model structure on cubical sets

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## Idea

There is a [[model category]] structure on the [[category]] $[\Box^{op},Set]$ of [[cubical set]]s whose [[homotopy theory]] is that of the [[classical model structure on simplicial sets]].

Using this version of the [[homotopy hypothesis]]-theorem, cubical sets are a way to describe the [[homotopy type]] of [[∞-groupoid]]s using of all the [[geometric shapes for higher structures]] the [[cube]].

## Definition

There is an evident [[simplicial set]]-valued [[functor]]

$$
  \Box \to sSet
$$

from the [[cube category]] to [[sSet]], which sends the cubical $n$-cube to the simplicial $n$-cube

$$
  \mathbf{1}^n \mapsto (\Delta[1])^{\times n}
  \,.
$$

Similarly there is a canonical [[Top]]-valued functor

$$
  \Box \to Top
$$

$$
  \mathbf{1}^n \mapsto (\Delta^1_{Top})^n
  \,.
$$

The corresponding [[nerve and realization]] [[adjunction]]

$$
  (|-| \dashv Sing_\Box) : Top \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing_\Box}{\to}} Set^{\Box^{op}}
$$

is the cubical analogue of the simplicial nerve and realization discussed [above](#ForKanComplexes).

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

There is a [[model structure on cubical sets]] $Set^{\Box^{op}}$ whose

* weak equivalences are the morphisms that become weak equivalences under geometric realization $|-|$;

* cofibrations are the [[monomorphism]]s.

=--

This is ([Jardine, section 3](#Jardine)).

Explicitly, a set of generating cofibrations is given by the boundary inclusions $\partial \Box^n \to \Box^n$, and a set of generating acyclic cofibrations is given by the horn inclusions $\sqcap_{k,\epsilon}^n \to \Box^n$. This is ([Cisinski, Thm 8.4.38](#Cisinski)). Thus, as a consequence of Cisinski's work, the fibrations are exactly cubical Kan fibrations.

## Properties

### Homotopy theory
 {#HomotopyTheory}

The following theorem establishes a form of the [[homotopy hypothesis]] for cubical sets.

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

The [[unit of an adjunction|unit of the adjunction]]

$$
  A \to Sing_\Box(|A|)
$$

is a weak equivalence in $Set^{{\Box}^{op}}$ for every cubical set $A$.

The counit of the adjunction

$$
  |Sing_\Box X| \to X
$$

is a weak equivalence in $Top$ for every topological space $X$.

It follows that we have an [[equivalence of categories]] induced on the [[homotopy categories]]

$$
  Ho(Top) \simeq Ho(Set^{\Box^{op}})
  \,.
$$

=--

This is ([Jardine, theorem 29, corollary 30](#Jardine)).

In fact, by the discussion at _[[adjoint (∞,1)-functor]]_  it follow that the [[derived functors]] of the adjunction exhibit the [[simplicial localizations]] of cubical sets equivalent to that of simplicial sets, hence makes their [[(∞,1)-categories]] [[equivalence of (∞,1)-categories|equivalent]] (hence equivalent to [[∞Grpd]]).

## Other kinds of cubical sets

In [[cubical type theory]] one uses more structured notions of cubical set (symmetric, cartesian, De Morgan, etc.)  In most cases such categories have both a test model structure, which is equivalent to spaces, and a [[cubical-type model structure]] that corresponds to the interpretation of type theory.  In many cases these are *not* equivalent, and the cubical-type model structure does not model classical homotopy types.  See [[cubical-type model structure]] for more discussion.

## Joyal-type model structures

In complete analogy to [[simplicial sets]], there is also an analogue of the [[Joyal model structure]] on [[cubical sets]], with or without connection.  See the article [[model structures for cubical quasicategories]].

## Related concepts

* [[cubical set]]

* [[cubical set with connections]]

* [[test category]], [[strict test category]]

* [[Cisinski model structure]]

* [[model structures for cubical quasicategories]]

## References

### Cubical sets without connections

Using that the [[cube category]] is a [[test category]] a model structure on cubical sets follows as a special case of the [[model structure on presheaves over a test category]], due to

* [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]_, Asterisque __308__.
  {#Cisinksi}

Cisinski also derives explicit generating cofibrations and generating acyclic cofibrations using his theory of [[generalized Reedy category]], or _categories skelettiques_. See Section 8.4.

The model structure on cubical sets as above is given in detail in

* [[J. F. Jardine]], _Cubical homotopy theory: a beginning_ (2002) ([pdf](https://web.archive.org/web/20210507014113/http://hopf.math.purdue.edu/Jardine/cubical2.pdf))
{#Jardine}

### Cubical sets with max-connections

The following paper proves that [[cubical sets with connections]] (more specifically, max-connections) form a [[strict test category]] and therefore admit a [[cartesian model structure]] that is Quillen equivalent to the [[Kan–Quillen model structure]] on [[simplicial sets]]:

* [[Georges Maltsiniotis]], _La catégorie cubique avec connexions est une catégorie test stricte_, Homology, Homotopy and Applications 11:2 (2009), 309-326.  [doi](http://dx.doi.org/10.4310/hha.2009.v11.n2.a15).

### Cubical sets with max-connections and min-connections

The case of cubical sets with both max-connections and min-connections largely follows the case of cubical sets with max-connections, the corresponding category of cubes again being a [[strict test category]].
The relevant results are stated explicitly as Corollary 3 and Theorem 3 of

* Ulrik Buchholtz, Edward Morehouse, _Varieties of Cubical Sets_, [arXiv](https://arxiv.org/abs/1701.08189).

### Cartesian cubical sets

The case of the category of presheaves over the free finite product category with a bipointed object is treated in:

* [[Steve Awodey]], _Cartesian cubical model categories_ ([arXiv:2305.00893](https://arxiv.org/abs/2305.00893))

[[!redirects model structures on cubical sets]]


model structure on DblCat

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## Idea

There are various [[model category]]-[[structures]] on the category $DblCat$ of [[double categories]].

## Related concepts

* [[canonical model structure on 2-categories]]

## References

* {#FPPModel} [[Thomas M. Fiore]], [[Simona Paoli]], [[Dorette A. Pronk]], *Model structures on the category of small double categories*, _Algebraic & Geometric Topology_ **8** (2008) 1855–1959 [[doi:10.2140/agt.2008.8.1855](https://doi.org/10.2140/agt.2008.8.1855)]

* {#FPThomason} [[Thomas Fiore]], [[Simona Paoli]], *A Thomason model structure on the category of small $n$-fold categories*_Algebraic & Geometric Topology_ **10** (2010) 1933–2008 [[doi:10.2140/agt.2010.10.1933](http://doi.org/10.2140/agt.2010.10.1933)]

* {#MSV2Cat} [[Lyne Moser]], Maru Sarazola, Paula Verdugo, *A $2Cat$-inspired model structure for double categories*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **LXIII**2 (2022) [[arxiv:2004.14233](https://arxiv.org/abs/2004.14233), journal:[pdf](http://cahierstgdc.com/wp-content/uploads/2022/04/MoserSarazolaVerdugo-LXIII-2.pdf)]

* {#CampbellGreg} [[Alexander Campbell]], *The gregarious model structure for double categories* (2020) [[talk slides pdf](https://acmbl.github.io/greg_slides.pdf)]

* {#MSVHInvar} [[Lyne Moser]], Maru Sarazola, Paula Verdugo,  *A model structure for weakly horizontally invariant double categories*, Algebraic & Geometric Topology **23** 4 (2023) 1725-1786 [[arxiv:2007.00588](https://arxiv.org/abs/2007.00588), journal:[pdf](https://msp.org/agt/2023/23-4/agt-v23-n4-p08-p.pdf)]

Review:

* [[Lyne Moser]], *Model Structures for Double Categories*, [talk at](CQTS#MoserOct2023) *[Workshop on Homotopy Theory and Applications](CQTS#WorkshopOnHomotopyTheory2023)*, [[CQTS]] (Oct 2023) [video:[YT](https://youtu.be/8j1WvFesnQM)]

[[!redirects model structures on DblCat]]


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## Statement

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

The [[category]] of [[Delta-generated topological spaces]] carries the [[mathematical structure|structure]] of a [[cofibrantly generated model category]] with the same generating (acyclic) cofibrations as for the [[classical model structure on topological spaces]] and such that the [[coreflective subcategory|coreflection]] into all [[topological spaces]] ([this Prop.](Delta-generated+topological+space#AdjunctionBetweenTopologicalSpacesAndDiffeologicalSpaces)) is a [[Quillen equivalence]] with the [[classical model structure on topological spaces]]:

$$
  Top_{Qu}
  \underoverset
    {\underset{Cdfflg}{\longrightarrow}}
    {\hookleftarrow}
    {\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;}
  D Top_{Qu}
$$

=--

([Haraguchi 13, Theorem 3.3](#Haraguchi13))

By the same kind of argument ([Gaucher 2007](#Gaucher07)), this factors by a [[Quillen equivalence]] through the [[model structure on compactly generated topological spaces]]:

$$
  Top_{Qu}
  \underoverset
    { \underset{ k }{\longrightarrow} }
    { {\hookleftarrow} }
    { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; }
  k Top_{Qu}
  \underoverset
    { \underset{ D }{\longrightarrow} }
    { {\hookleftarrow} }
    { \;\;\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;\;\; }
  D Top_{Qu}
  \,.
$$

## Related concepts

* [[model structure on topological spaces]]

  * [[classical model structure on topological spaces]] ([[classical model structure on pointed topological spaces|on pointed spaces]])

