] on $V^{\mathbf{B}G}$ along the [[forgetful functor]]

$$
 U :  V^{\mathbf{B}G} \to V
$$

exists.

It follows that in this case the category of collections $Coll(V)$ is a [[cofibrantly generated model category]] where a morphisms is a fibration or weak equivalence if it is so degreewise in $V$, respectively.

=--

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

A $V$-operad is called **$\Sigma$-cofibrant** if its underlying collection is cofibrant in the above model stucture

=--

A $V$-operad $P$ is called _reduced_ if $P(0)$ is the tensor unit, $P(0) = I$.  A morphism of reduced operads is one that is the identity on the 0-component.

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

If

* $V$ is [[cofibrantly generated model category|cofibrantly generated]]

* $I$ is cofibrant;

* the [[model structure on an over category|model structure on the overcategory]] $V/I$ has a [[symmetric monoidal functor|symmetric monoidal]] fibrant [[resolution|replacement]] functor;

* $V$ admits a commutative Hopf interval object.

Then there exists a [[cofibrantly generated model category]] structure on the category of reduced $V$-[[operad]]s, in which

* a morphism $P \to Q$ is a weak equivalence (resp. fibration) precisely if for all $n \gt 0$ the morphisms $P(n) \to Q(n)$ are weak equivalences (resp. fibrations) in $V$.

=--

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

This is [BergerMoerdijk, theorem 3.1](#BergerMoerdijkHomotopy).

=--

If $V$ is even a [[cartesian closed category]], a stronger statement is possible:

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

Let $V$ be a [[cartesian closed category]], such that

* $V$ is [[cofibrantly generated model category|cofibrantly generated]] and the [[terminal object]] is cofibrant;

* $V$ has a symmetric monoidal fibrant replacement functor.

Then there exists a cofibrantly generated model structure on the category of $V$-operads, in which a morphism $P \to Q$ is a weak equivalence (resp. fibration) precisely if for all $n \geq 0$ the morphisms $P(n) \to Q(n)$ are weak equivalences (resp. fibrations) in $V$.

=--

## Examples

The conditions of the above theorems are satisfied for

* $V = $ [[sSet]] with the [[classical model structure on simplicial sets]].

  The induced model structure on $sSet$-operads is [[Quillen equivalence|Quillen equivalent]] to the [[model structure on dendroidal sets]].

* $V = $ [[Top]] the equivalent [[classical model structure on topological spaces]] ([[compactly generated topological spaces]]);

  The _homotopy algebras_ over a simplicial/topological operad as defined by Boardman and Vogt (see references below), are algebras for cofibrant replacements of these operads in this model structure. This is essentially the statement of theorem 4.1 in ([Vogt](#Vogt))

* $V = Ch_\bullet$, the [[model structure on chain complexes]];

* $V = sSh(C)$ the [[model structure on simplicial sheaves]] on some [[site]] $C$.

In these contexts,

* the [[associative operad]] is admissible $\Sigma$-cofibrant

  ([BergerMoerdijk, page 15](#BergerMoerdijkHomotopy))

* the [[commutative operad]] is far from being $\Sigma$-cofibrant.

This means we have rectification theorems for [[A-∞ algebras]] but not for [[E-∞ algebras]]. See [[model structure on algebras over an operad]] for more.

## Properties

### Cofibrancy

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

Every cofibrant operad is also $\Sigma$-[cofibrant](#SigmaCofibrancy).

=--

This is ([BergerMoerdijk, prop. 4.3](#BergerMoerdijkHomotopy)).

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

The relevance of this is in section [Homotopy algebras](#HomotopyAlgebras): this property enters the proof of the statement that the [[model structure on algebras over an operad]] over a $\Sigma$-cofibrant resolution is already Quillen equivalent to that of a full cofibrant resolution.

Many resolutions of operads that appear in the literature are in fact just $\Sigma$-cofibrant.

=--

### Resolutions {#Resolutions}

We now discuss the construction and properties of cofibrant [[resolution]]s of operads and their algebras.