    * [[model structure on compactly-generated topological spaces]]

    * [[model structure on Delta-generated topological spaces]]

  * [[Strøm model structure]]

* [[model structure on diffeological spaces]]

## References

* {#Gaucher07} [[Philippe Gaucher]], p. 7 of: *Homotopical interpretation of globular complex by multipointed d-space*, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 ([arXiv:0710.3553](https://arxiv.org/abs/0710.3553))

* {#Haraguchi13} [[Tadayuki Haraguchi]], _On model structure for coreflective subcategories of a model category_, Math. J. Okayama Univ. 57 (2015), 79–84 ([arXiv:1304.3622](http://arxiv.org/abs/1304.3622), [doi:10.18926/mjou/53040](http://doi.org/10.18926/mjou/53040), MR3289294, Zbl 1311.55027)

See also:

* {#SterlingKapulkin23} [[Sterling Ebel]], [[Chris Kapulkin]], Thm. 4.9 in: *Synthetic approach to the Quillen model structure on topological spaces* [[arXiv:2310.14235](https://arxiv.org/abs/2310.14235)]

[[!redirects model structure on Delta-generated spaces]]

[[!redirects model structure on D-topological spaces]]


model structure on dendroidal complete Segal spaces

see _[[model structure for dendroidal complete Segal spaces]]_


model structure on dendroidal sets

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## Idea

The [[Denis-Charles Cisinski|Cisinski]]--[[Ieke Moerdijk|Moerdijk]] [[model category]] structure on the [[category]] [[dSet]] of [[dendroidal sets]] models [[(∞,1)-operads]] in generalization of the way the [[Andre Joyal|Joyal]] [[model structure on simplicial sets]] models [[(∞,1)-category|(∞,1)-categories]].

## Overview

We have the following diagram of [[model category|model categories]]:

$$
  \array{
    SSet\text{-}Operad
     &\stackrel{\simeq}{\to}&
    dSet
     &\stackrel{\simeq}{\to}&
    dSpaces
    &&&&& models\;for\;(\infty,1)\text{-}operads
    \\
    \uparrow && \uparrow && \uparrow
    \\
    SSet\text{-}Cat
     &\stackrel{\simeq}{\to}&
    sSet
     &\stackrel{\simeq}{\to}&
    sSpaces
    &&&&&
    models\;for\;(\infty,1)\text{-}categories
  }
  \,,
$$

where the entries are

* the category $SSet Cat$ of [[simplicially enriched category|simplicially enriched categories]] equipped with the [[Julie Bergner|Bergner]] model structure;

* the category [[SSet]] of [[simplicial set]]s equipped with the [[model structure on simplicial sets|Joyal model structure]] for [[quasi-category|quasi-categories]];

* the category $dSet$ of [[dendroidal set]]s

and where

* the horizontal morphisms are [[Quillen equivalence]]s

* the vertical morphisms are [[reflective sub-(infinity,1)-category|homotopy full embeddings]].

## Definition

### Special morphisms ###

Recall from the entry on [[dendroidal set]]s the definition of inner and outer faces, boundaries and inner and outer horns.

The following definition are the obvious generalizations of the corresponding notions for the [[model structure on simplicial sets]], in particular for the [[model structure for quasi-categories]].

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

The class of morphisms in $dSet$ generated from the inner horn inclusions $\Lambda^e \Omega[T] \to \Omega[T]$ under

* [[pushout]]

* [[transfinite composition]]

* [[retract]]s

is called the **inner [[anodyne extensions]]**.

=--

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

The class of morphisms in $dSet$ generated from boundary inclusions
$\delta \Omega[T] \to \Omega[T]$ under

* [[pushout]]

* [[transfinite composition]]

* [[retracts]]

is called the **[normal monomorphisms](http://ncatlab.org/nlab/show/dendroidal%20set#NormalMonomorphisms)**.

=--

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

A [[morphism]] $A \to B$ in $dSet$ is an **inner Kan fibration** if it has the [[right lifting property]] with respect to all inner horn inclusions.

$$
  \array{
    \Lambda^e[T] &\to& A
     \\
    \downarrow && \downarrow
    \\
    \Omega[T] &\to& B
  }
$$

or equivalently with respect to the class of inner anodyne extensions.

=--

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

A dendroidal set $X$ is an **inner Kan complex** or **quasi-operad** if the canonical morphism $X\to {*}$ to the [[terminal object]] is an inner Kan fibration.

=--

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

A morphism $A \to B$ of dendroidal sets is an **acyclic fibration** if it has the [[right lifting property]] with respect to all normal monomorphisms.

=--

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

A morphism $f : X \to Y$ of dendroidal sets is called an **isofibration** if it

1. it is an inner Kan fibration;

1. the morphism of operads $\tau_d(f) : \tau_d(X) \to \tau_d(Y)$ is a fibration in the [[canonical model structure on operads]], hence the underlying [[functor]] of categories is an [[isofibration]].

=--

### The model structure

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

On the category of [[dendroidal sets]] let

* the cofibrations be the [normal monomorphisms](#NormalMonomorphism);

* the fibrant objects are the weak Kan complexes/[quasi-operads](#QuasiOperad);

* the fibrations between fibrant objects are precisely the [isofibrations](#Isofibration).

* the weak equivalences form the smallest class of maps that satisfy

  * [[category with weak equivalences|2-out-of-3]]

  * every [inner anodyne extension](#InnerAnodyneExtension) is a weak equivalence

  * every acyclic fibration between quasi-operads is a weak equivalence.

=--

## Properties
  {#Properties}

### Establishing the model structure

#### Statement
 {#StatementOfTheModelStructure}

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

The above choices of cofibrations, fibrations and weak equivalences equips the category $dSet$ of dendroidal sets with the structure of a [[model category]] $dSet_{CM}$. This model structure is

* a left [[proper model category]]

* a [[cofibrantly generated model category]]

* a [[combinatorial model category]]

* a [[symmetric monoidal category|symmetric]] [[monoidal model category]] (with respect to the [Boardman-Vogt tensor product](http://ncatlab.org/nlab/show/dendroidal%20set#BoardmanVogtTensorProduct)).

=--

This is ([Cisinski Moerdijk, theorem 2.4 and prop. 2.6](#CisinskiMoerdijk09)). We indicate the proof [below](#ProofOfTheModelStructure).

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

The [[cofibrantly generated model category|generating cofibrations]] $I$ are the boundary inclusion of [[trees]]

$$
  I = \{\partial \Omega[T] \hookrightarrow \Omega[T]\}
  \,.
$$

A set of generating acyclic cofibrations is guaranteed to exist, but no good explicit characterization is known to date.

=--

This is ([CisMoe, cor. 6.17](#CisinskiMoerdijk09)).

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

A morphism $j : X \to Y$ between cofibrant objects in $dSet$ is a weak equivalence precisely if for all fibrant objects $A$ the morphism

$$
  \tau : dSet(Y,A) \to \tau dSet(X,A)
$$

is an [[equivalence of categories]], where $\tau : SSet \to Cat$ is the [[left adjoint]] to the [[nerve]].

=--

This is in ([Moerdijk, section 8.4](#Moerdijk)).

#### Proof
 {#ProofOfTheModelStructure}

We state a list of lemmas to establish theorem \ref{TheModelStructureExistenceAndBasicProperties}.

The **strategy** is the following: we [[over topos|slice]] $dSet$ over a normal [[resolution]] $E_\infty$ (a model of the [[E-∞ operad]]) of the terminal object. Since over a normal object all morphisms are normal, we have a chance to establish a _[[Cisinski model structure]]_ on $dSet_{/E_\infty}$. This indeed exists, prop. \ref{ModelStructureOnSlices} below, and the model structure on $dSet$ itself can then be [[transferred model structure|transferred]] along the [[inverse image]] of the [[etale geometric morphism]] $dSet_{/E_\infty} \to dSet$, prop. \ref{TransferOfTheModelStructure}.

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

For $A \to B$ an inner [[anodyne extension]] and $X \to Y$ a normal monomorphism of [[dendroidal sets]], the [[pushout-product axiom|pushout-product morphism]]

$$
  A \otimes Y \cup B \otimes X \to B\otimes Y
$$

(with "$\otimes$" the [[Boardman-Vogt tensor product]]) is an inner anodyne extension.

=--

This is ([Cis-Moer, prop. 3.1](#CisinskiMoerdijk09)).

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

Write $J := \{0 \stackrel{\simeq}{\to} 1\}$ for the [[codiscrete groupoid]] on two objects. We use the same symbol for its image along $N_d i_! : Cat \hookrightarrow dSet$.

The **$J$-anodyne extensions** in $dSet$ are the morphisms generated under [[pushouts]], [[transfinite composition]] and [[retracts]] from the the inner [[anodyne extensions]] and from the [[pushout-product axiom|pushout products]] of $\{e\} \to J$ with the tree boundary inclusion

$$
  \{
    \partial \Omega[T] \otimes J \; \cup \;
    \Omega[T] \otimes \{0\}
    \to
    \Omega[T] \otimes J
  \}_{e \in \{0,1\}, T \in \Omega}
  \,.
$$

Call a morphism a **$J$-fibration** if it has the [[right lifting property]] against $J$-anodyne extensions.

=--

This is ([Cis-Moer, 3.2](#CisinskiMoerdijk09)).

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

For $A \to B$ a $J$-anodyne extension and $X \to Y$ a normal monomorphism of [[dendroidal sets]], the [[pushout-product axiom|pushout-product morphism]]

$$
  A \otimes Y \cup B \otimes X \to B\otimes Y
$$

is a $J$-anodyne extension.

=--

This is ([Cis-Moer, prop. 3.3](#CisinskiMoerdijk09)).

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

For $X \in dSet$, the BV-tensor product with

$$
  \{0\} \coprod \{1\} \to J \to \eta
$$

defines an [[interval object]]

$$
  X \coprod X \to J \otimes X \to X
  \,.
$$

This is compatible with slicing, in that for any $B \in sSet$ and $a : X \to B$ a given morphism, we have an interval object in the [[over category]] $dSet_{/B}$ given by a diagram

$$
  \array{
    X \coprod X &\to& J \otimes X &\to& X
    \\
    & {}_{(a,a)}\searrow & \downarrow & \swarrow_{a}
    \\
    && B
  }
  \,.
$$

If in the above $B$ is normal and $a : X \to B$ is $J$-anodyne, then the [[left homotopies]] given by the above cylinder
defines a notion of [[left homotopy]] in $dSet_{/B}$.

Say a morphism $f : A \to A'$ in $dSet_{/B}$ is a **$B$-equivalence** if it is a homotopy equivalence with respect to this notion of homotopy, hence if for all $J$-fibrations $X \to B$ the induced morphism

$$
  [A',X]_{\sim_B} \to [A,X]_{\sim B}
$$

is a bijection.

=--

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

Let $B \in dSet$ be normal. Then $dSet_{/B}$ carries a [[left proper model category|left proper]] [[cofibrantly generated model category]] for which

* the weak equivalences are the $B$-equivalences;

* the cofibrations are the monomorphisms;

* the fibrant objects are the $J$-fibrations into $B$;

* a morphism between fibrant objects is a fibration precisely if its image in $dSet$ is a $J$-fibration.

=--

This is ([Cis-Moer, prop. 3.5](#CisinskiMoerdijk09)).

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

Notice that

1. every monomorphism over a normal object $B$ is normal,

1. the [[over topos]] (see there) $PSh(\Omega)_{/B}$ may be identified with presheaves on the slice site

   $$
     PSh(\Omega)_{/B} \simeq PSh(\Omega_{/B})
     \,.
   $$

Therefore for normal $B$ the slice $dSet_{/B}$, as opposed to $dSet$ itself, has a chance to carry a [[Cisinski model structure]]. And this is indeed the case: one checks that

* $J \otimes (-)$ is a _functorial cyclinder_ in the sense of _[this definition](http://ncatlab.org/nlab/show/Cisinski+model+structure#FunctorialCylinder)_;

* the class of $J$-anodyne extensions is a class of corresponding anodyne extensions in the sense of _[this definition](http://ncatlab.org/nlab/show/Cisinski+model+structure#AnodyneExtensions)_

* so that together these form a _homotopical structure_ on $dSet_{/B}$ in the sense of _[this definition](http://ncatlab.org/nlab/show/Cisinski+model+structure#HomotopicalStructure)_.

The statement then is a special case of _[this theorem](http://ncatlab.org/nlab/show/Cisinski+model+structure#ModelStructureFromHomotopicalStructure)_ at _[[Cisinski model structure]]_.

=--

Let $E_\infty \in dSet$ be any normal dendroidal set such that the terminal morphism $E_\infty \to *$ is an acyclic fibration in that it has the [[right lifting property]] against the tree boundary inclusions.

By the general discussion at [[over topos]] we have an [[adjunction]]

$$
  (p_! \dashv p^*)
  \,.
  dSet
    \stackrel{\overset{p_!}{\leftarrow}}{\underset{p^*}{\to}}
  dSet_{/E_\infty}
  \,,
$$

where $p_!$ simply forgets the map to $E_\infty$ and where $p^*$ forms the [[product]] with $E_\infty$

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

The _[[transferred model structure]]_ on $dSet$ along the [[right adjoint]] $p^*$ of the model structure from prop. \ref{ModelStructureOnSlices} exists.

This is the model structure characterized in theorem \ref{TheModelStructureExistenceAndBasicProperties}.

=--

This appears as ([Cis-Moer, prop. 3.12](#CisinskiMoerdijk09)).

So far this establishes the existence of the model structure and that every dendroidal inner Kan complex is fibrant. Below in _[characterization of the fibrant objects](#CharacterizationOfTheFibrantObjects)_ we consider the converse statement: that the fibrant objects are precisely the inner Kan complexes.

#### Characterization of the fibrations
 {#CharacterizationOfTheFibrantObjects}

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

A dendroidal set is $J$-fibrant, def. \ref{JFibrations}, hence fibrant in $dSet_{CM}$, precisely if it is an inner Kan complex.

A morphism $f : X \to Y$ in $dSet$ between inner Kan complexes is a $J$-fibration, hence a fibration in $dSet_{CM}$, precisely if

1. it is an [[inner Kan fibration]];

1. on [[homotopy categories]] $\tau i^* f$ is an [[isofibration]].

=--

This is ([Cisinski Moerdijk, theorem 5.10](#CisinskiMoerdijk09)).

### Monoidal model category structure
 {#MonoidalModelCategoryStructure}

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

With respect to the [[Boardman-Vogt tensor product]] on [[dendroidal sets]], the model structure $dSet_{CM}$ is a [[symmetric monoidal category|symmetric]] [[monoidal model category]].

=--

This is ([Cis-Moer, prop. 3.17](#CisinskiMoerdijk09)).

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

With respect to the [[internal hom]] corresponding to the [[Boardman-Vogt tensor product]], $dSet_{CM}$ is a $dSet_{CM}$-[[enriched model category]].

=--

### Other enrichments of the underlying category
 {#Enrichment}

+-- {: .num_prop }
###### Proposition
**(compatibility with the Joyal model structure)**

Let $|$ be the tree with a single leaf and no vertex. Then the [[overcategory]] $dSet/\Omega[|]$ is canonically isomorphic to [[sSet]].

The model structure on [[sSet]] induced this way as the [[model structure on an overcategory]] from the model structure on $dSet$ coincides with the
[[model structure for quasi-categories]].

=--

This is for instance ([Moerdijk, proposition 8.4.3](#Moerdijk)).

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

This model category is naturally an $sSet_{Joyal}$-[[enriched model category]], where
$sSet_{Joyal}$ is the [[model structure for quasi-categories]].

=--

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

This follows from the fact, cor. \ref{EnrichmentOverItself}, that $dSet_{CM}$ is a [[monoidal model category]]
and the fact that the
functor $i^*: dSet \to sSet_{Joyal}$ is a [[right Quillen functor]].

=--

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

However, $dSet_{CM}$ is _not_ an [[enriched model category]] over $sSet_{Quillen}$, the _standard_ [[model structure on simplicial sets]] (but see _[[model structure for dendroidal Cartesian fibrations]]_). But it comes close, as the following propositions show.

=--

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

Write

$$
  \mathcal{Hom} : dSet^{op} \times dSet \to dSet
$$

for the [[internal hom]] corresponding to the [[Boardman-Vogt tensor product]].

For $A$ normal and $X$ an inner Kan dendroidal set, write

$$
  \mathcal{hom}(A,X) := i^* \mathcal{Hom}(A,X)
$$

for the underlying [[quasi-category]], and write

$$
  k(A,X) := Core(\mathcal{hom}(A,X)) \in KanCplx
$$

for the maximal [[Kan complex]] inside the [[quasi-category]] inside the [[internal hom]].

Write

$$
  -^{(-)} : sSet^{op} \times dSet \to dSet
$$

for the corresponding [[powering]], characterized by the existence of a
[[natural isomorphism]]

$$
  Hom_{sSet}(K, k(A,X)) \simeq Hom_{dSet}(A, X^{(K)})
  \,.
$$

=--

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

For $p : X \to Y$ a fibration between fibrant dendroidal sets (hence an [[inner Kan fibration]] and an [[isofibration]] on the underlying [[homotopy category]]), and for $A \to B$ a normal monomorphism, the induced morphism

$$
  k(B,X) \to k(B,Y) \times_{k(A,Y)} k(A,X)
$$

is a [[Kan fibration]] between [[Kan complexes]].

=--

This is  ([Cis-Moer, prop. 6.7](#CisinskiMoerdijk09)).

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

If $A \to B$ is the above is an [[anodyne extension]] (acyclic monomorphism) of simplicial sets, then

$$
  X^{(B)} \to Y^{(B)} \times_{Y^{(A)}} X^{(A)}
$$

is an acyclic fibration in $dSet_{CM}$.
=--

This is  ([Cis-Moer, cor. 6.9](#CisinskiMoerdijk09)).

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

For $A$ normal and $X$ fibrant, the Kan complex

$$
  k(A,X)
  \simeq
  \mathbb{R}Hom(A,X)
$$

is the correct
[[derived hom-space]] of $dSet_{CM}$.

=--

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

One checks that $n \mapsto X^{(\Delta[n])}$ is a fibrant resolution of
$X$ in the [[Reedy model structure]] $[\Delta^{op}, dSet_{CM}]_{Reedy}$.
By the discussion at _[[simplicial model category]]_ and _[[derived hom-space]]_ the latter is therefore given by the simplicial set

$$
  n \mapsto Hom_{dSet}(A, X^{(\Delta[n])})
  \,.
$$

By the [[tensoring]]-definition of $X^{(\Delta[n])}$ this is isomorphic to

$$
  \cdots = Hom_{sSet}(\Delta[n], k(A,X)) = k(A,X)_n
  \,.
$$

=--

### Relation to other model structures

We discuss the relation of the model structure on dendroidal sets to other model category structures for operads.