> (assumptions now as at [[model structure on algebras over an operad]])

First we describe _free_ operads, and then _[[Boardman-Vogt resolution]]s_ of operads, obtained from the construction of the free ones by adding labels for _lengths_ in an [[interval object]]

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

The category of $C$-coloured operads is itself the category of [[algebra over an operad|algebras over]] a [[non-symmetric operad]]. See [[coloured operad]] for more. Thus the above theorem provides conditions under which $C$-coloured operads carry a model structure in which fibrations and weak equivalences are those morphisms of operads $P \to Q$ that are  degreewise fibrations and weak equivalences in $\mathcal{E}$.

=--

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

We shall from now on call an operad $P$ **cofibrant** if the morphism $I_C \to P$ from the initial $C$-[[coloured operad]] has the [[left lifting property]] against degreewise acyclic fibrations of operads (irrespective of whether the above conditions for the existence of the model structure hold).

=--

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

The [[forgetful functor]] from $C$-colored operads to pointed $C$-colored collections has a [[left adjoint]]

$$
  (F^*_C \dashv U_C)
  :
  Oper_C(\mathcal{E})
  \stackrel{\leftarrow}{\to}
  Coll_C^*(\mathcal{E})
  \,.
$$

=--

This is  ([BergerMoerdijk, theorem 3.2](#BergerMoerdijkAlgebras)).

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

For each well-pointed  $\Sigma$-[cofibrant](#SigmaCofibrancy) $C$-coloured operad $P$, the $(F^*_C \dashv U_C)$-counit factors as a cofibration followed by a weak equivalence

$$
  F_C^*(P) \hookrightarrow W(H,P) \stackrel{\simeq}{\to} P
$$

of $C$-coloured operads, [[natural transformation|naturally]] in $P$ and $H$.

If $P \to Q$ is a $\Sigma$-cofibration between well-pointed $\Sigma$-cofibrant $C$-coloured operads, then the induced map $W(H,P) \to W(H,Q)$ is a cofibration of cofibrant $C$-coloured operads.

=--

This is  ([BergerMoerdijk, theorem 3.5](#BergerMoerdijkAlgebras)).

Here $W(H,P)$ is also called the **coloured [[Boardman-Vogt resolution]]** of $P$.

An [[algebra over an operad]] over $W(H,P)$ is called a **$P$-algebra up to homotopy**.

### Homotopy algebras over an operad {#HomotopyAlgebras}

We discuss model structures on algebras over resolutions of operads. A more  detailed treatment is at [[model structure on algebras over an operad]].

With $V$ as above, say

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

A $V$-operad $P$ is _admissible_ if the category of $P$-[[algebra over an operad|algebras]] carries a [[transferred model structure]] from the [[free functor]]/[[forgetful functor]] [[adjunction]]

$$
  F_P : V \stackrel{\leftarrow}{\to} Alg_P : U_P
  \,.
$$

=--

Under mild assumptions on $V$, cofibrant operads are admissible.

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

For an arbirtrary $V$-operad $P$, [[generalized the|the]] category of **homotopy $P$-algebras** is the category of $\hat P$-algebras for some cofibrant replacement $\hat P$ of $P$.

=--

Indeed, this is well defined up to [[Quillen equivalence]]:

[BerMor03, corollary 4.5](http://arxiv.org/PS_cache/math/pdf/0206/0206094v3.pdf#page=15).

Moreover, for this it is sufficient that $\hat P$ be _$\Sigma$-[cofibrant](#SigmaCofibrancy)_ .

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

If $V$ is a [[left proper model category]] with cofibrant unit, then for $\hat P$ a $\Sigma$-[cofibrant](#SigmaCofibrancy) [[resolution]] of $P$ (not necessarily fully cofibrant!) the category of $\hat P$ algebras is [[Quillen equivalence|Quillen equivalent]] to that of homotopy $P$-algebras.