See the _[[table - models for (infinity,1)-operads]]_ for an overview.

#### Model structure for quasi-categories

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

The [[adjunction]]

$$
  (j_! \dashv j^*) :
   dSet
    \stackrel{\overset{j_!}{\leftarrow}}{\underset{j^*}{\to}}
   sSet
$$

induced from the inclusion $j : \Delta \hookrightarrow \Omega$
constitutes a [[Quillen adjunction]]
between the above model structure on dendroidal sets, and the
[[model structure for quasi-categories]].

=--

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

By the proof of ([Cisinski-Moerdijk, cor. 2.10](#CisinskiMoerdijk09)) the model structure for quasi-categories is in fact the restriction, along $j_!$, of the model structure on dendroidal sets. Therefore $j_!$ is left Quillen.

=--

#### Model structure for dendroidal complete Segal spaces
 {#RelationToModelStructureForDendroidalCompleteSegal}

There is a [[Quillen equivalence]] to the [[model structure for dendroidal complete Segal spaces]] (see there). A crucial step in the proof is the following expression of the acyclic cofibrations on $dSet_{CM}$ in terms of the dendroidal interval $J_d$ as follows.

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

The class of acyclic cofibrations between normal dendroidal sets is the smallest
class of morphisms between normal dendroidal sets

* which contains the $J$-anodyne extensions;

* with left cancellation property: if a composite $\stackrel{i}{\to} \stackrel{j}{\to}$ is in the class and $i$ is, then so is $j$.

=--

([Cis-Moer 09, prop. 3.16](#CisinskiMoerdijk09))

#### Model structure on simplicial operads
 {#RelationToModelStrictureOnSimplicialOperads}

There exists also a [[model structure on simplicial operads]], which is
[[Quillen equivalence|Quillen equivalent]] to the model structure on dendroidal sets.

This Quillen equivalence is an operadic generalization of the Quillen equivalence between the [[model structure on sSet-categories]] and the [[model structure for quasi-categories]].

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

There exists an [[adjunction]]

$$
  (W_! \dashv hcN_d)
   :
  sSet Operad
    \stackrel{\overset{W_!}{\leftarrow}}{\underset{hcN_d}{\to}}
  dSet
  \,,
$$

Whise [[right adjoint]] is the [[dendroidal homotopy coherent nerve]].

This is a [[Quillen equivalence]] between the model structure on dendroidal sets, and the [[model structure on simplicial operads]].

=--

This is  ([Cisinski-Moerdijk 11, theorem 815](#CisinskiMoerdijk11)).

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

Under the inclusions (see the discussion at _[[dendroidal set]]_)

$$
  \array{
     sSet Cat &\hookrightarrow & sSet Operad
     \\
     && {}^{\mathllap{hcN_d}}\downarrow \uparrow^{\mathrlap{W_!}}
     \\
     sSet \simeq dSet/\eta &\hookrightarrow & dSet
  }
$$

this restricts to the Quillen equivalence
between the
[[model structure on sSet-categories]] and the [[model structure for quasi-categories]] discussed at _[[relation between quasi-categories and simplicial categories]]_.

=--

#### Model structure on $Set$-Operads

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

Write [[Operad]] for the category of [[symmetric operads]] over [[Set]].

The [[dendroidal set|dendroidal nerve adjunction]]

$$
  (\tau_d \dashv N_d) :
  Operad
   \stackrel{\overset{\tau_d}{\leftarrow}}{\underset{N_d}{\to}}
  dSet
$$

is a [[Quillen adjunction]] between the model structure on dendroidal sets, def. \ref{ModelStructureOnDendroidalSets}, and the [[canonical model structure on Operad]].

Moreover, $N_d$ detects and preserves weak equivalences, while $\tau_d$ preserves weak equivalences.

=--

This is ([Cisinski-Moerdijk 09, prop. 2.5](#CisinskiMoerdijk09)).

#### Model structure for symmetric monoidal $(\infty,1)$-categories

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

There is a [[Quillen adjunction]]

$$
  dSet_{CM}
   \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}}
  dSet_{He}
$$

which exhibits the [[model structure for dendroidal left fibrations]] as a [[Bousfield localization of model categories|left Bousfield localization]] of the Cisinski-Moerdijk model structure on dendroidal sets.

=--

See ([Heuts, remark 6.8.0.2](#Heuts)).

## Related concepts

* [[canonical model structure on Operad]],

* [[model structure on operads]]

* [[model structure for dendroidal complete Segal spaces]]

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

## References

A useful discussion of of the model structure on dendroidal sets is section 8 of

* [[Ieke Moerdijk]], _Lectures on dendroidal sets_ , lectures given at the Barcelona workshop on _[Simplicial methods in higher categories](http://www.crm.es/HigherCategories/)_ (2008) ([preliminary writeup](http://www.crm.es/Publications/quaderns/Quadern45-1.pdf))
 {#Moerdijk}

An expanded and polished version has meanwhile been written up by Javier Guitiérrez and should appear in print soon. An electronic copy is probably available on request.

The model structure was originally given in

* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets as models for homotopy operads_ ([arXiv:0902.1954](http://arxiv.org/abs/0902.1954))
 {#CisinskiMoerdijk09}

making heavy use of results on [[Cisinski model structures]] from

* [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]_, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf))
 {#Cisinski}

A detailed discussion of the fibrant objects in the model structure is in

* [[Ieke Moerdijk]] [[Ittay Weiss]], _On inner Kan complexes in the category of dendroidal sets_ ([web](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6W9F-4VM2KC8-1&_user=457046&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1115574112&_rerunOrigin=google&_acct=C000021878&_version=1&_urlVersion=0&_userid=457046&md5=5eb2307e02ed1aa9e6fe2c7809346546))

The proof of the Quillen equivalence between the model structure on dendroidal sets and that on $sSet$-operads is given in

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

The relation to the [[model structure for dendroidal Cartesian fibrations]] and the [[model structure for dendroidal left fibrations]] is discussed in

* [[Gijs Heuts]], _Algebras over infinity-Operads_ ([arXiv:1110.1776](http://arxiv.org/abs/1110.1776))


model structure on dg-algebras

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

#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] structure on a category of [[differential graded algebras]] or more specifically on a [[category of differential graded-commutative algebras]] tends to present an [[(∞,1)-category]] of [[∞-algebra over an (∞,1)-algebraic theory|∞-algebras]].

For dg-algebras bounded in negative or positive degrees, the [[monoidal Dold-Kan correspondence]] asserts that their model category structures are [[Quillen equivalence|Quillen equivalent]] to the corresponding [[model structure on simplicial algebras|model structure on (co)simplicial algebras]]. This case plays a central role in [[rational homotopy theory]].

The case of model structures on unbounded dg-algebras may be thought of as induced from this by passage to the [[derived geometry]] modeled on formal duals of the bounded dg-algebras. This is described at [[dg-geometry]].

## General

The category of [[dg-algebra]]s is that of [[monoid]]s in a [[category of chain complexes]]. Accordingly general results on a [[model structure on monoids in a monoidal model category]] apply.

Below we spell out special cases, such as restricting to [[commutative monoids]] when working over a [[ground field]] of [[characteristic zero]], or restricting to non-negatively graded cochain dg-algebras.

## On connective dgc-algebras

We discuss the projective model structure on
[[differential graded-commutative algebras|differential non-negatively graded-commutative algebras]]. This was originally introduced in [Bousfield-Gugenheim 76](#BousfieldGugenheim76) as a [[model category]] for [[Dennis Sullivan]]'s approach to [[rational homotopy theory]].

### Definition

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

For $k$ a [[field]] of [[characteristic zero]], write

\[
  \label{CategoryOfdgcAlgebras}
  dgcAlg^{\geq 0}_{k}
  \;\in\;
  Categories
\]

for the [[category]] of [[associative unital algebra|unital]] [[differential graded-commutative algebras]] over $k$ in non-negative degrees, equivalently the category of [[commutative monoids]] in the [[symmetric monoidal category]] $Ch^{\geq 0}(k)$ of [[cochain complexes]] in non-negative degrees, equipped with the [[tensor product of chain complexes]].

=--

([Gelfand-Manin 96, V.3.1](#GelfandManin96))

+-- {: .num_example}
###### Example
**([[initial object|initial]] and [[terminal object]])**

In $dgcAlg^{\geq 0}_{k}$ (eq:CategoryOfdgcAlgebras):

1. the [[initial object]] is the ground field algebra $k$;

1. the [[terminal object]] is the zero algebra $0$ (which is indeed a unital algebra).

=--

(Beware that this is incorrectly stated in [Gelfand-Manin 96, p. 335](#GelfandManin96))

More generally:

+-- {: .num_example}
###### Example
**([[coproducts]] and [[products]])**

In $dgcAlg^{\geq 0}_{k}$ (eq:CategoryOfdgcAlgebras):

1. the [[coproduct]] is given by the [[tensor product of algebras]];

   (see at _[pushouts of commutative monoids](category+of+monoids#PushoutOfCommutativeMonoids)_)

1. the [[product]] is given by [[direct sum]] on [[underlying]] [[graded vector spaces]]

   (since the [[forgetful functor]] is a [[right adjoint]]).

=--

+-- {: .num_defn #dgcCochainAlgebraInNonNegDegreeOfFiniteType}
###### Definition
**(finite type)**

Say that a [[dgc-algebra]] $A \in dgcAlg^{\geq 0}_k$ (def. \ref{dgcCochainAlgebrasInNonNegativeDegrees}) is of _[[finite type]]_ if its [[forgetful functor|underlying]] [[chain complex]] is in each degree of [[finite number|finite]] [[dimension]] as a $k$-[[vector space]].

=--

+-- {: .num_defn #ProjectiveModelStructureOnCdgAlg}
###### Definition
**(projective model structure on rational connective dgc-algebras)**

Write $(dgcAlg^{\geq 0}_k)_{proj}$ for the catgory of [[dgc-algebras]] from def. \ref{dgcCochainAlgebrasInNonNegativeDegrees} equipped with the following [[classes]] of morphisms:

* _[[weak equivalences]]_ are the [[quasi-isomorphism]];

* _[[fibrations]]_ are the degreewise [[surjections]];

=--

([Bousfield-Gugenheim 76, Def. 4.2](#BousfieldGugenheim76), [Gelfand-Manin 96, Def. V.3.3](#GelfandManin96))

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

The category $(dgcAlg^{\geq 0}_k)_{proj}$ from def. \ref{ProjectiveModelStructureOnCdgAlg} is a [[model category]], to be called the _projective model structure_.

=--

([Bousfield-Gugenheim 76, Theorem 4.3](#BousfieldGugenheim76), [Gelfand-Manin 96, Theorem V.3.4](#GelfandManin96))

+-- {: .num_remark }
###### Remark
**([[category of fibrant objects]])**

Evidently every [[object]] in $(dgcAlg^{\geq 0}_k)_{proj}$ (Def. \ref{ProjectiveModelStructureOnCdgAlg}, prop. \ref{IndeedProjectiveModelStructureOnCdgAlg}) is a [[fibrant object]]. Therefore these model categories structures are in particular also structures of a [[category of fibrant objects]].

=--

The nature of the cofibrations is discussed [below](#SullivanAlgebras).

### Properties

#### Cofibrations and Sullivan algebras
 {#SullivanAlgebras}

+-- {: .num_defn }
###### Definition
**(sphere and disk algebras)**

Write $k[n]$ for the graded vector space which is the ground field $k$ in degree $n$ and 0 in all other degrees. For $n \in \mathbb{N}$, consider the [[semifree dgc-algebras]]

$$
  S(n) \coloneqq (\wedge^\bullet k[n], 0)
$$

and for $n \geq 1$ the [[semifree dgc-algebras]]

$$
  D(n)
   \coloneqq
   \left\lbrace
     \array{
         0 & (n = 0)
         \\
        (\wedge^\bullet (k[n] \oplus k[n-1]), 0) & (n \gt 0)
     }
   \right.
$$

for which the differential sends the generator of $k[n-1]$ to that of $k[n]$

Write

$$
  i_n \colon S(n) \to D(n)
$$

for the obvious morphism that takes the generator in degree $n$ to the generator in degree $n$ (and for $n = 0$ it is the unique morphism from the [[initial object]] $(0,0)$).

For $n \gt 0$ write

$$
  j_n \colon k[0] \to D(n)
  \,.
$$

=--

+-- {: .num_prop }
###### Proposition
**(generating cofibrations)**

The sets

$$
  I = \{i_n \}_{n \geq 1} \cup \{k[0] \to S(0), S(0) \to k[0]\}
$$

and

$$
  J = \{j_n \}_{n \gt 1}
$$

are sets of generating cofibrations and acyclic cofibrations, respectively, exhibiting the model category $(dgcAlg^{\geq 0}_k)_{proj}$ from prop. \ref{IndeedProjectiveModelStructureOnCdgAlg} as a [[cofibrantly generated model category]].

=--

review includes ([Hess 06, p. 6](#Hess06))

In this section we describe the cofibrations in the model structure on $(dgcalg^{\geq 0}_k)_{proj}$ (def. \ref{ProjectiveModelStructureOnCdgAlg}, prop. \ref{IndeedProjectiveModelStructureOnCdgAlg}). Notice that it is these that are in the image of the dual [[monoidal Dold-Kan correspondence]].

Before we characterize the cofibrations, first some notation.

For $V$ a $\mathbb{Z}$-[[graded vector space]] write
$\wedge^\bullet V$ for the [[Grassmann algebra]] over it. Equipped with the trivial differential $d = 0$ this is a [[semifree dga]] $(\wedge^\bullet V, d=0)$.

With $k$ our ground field we write $(k,0)$ for the corresponding dg-algebra, the tensor unit for the standard [[monoidal category|monoidal structure]] on $dgAlg$. This is the [[Grassmann algebra]] on the 0-vector space
$(k,0) = (\wedge^\bullet 0, 0)$.

+-- {: .num_defn }
###### Definition
**(Sullivan algebras)**

A **[[relative Sullivan algebra]]** is a [[morphism]] of dg-algebras that is an inclusion

$$
  (A,d) \to (A \otimes_k \wedge^\bullet V, d')
$$

for $(A,d)$ some dg-algebra and for $V$ some graded vector space, such that

* there is a [[well ordered set]] $J$

* indexing a basis $\{v_\alpha \in V| \alpha \in J\}$ of $V$;

* such that with $V_{\lt \beta} = span(v_\alpha | \alpha \lt \beta)$ for all basis elements $v_\beta$ we have that

  $$
    d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta}
    \,.
  $$

This is called a **minimal** [[relative Sullivan algebra]] if in addition the condition

$$
  (\alpha \lt \beta) \Rightarrow (deg v_\alpha  \leq deg v_\beta)
$$

holds. For a Sullivan algebra $(k,0) \to (\wedge^\bullet V, d)$ relative to the tensor unit we call the [[semifree dga]] $(\wedge^\bullet V,d)$ simply a **Sullivan algebra**.  And a **minimal Sullivan algebra** if $(k,0) \to (\wedge^\bullet V, d)$ is a minimal relative Sullivan algebra.

=--

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

Sullivan algebras were introduced by [[Dennis Sullivan]] in his development of [[rational homotopy theory]]. This is one of the key application areas of the model structure on dg-algebras.

=--

+-- {: .num_remark }
###### Remark
**($L_\infty$-algebras)**

Because they are [[semifree dgas]], Sullivan dg-algebras $(\wedge^\bullet V,d)$ are (at least for degreewise finite dimensional $V$) [[Chevalley-Eilenberg algebra]]s of [[L-∞-algebra]]s.

The co-commutative differential co-algebra encoding the corresponding [[L-∞-algebra]] is the free cocommutative algebra $\vee^\bullet V^*$ on the degreewise dual of $V$ with differential $D = d^*$, i.e. the one given by the formula

$$
  \omega(D(v_1 \vee v_2 \vee \cdots v_n)) =
  - (d \omega) (v_1, v_2, \cdots,  v_n)
$$

for all $\omega \in V$ and all $v_i \in V^*$.

=--

+-- {: .num_prop }
###### Proposition
**(cofibrations are relative Sullivan algebras)**

The cofibrations in $(dgcAlg^{\geq 0}_{k})_{proj}$ are precisely the [[retracts]] of [[relative  Sullivan algebras]] $(A,d) \to (A\otimes_k \wedge^\bullet V, d')$.

Accordingly, the cofibrant objects in $(dgcAlg^{\geq 0}_{k})_{proj}$ are precisely the Sullivan algebras $(\wedge^\bullet V, d)$

=--

([Bousfield-Gugenheim 76, Prop. 7.11](#BousfieldGugenheim76)[Gelfand-Manin 96., Prop. V.5.4](#GelfandManin96))

#### Simplicial hom-complexes
 {#HomComplexes}

We discuss [[simplicial set|simplicial]] [[mapping spaces]] between [[dgc-algebras]]. These _almost_ make the projective model structure $(dgcAlg^{\geq 0}_k)_{proj}$ from prop. \ref{IndeedProjectiveModelStructureOnCdgAlg} into a [[simplicial model category]], except that the [[tensoring]]/[[powering]] [[isomorphism]] holds only for [[finite set|finite]] [[simplicial sets]] or else on [[dgc-algebras]] of [[finite type]]. Still, this has useful implications, for instance it implies that the [[reduced suspension]] and [[loop space]] [[adjunction]] on [augmented algebras|augmented]] [[dg-algebras]] is a [[Quillen adjunction]].

+-- {: .num_defn #MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees}
###### Definition
**(simplicial mapping spaces)**

For $A,B \in dgcAlg^{\geq 0}_k$ (def. \ref{dgcCochainAlgebrasInNonNegativeDegrees}), let

$$
  Maps(A,B)
  \in
  sSet
$$

be the [[simplicial set]] whose [[n-simplices]] are the dg-algebra [[homomorphisms]] from $A$ into the [[tensor product]] of $B$ with the de Rham complex of [[polynomial differential forms on the n-simplex]] $\Omega_{poly}^\bullet(\Delta^n)$.

$$
  Maps(A,B)_n
   \;\coloneqq\;
  Hom_{dgcAlg^{\geq 0}_k}
  \left(
    A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B
  \right)
$$

and whose face and degeneracy maps are the obvious ones induced from the fact that $\Omega_{poly}^\bullet \colon \Delta^{op} \to dgcAlg^{\geq 0}_k$ is canonically a [[simplicial object]] in dgc-algebras.

We also call this the _simplicial [[mapping space]]_ from $A$ to $B$. This construction naturally extends to a [[functor]]

$$
  Maps(-,-)
  \;\colon\;
  (dgcAlg^{\geq 0}_k)^{op}
  \times
  dgcAlg^{\geq 0}_k
    \longrightarrow
   dgcAlg^{\geq 0}_k
$$

from the [[product category]] of the [[opposite category]] of [[dgc-algebras]] with the category itself.