=--

For instance the [[associative operad]] is $\Sigma$-cofibrant, so that by the above every $A-\infty$-algebra may be rectified to an ordinary monoid.

See around [BerMor03, remark 4.6](http://arxiv.org/PS_cache/math/pdf/0206/0206094v3.pdf#page=15).

For more see [[model structure on algebras over an operad]].

### Relation to dendroidal sets

For enrichment in $\mathcal{E} = $ [[Top]] or [[sSet]],
the [[dendroidal homotopy coherent nerve]] induces a [[Quillen equivalence]]
between the model structure on coloured [[topological operads]]/[[simplicial operads]] and the [[model structure on dendroidal sets]]. (See there for more details.)

## Related concepts

* [[model structure on enriched categories]]

* [[(∞,1)-operad]], **model structure on operads**

  * [[canonical model structure on Operad]], [[model structure on dendroidal sets]]

  * [[algebra over an (∞,1)-operad]], [[model structure on algebras over an operad]]

    * [[module over an algebra over an (∞,1)-operad]], [[model structure on modules over an algebra over an operad]]

* [[A-∞ operad]]

* [[E-∞ operad]]

## References

An influential article in which many of the homotopical and $(\infty,1)$-categorical aspects of operad theory originate is

* [[Michael Boardman]] and [[Rainer Vogt]], _Homotopy invariant algebraic structures on topological spaces_ , Lect. Notes Math. 347 (1973).

An early notion of [[resolution]] of operads in [[chain complex]]es is given in section 3 of

* [[Martin Markl]], _Models for Operads_ ([arXiv](http://arxiv.org/abs/hep-th/9411208))

Cofibrant [[Boardman-Vogt resolution]]s of operads are discussed in

* [[Rainer Vogt]], _Cofibrant operads and universal $E_\infty$-operads_ ,
Bielefeld SB 343 (1999), to appear in Topology Appl.

A systematic study of [[model category]] structures on monochromatic [[symmetric operads]] and their algebras is in

* {#BergerMoerdijkHomotopy} [[Clemens Berger]], [[Ieke Moerdijk]], _Axiomatic homotopy theory for operads_ Comment. Math. Helv. 78 (2003), 805–831. ([arXiv:math/0206094](http://arxiv.org/abs/math/0206094))

The generalization to a model structure on [[coloured operads|coloured symmetric operads]] ([[symmetric multicategories]]) is discussed in

* {#CisinskiMoerdijk11} [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets and simplicial operads_ ([arXiv:1109.1004](http://arxiv.org/abs/1109.1004))

and independently in

* [[Marcy Robertson]], _The Homotopy Theory of Simplicially Enriched Multicategories_ ([arXiv](http://arxiv.org/abs/1111.4146))

And the generalization to [[colored operads]] over more general suitable enriching categories is in

* {#Caviglia14} [[Giovanni Caviglia]], _A Model Structure for Enriched Coloured Operads_ ([arXiv:1401.6983](http://arxiv.org/abs/1401.6983))

(generalizing a corresponding [[model structure on enriched categories]]).

An explicit construction of cofibrant resolution in this model structure and its relation to the original constructon of the [[Boardman-Vogt resolution]] is in

* {#BergerMoerdijkResolution} [[Clemens Berger]], [[Ieke Moerdijk]], _The Boardman-Vogt resolution of operads in monoidal model categories_ , Topology 45 (2006), 807–849. ([pdf](http://math.unice.fr/~cberger/BV.pdf))

The induced model structures and their properties on [[algebras over operads]] are discussed in

* {#BergerMoerdijkAlgebras} [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_ ([arXiv:math/0512576](http://arxiv.org/abs/math/0512576))

The [[model structure on dg-operads]] is discussed in

* {#Hinich} [[Vladimir Hinich]],  _Homological algebra of homotopy algebras_ Communications in algebra, 25(10). 3291-3323 (1997)([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), _Erratum_ ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453)))

[[!redirects model structure on topological operads]]