Observe that

$$
  Hom_{dgcAlg^{\geq 0}_k}
  \left(
    A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B
  \right)
  \;\simeq\;
  {}_{\Omega^\bullet_{poly}}Hom_{dgcAlg^{\geq 0}_k}
  \left(
     \Omega^\bullet_{poly}(\Delta^n) \otimes_k A
     \,,\,
     \Omega^\bullet_{poly}(\Delta^n) \otimes_k B
  \right)
  \,,
$$

where on the right we have those dg-algebra homomorphism which in addition preserves the left [[dg-module]] structure over $\Omega^\bullet_{poly}(\Delta^n)$. This induces for any three $A,B,C \in dgcAlg^{\geq 0}_k$ a [[composition]] homomorphism of [[simplicial sets]] out of the [[Cartesian product]] of mapping spaces

$$
  \circ^{sSet}_{A,B,C}
  \;\colon\;
  Maps(A,B) \times Maps(B,C)
    \longrightarrow
  Maps(A,C)
  \,.
$$

=--

([Bousfield-Gugenheim 76, 5.1](#BousfieldGugenheim76))

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

The set of 0-simplices of of the [[mapping space]] $Maps(A,B)$ in def. \ref{MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees} is [[natural isomorphism|naturally isomorphic]] to the ordinary [[hom-set]] of dg-algebras:

$$
  Maps(A,B)_0 \simeq Hom_{dgcAlg^{\geq 0}_k}(A,B)
$$

and under this identification the two notions of [[composition]] agree.

=--

Definition \ref{MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees} makes $dgcAlg^{\geq 0}_k$ an [[sSet]]-[[enriched category]] ("[[simplicial category]]"). The follows says that it is also [[powering|powered]], not over all of $sSet$, but over finite simplicial sets:

+-- {: .num_prop #PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets}
###### Proposition
**(powering over finite simplicial sets)**

For $A, B \in dgcAlg^{\geq 0}_k$ and $S \in $ [[sSet]], there is a [[natural transformation]]

$$
  Hom_{dgcAlg^{\geq}_k}(A, \Omega^\bullet_{poly}(S) \otimes_k B)
   \longrightarrow
  Hom_{sSet}( S, Maps(A,B) )
$$

from the [[hom-set]] of [[dgc-algebras]] into the [[tensor product]] with the [[polynomial differential forms on n-simplices]] from def. \ref{MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees} to the [[hom-set]] in [[simplicial sets]] into the simplicial [[mapping space]] from def. \ref{MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees}.

Moreover, this morphism is an [[isomorphism]] if one of the following conditions holds:

* $S$ is a [[finite set|finite]] [[simplicial set]];

* $B$ is of [[finite type]] (def. \ref{dgcCochainAlgebraInNonNegDegreeOfFiniteType}).

=--

([Bousfield-Gugenheim 76, lemma 5.2](#BousfieldGugenheim76))

+-- {: .num_prop}
###### Proposition
**(pullback powering axiom)**

Let $i \colon V \to W$ and $p \colon X \to Y$ be two [[morphisms]] in $dgcAlg^{\geq 0}_k$. Then their [[pullback power]] with respect to the simplicial [[mapping space]] functor (def. \ref{MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees})

$$
  p^i
  \;\colon\;
  Maps(W,X)
   \longrightarrow
  Maps(V,X)
   \underset{Maps(V,Y)}{\times}
  Maps(W,Y)
$$

is

1. a [[Kan fibration]] if $i$ is a cofibration and $p$ a fibration in the projective [[model category]] structure from prop. \ref{IndeedProjectiveModelStructureOnCdgAlg};

1. in addition a [[weak homotopy equivalence]] (i.e. a weak equivalence in the [[classical model structure on simplicial sets]]) if at least one of $i$ or $p$ is a weak equivalence in the projective model structure from prop. \ref{IndeedProjectiveModelStructureOnCdgAlg}.

=--

([Bousfield-Gugenheim 76, prop. 5.3](#BousfieldGugenheim76))

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

Prop. \ref{PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets} _would_ say that $(dgcAlg^{\geq 0}_k)_{proj}$ is a [[simplicial model category]] with respect to the simplicial enrichment from def. \ref{MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees} were it not for the fact that prop. \ref{PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets} gives the [[powering]] only over finite simplicial sets.

=--

#### Relation to simplicial sets

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

The [[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 connective dgc-algebras]]

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

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

=--

#### Relation to cosimplicial commutative algbras

The [[monoidal Dold-Kan correspondence]] gives a [[Quillen equivalence]] to the [[model structure on cosimplicial algebras|projective model structure on cosimplicial commutative algebras]] $(cAlg_k^{\Delta})_{proj}$.

#### Preservation of weak equivalences under pushout
 {#PreservationOfWeakEquivalencesUnderPushout}

+-- {: .num_prop #PushoutAlongRelativeSullivanModelsPreservesQuasiIsos}
###### Proposition
**([[pushout]] along [[relative Sullivan models]] preserves [[quasi-isomorphisms]])**

In the projective model structure on connective dgc-algebras (Def. \ref{ProjectiveModelStructureOnCdgAlg}), the operation of [[pushout]] along a [[relative Sullivan model]] preserves weak equivalences (quasi-isomorphisms).

=--

This is [Felix-Halperin-Thomas 00, Lemma 14.2, using Prop. 6.7 (ii)](#FelixHalperinThomas00).

The same statement for augmented dgc-algebras is in [Baues 88, Section I.8, Lemma 8.16](#Baues88).

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

Lemma 14.2 in [Felix-Halperin-Thomas 00](#FelixHalperinThomas00) is stated under the additional assumption that the dgc-algebras being pushed put have $H^0(-) = k$. But this is only used to find the stronger statement that the pushout is itself a Sullivan model. The argument via Prop. 6.7 that the pushout is a quasi-iso does not use this assumption.

=--

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

Prop. \ref{PushoutAlongRelativeSullivanModelsPreservesQuasiIsos} comes close to saying that the projective model structure on connective dgc-algebras is a [[left proper model category]], but not quite: The class of all [[cofibrations]] is larger than that of [[relative Sullivan algebras]], it includes also their [[retracts]] (see e.g [Hess 06](#Hess06)).

=--

#### Change of scalars

\begin{proposition}\label{ExtensionOfScalarsQuillenAdjunction}
  Let $k$ be a [[field]] of [[characteristic zero]], with
  $\mathbb{Q} \xhookrightarrow{i} k$ the corresponding inclusion of the [[rational numbers]]. Then the iduced [[extension of scalars]]$\;\dashv\;$[[restriction of scalars]]-[[adjunction]]

$$
  \big(
    dgcAlg^{\geq 0}_k
  \big)_{proj}
    \underoverset
      {\underset{res_{\mathbb{Q}}}{\longrightarrow}}
      {\overset{ (-) \otimes_{\mathbb{Q}} k }{\longleftarrow}}
      {\bot_{\mathrlap{Qu}}}
  \big(
    dgcAlg^{\geq 0}_{\mathbb{Q}}
  \big)_{proj}
$$

is a [[Quillen adjunction]] between the respective projective model categories (Def. \ref{ProjectiveModelStructureOnCdgAlg}).

\end{proposition}
([Bousfield & Gugenheim 1976, Lemma 11.6](#BousfieldGugenheim76))
\begin{proof}
  It is immediate that [[restriction of scalars]] is a [[right Quillen functor]]:

1. It preserves [[fibrations]], since these are the [[surjections]] of [[underlying]] [[sets]], and restriction of scalars does not change the underlying sets.

1. It preserves [[weak equivalences]], since these are the [[isomorphisms]] on [[cochain cohomology]] [[cohomology groups|groups]] of [[underlying]] [[cochain complexes]], and, again, restriction of scalars does not change the underlying sets of the underlying cochain complexes.

\end{proof}

#### Commutative vs. non-commutative dg-algebras
 {#CommVsNoncomm}

> this needs harmonization

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

The [[forgetful functor]]

$$
  F dgcAlg_k \to dgAlg_k
$$

from (graded-)commutative [[dg-algebra]]s to dg-algebras is the [[right adjoint]] part of a [[Quillen adjunction]]

$$
  Ab
  \colon
  dgAlg
   \stackrel{\leftarrow}{\to}
  CdgAlg
  :
  F
$$

=--

> boundedness?

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

The forgetful functor clearly preserves fibrations and cofibrations. It has a [[left adjoint]], the free abelianization functor $Ab$, which sends a [[dg-algebra]] $A$ to its quotient $A/[A,A]$.

=--

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

Let the ground [[ring]] $k$ be a [[field]] of [[characteristic zero]]. Then every [[dg-algebra]] $A$ which has the structure of an [[algebra over an operad|algebra over]] the [[E-k-operad|E-∞ operad]] has a [[dg-algebra]] morphism $A \to A_c$ to a commutative dg-algebra $A_c$  which is

* a morphism of [[E-k-operad|E-∞ algebras]] (where $A_c$ has the obvious [[E-k-operad|E-∞ algebras]] structure)

* a weak weak equivalence in the model structure on dg-algebras (i.e. a [[quasi-isomorphism]] of the underlying cochain complexes).

=--

This is in ([Kriz-May 95, II.1.5](#KrizMay95)).

So this says that the weak equivalence classes of the commutative dg-algebras in the model category of all dg-algebras already exhaust the most general non-commutative but  homotopy-commutative dg-algebras.

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

Discussion of a restricted kind of homotopy-faithfulness of the forgetful functor from the homotopy theory of commutative to not-necessarily commutative dg-algebras is in ([Amrani 14](#Amrani14)).

=--

## Unbounded dg-algebras
 {#Unbounded}

We discuss now the case of unbounded dg-algebras. For these there is no longer the [[monoidal Dold-Kan correspondence]] available. Instead, these can be understood as arising naturally as function $\infty$-algebras in the [[derived geometry|derived]] [[dg-geometry]] over formal duals of bounded dg-algebras, see [[function algebras on ∞-stacks]].

### Gradings and conventions {#GradingsAndConventions}

In [[derived geometry]] two categorical gradings interact: a [[cohesive (∞,1)-topos|cohesive]] $\infty$-groupoid $X$ has a space of [[k-morphism]]s $X_k$ for all non-negative $k$, and each such has itself a _[[simplicial T-algebra]]_ of functions with a component in each non-positive degree. But the directions of the face maps are opposite. We recall the grading situation from [[function algebras on ∞-stacks]].

Functions on a bare $\infty$-groupoid $K$, modeled as a [[simplicial set]], form a [[cosimplicial algebra]] $\mathcal{O}(K)$, which under the [[monoidal Dold-Kan correspondence]] identifies with a cochain [[dg-algebra]] (meaning: with positively graded differential) in non-negative degree

$$
  \left(
    \array{
        \vdots
       \\
       \downarrow \downarrow \downarrow \downarrow
        \\
        K_2
       \\
       \downarrow^{\partial_0} \downarrow^{\partial_1} \downarrow^{\partial_2}
       \\
       K_1
       \\
       \downarrow^{\partial_0} \downarrow^{\partial_1}
       \\
       K_0
     }
  \right)
   \;\;\;\;\;
   \stackrel{\mathcal{O}}{\mapsto}
   \;\;\;\;\;
  \left(
  \array{
    \vdots
    \\
    \uparrow \uparrow \uparrow \uparrow
    \\
    \mathcal{O}(K_2)
    \\
    \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \uparrow^{\partial_2^*}
    \\
    \mathcal{O}(K_1)
    \\
    \uparrow^{\partial_0^*} \uparrow^{\partial_1^*}
    \\
    \mathcal{O}(K_0)
  }
  \right)
   \;\;\;\;\;
  \stackrel{\sim}{\leftrightarrow}
   \;\;\;\;\;
  \left(
    \array{
     \cdots
     \\
     \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}}
     \\
     A_2
     \\
     \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}}
     \\
     A_1
     \\
     \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}}
     \\
     A_0
     \\
     \uparrow
     \\
     0
     \\
     \uparrow
     \\
     0
     \\
     \uparrow
     \\
     \vdots
   }
  \right)
  \,.
$$

On the other hand, a representable $X$ has itself a _[[simplicial T-algebra]]_ of functions, which under the monoidal Dold-Kan correspondence also identifies with a cochain dg-algebra, but then necessarily in non-positive degree to match with the above convention. So we write

$$
  \mathcal{O}(X)
   \;\;\;\;\;
  =
   \;\;\;\;\;
  \left(
   \array{
     \mathcal{O}(X)_0
     \\
     \uparrow \uparrow
     \\
     \mathcal{O}(X)_{-1}
     \\
     \uparrow \uparrow \uparrow
     \\
     \mathcal{O}(X)_{-2}
     \\
     \uparrow \uparrow \uparrow \uparrow
     \\
     \vdots
   }
  \right)
   \;\;\;\;\;
  \stackrel{\sim}{\leftrightarrow}
   \;\;\;\;\;
  \left(
    \array{
        \vdots
        \\
        \uparrow
        \\
        0
        \\
        \uparrow
        \\
        0
        \\
        \uparrow
        \\
       \mathcal{O}(X)_0
       \\
       \uparrow
       \\
       \mathcal{O}(X)_{-1}
       \\
       \uparrow
       \\
       \mathcal{O}(X)_{-2}
       \\
       \uparrow
       \\
       \vdots
    }
  \right)
  \,.
$$

Taking this together, for $X_\bullet$ a general [[∞-stack]], its function algebra is generally an _unbounded_ cochain dg-algebra with mixed contributions as above, the simplicial degrees contributing in the positive direction, and the homological resolution degrees in the negative direction:

$$
  \mathcal{O}(X_\bullet)
   \;\;\;\;\;
   =
   \;\;\;\;\;
  \left(
     \array{
        \vdots
        \\
        \uparrow
        \\
        \bigoplus_{k-p = q} \mathcal{O}(X_k)_{-p}
        \\
        \uparrow
        \\
        \vdots
        \\
        \uparrow^d
        \\
        \mathcal{O}(X_1)_0 \oplus \mathcal{O}(X_2)_{-1} \oplus
          \mathcal{O}(X_3)_{-2} \oplus \cdots
        \\
        \uparrow^{d}
        \\
        \mathcal{O}(X_0)_0 \oplus \mathcal{O}(X_1)_{-1} \oplus
          \mathcal{O}(X_2)_{-2} \oplus \cdots
        \\
        \uparrow^{d}
        \\
        \mathcal{O}(X_0)_{-1} \oplus \mathcal{O}(X_1)_{-2} \oplus
          \mathcal{O}(X_2)_{-3}\oplus \cdots
        \\
        \uparrow^{d}
        \\
        \vdots
     }
  \right)
  \,.
$$

### Definition
 {#ForUnboundedDGAlgebrasDefinition}

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

For $k$ a [[field]] of [[characteristic]] 0 let

$$
  cdgAlg = CMon(Ch_\bullet(k))
$$

be the category of undounded commutative dg-algebras. With fibrations the degreewise surjections and weak equivalences the [[quasi-isomorphisms]] this is a

* [[model category]]

which is

* [[proper model category|proper]];

* [[combinatorial model category|combinatorial]].

=--

The existence of the model structure follows from the general discussion at [[model structure on dg-algebras over an operad]].

Properness and combinatoriality is discussed in ([ToënVezzosi](#ToenVezzosi)):

* in lemma 2.3.1.1 they state that $cdgAlg_+$ constitutes the first two items in a triple which they call an _HA context_ .

* this implies their assumption 1.1.0.4 which asserts properness and
  combinatoriality

Discussion of cofibrations in $dgAlg_{proj}$ is in ([Keller](#Keller)).

### Properties

#### Properness
 {#Properness}

Let $cdgAg_k$ be the projective model structure on commutative unbounded dg-algebras from above.

This is a [[proper model category]]. See MO discussion [here](http://mathoverflow.net/q/204414/381).

#### Derived tensor product

Let $cdgAg_k$ be the projective model structure on commutative unbounded dg-algebras from above

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

For cofibrant $A \in cdgAlg_k$, the functor

$$
  A\otimes_k (-) : k Mod \to A Mod
$$

preserves [[quasi-isomorphism]]s.

For $A,B \in cdgAlg_k$, their [[derived functor|derived]] [[coproduct]] in $k Mod$ coincides in the [[homotopy category]] with the derived [[tensor product]] in $k Mod$: the morphism

$$
  A \coprod_k^{L} B \stackrel{}{\to} A \otimes_k^L B
$$

is an isomorphism in $Ho(k Mod)$.

=--

This follows by the above with ([ToënVezzosi, assumption 1.1.0.4, and page 8](#ToenVezzosi)).

#### Derived hom-functor {#SimplicialHomObjects}

The model structure on unbounded dg-algebras is _almost_ a [[simplicial model category]]. See the section _[simplicial enrichment](model+structure+on+dg-algebras+over+an+operad#SimplicialEnrichment)_ at _[[model structure on dg-algebras over an operad]]_ for details.

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

Let $k$ be a [[field]] of [[characteristic]] 0. Let
$\Omega^\bullet_{poly} : sSet \to (cdgAlg_k)^{op}$ be the functor that assigns polynomial [[differential forms on simplices]].

For $A,B \in dgcAlg_k$ define the [[simplicial set]]

$$
cdgAlg_k(A,B) : ([n] \mapsto Hom_{cdgAlg_k}(A, B \otimes_k \Omega^\bullet_{poly}(\Delta[n]))
  \,.
$$

This extends to a functor

$$
  cdgAlg_k(-,-) : cdgAlg_k^{op} \times cdgAlg_k \to sSet
  \,.
$$

=--

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

The functor $cdgAlg_k(-,-)$ satisfies the dual of the [[pushout-product axiom]]: for $i : A \to B$ any cofibration in $cdgAlg_k$ and $p : X \to Y$ any fibration, the canonical morphism

$$
  (i^*, p_*) : cdgalg_k(A,B) \to
    cdgAlg_k(A,X)
      \times_{cdgAlg_k(A,Y)} cdgAlg_k(B,Y)
$$

is a [[Kan fibration]], which is acyclic if $i$ or $p$ is.

=--

This implies in particular that for $A$ cofibrant, $cdgAlg_k(A,B)$ is a [[Kan complex]].

The proof works along the lines of ([Bousfield-Gugenheim 76, prop. 5.3](#BousfieldGugenheim76)). See also the discussion at [[model structure on dg-algebras over an operad]].

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

We give the proof for a special case. The general case is analogous.

We show that for $A$ cofibrant, and for any $B$ (automatically fibrant), $cdgAlg_k(A,B)$ is a [[Kan complex]].

By a standard fact in [[rational homotopy theory]] (due to [Bousfield-Gugenheim 76](#BousfieldGugenheim76), discussed at [[differential forms on simplices]]) we have that $\Omega^\bullet_{poly} : sSet \to (cdgAlg^+_k)^{op}$ is a left [[Quillen adjunction|Quillen functor]], hence in particular sends acyclic cofibrations to acyclic cofibrations, hence acyclic [[monomorphism]]s of [[simplicial set]]s to acyclic fibrations of [[dg-algebra]]s.

Specifically for each [[horn]] inclusion $\Lambda[n]_k \hookrightarrow \Delta[n]$ we have that the restriction map $\Omega^\bullet_{poly}(\Delta[n]) \to \Omega^\bullet_{poly}(\Lambda[n]_k)$ is an acyclic fibration in $cdgAlg_k^*$, hence in $cdgAlg_k$.

A $k$-horn in $cdgAlg_k(A,B)$ is a morphism $A \to B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k)$. A filler for this horn is a lift $\sigma$ in

$$
  \array{
    && B \otimes \Omega^\bullet_{poly}(\Delta[n])
    \\
    & {}^{\mathllap{\sigma}}\nearrow & \downarrow
    \\
    A &\to& B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k)
  }
  \,.
$$

If $A$ is cofibrant, then such a lift does always exist.

=--

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

For $A \in cdgAlg$ cofibrant, $cdgAlg_k(A,B)$ is the correct [[derived hom-space]]

$$
  cdgAlg_k(A,B)
  \simeq
  \mathbb{R}Hom(A,B)
  \,.
$$

=--

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

By the assumption that $A$ is cofibrant and according to the facts discussed at [[derived hom-space]], we need to show that

$$
  s B : [n] \mapsto B\otimes_k \Omega^\bullet_{poly}(\Delta[n])
$$

is a [[simplicial resolution|resolution]], or _simplicial frame_ for $B$. (Notice that every object is fibrant in $cdgAlg_k$).

Since polynomial differential forms are acyclic on simplices (discussed [here](http://nlab.mathforge.org/nlab/show/differential+forms+on+simplices)) it follows that

$$
  const B \to s B
$$

is degreewise a weak equivalence. It remains to show that $s A$ is fibrant in the [[Reedy model structure]] $[\Delta^{op}, cdgAlg_k]_{Reedy}$.

One finds that the matching object is given by

$$
  (match s B)_k = B \otimes \Omega^\bullet_{poly}(\partial \Delta[k])
  \,.
$$

Therefore $s B$ is Reedy fibrant if in each degree the morphism

$$
  (s B_k \to (match s B)_k ) = (\Omega^\bullet_{poly}(\partial \Delta[k] \hookrightarrow \Delta[k]))
$$

is a fibration. But this follows from the fact that $\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}$ is a left [[Quillen adjunction|Quillen functor]] (as discussed at [[differential forms on simplices]]).
=--

#### Derived copowering over $sSet$ {#DerivedCopowering}

We discuss a concrete model for the $(\infty,1)$-copowering of
$(cdgAlg_k)^\circ$ over [[∞Grpd]] in terms of an operation of $cdgAlg_k$ over [[sSet]].

First notice a basic fact about ordinary commutative algebras.

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

In $CAlg_k$ the [[coproduct]] is given by the [[tensor product]] over $k$:

$$
  \left(
  \array{
    A
      &\stackrel{i_A}{\to}&
    A \coprod B
      &\stackrel{i_B}{\leftarrow}&
    B
   }
  \right)
  \simeq
  \left(
    \array{
      A
        &\stackrel{Id_A \otimes_k e_B}{\to}&
      A \otimes_k B
        & \stackrel{e_A \otimes Id_B}{\leftarrow}&
      B
   }
  \right)
$$

=--

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

We check the [[universal property]] of the coproduct: for $C \in CAlg_k$ and $f,g : A,B \to C$ two morphisms, we need to show that there is a unique morphism $(f,g) : A \otimes_k B \to C$  such that the diagram

$$
  \array{
    A
     &\stackrel{Id_A \otimes e_B}{\to}&
    A \otimes_k B
     &\stackrel{e_A \otimes Id_B}{\leftarrow}&
    B
    \\
    & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{(f,g)}} & \swarrow_{\mathrlap{g}}
    \\
    && C
  }
$$

commutes.
For the left triangle to commute we need that $(f,g)$ sends elements of the form $(a,e_B)$ to $f(a)$. For the right triangle to commute we need that $(f,g)$ sends elements of the form $(e_A, b)$ to $g(b)$. Since every element of $A \otimes_k B$ is a product of two elements of this form

$$
  (a,b) = (a,e_B) \cdot (e_A, b)
$$

this already uniquely determines $(f,g)$ to be given on elements by the map

$$
  (a,b) \mapsto f(a) \cdot g(b)
  \,.
$$

That this is indeed an $k$-algebra homomorphism follows from the fact that $f$ and $g$ are

=--

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

For these derivations it is crucial that we are working with commutative algebras.

=--

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

We have that the [[copower]]ing of $A$ with the map of sets from two points to the single point

$$
  (* \coprod * \to *) \cdot A
  \simeq
  (
    A \otimes_k A
     \stackrel{\mu}{\to}
    A
  )
$$

is the product morphism on $A$. And that the tensoring with the map from the empty set to the point

$$
  (\emptyset \to *)\cdot A
  \simeq
  (k \stackrel{e_A}{\to} A)
$$

is the unit morphism on $A$. Generally, for $f : S \to T$ any map of sets we have that the tensoring

$$
  (S \stackrel{f}{\to} T) \cdot A
  =
  A^{\otimes_k |S|}
  \to
  A^{\otimes_k |T|}
$$

is the morphism between [[tensor power]]s of $A$ of the cardinalities of $S$ and $T$, respectively, whose component over a copy of $A$ on the right corresponding to $t \in T$ is the iterated product $A^{\otimes_k |f^{-1}\{t\}|} \to A$ on as many tensor powers of $A$ as there are elements in the preimage of $t$ under $f$.

=--

The analogous statements hold true with $CAlg_k$ replaced by $cdgAlg_k$: for $S \in sSet$ and $ A \in cdgAlg_k$ we obtain a simplicial cdg-algebra

$$
  S \cdot A \in cdgAlg_k^{\Delta^{op}}
$$

by the ordinary degreewise [[copower]]ing over [[Set]], using that $cdgAlg_k$ has coproducts (equal to the tensor product over $k$).

This is equivalently a commutative monoid in simplicial unbounded chain complexes

$$
  cdgAlg_k^{\Delta^{op}} \simeq CMon(Ch^\bullet(k)^{\Delta^{op}})
  \,.
$$

By the logic of the [[monoidal Dold-Kan correspondence]] the symmetric [[lax monoidal functor|lax monoidal]] [[Moore complex]] functor (via the [[Eilenberg-Zilber map]]) sends this to a commutative [[monoid]] in non-positively graded cochain complexes in unbounded cochain complexes

$$
  C^\bullet(S \cdot A) \in CMon(Ch^\bullet_-(Ch^\bullet(k)))
  \,.
$$

Since the [[total complex]] functor $Tot : Ch^\bullet(Ch^\bullet(k)) \to Ch^\bullet(k)$ is itself symmetric lax monoidal (...), this finally yields

$$
  Tot C^\bullet(S \cdot A) \in CMon(Ch^\bullet(k)) \simeq
    cdgAlg_k
$$

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

Define the functor

$$
  CC : sSet \times cdgAlg \to cdgAlg
$$

by

$$
  CC(S,A) := Tot C^\bullet(S \cdot A)
  \,.
$$

=--

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

We have

$$
  CC(Y,A)^n := \bigoplus_{k \geq 0} (A^{\otimes_k |Y_k| })_{n+k}
$$

=--

This appears essentially (...) as ([GinotTradlerZeinalian, def 3.1.1](#GinotTradlerZeinalian)).

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

The [[(∞,1)-limit|(∞,1)-copowering]] of $(dgcAlg_k)^\circ$ over [[∞Grpd]] is modeled by the [[derived functor]] of $CC$.

=--

This follows from ([GinotTradlerZeinalian, theorem 4.2.7](#GinotTradlerZeinalian)), which asserts that the [[derived functor]] of this tensoring is the unique [[(∞,1)-functor]], up to equivalence, satisfying the axioms of $(\infty,1)$-copowering.

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

The functor

$$
  CC : sSet \times cdgAlg_k \to cdgAlg_k
$$

preserves weak equivalences in both arguments.

=--

This is essentially due to ([Pirashvili](#Pirashvili)). The full statement is ([GinotTradlerZeinalian, prop. 4.2.1](#GinotTradlerZeinalian)).

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

This means that the assumption for the copowering models of higher order [[Hochschild cohomology]] are satsified in $cdgAlg_k$ which are described in the section <a href="http://nlab.mathforge.org/nlab/show/Hochschild+cohomology#PirashviliHigherOrder">Pirashvili's higher Hochschild homology</a> is satisfied:

this means that for $A \in cdgAlg$ and $S \in sSet$, $CC(S,A)$ is a model for the function $\infty$-algebra on the [[free loop space object]] of $Spec A$. See the section <a href="http://nlab.mathforge.org/nlab/show/Hochschild+cohomology#OvercdgAlgs">Higher order Hochschild homology modeled on cdg-algebras</a> for more details.

=--

#### Derived powering over $sSet$ {#DerivedPowering}

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

Let $S \in \infty Grpd$ be presented by a degreewise finite [[simplicial set]] (which we denote by the same symbol).

Then the [[homotopy limit]] in $cdgAlg_k$ over the $S$-shaped diagram constant on $k$ is given by $\Omega^\bullet_{poly}(S)$.

$$
  \mathbb{R}{\lim_{\leftarrow}}_S const k \simeq \Omega^\bullet_{poly}(S)
  \,.
$$

=--

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

We show dually that for degreewise finite $S$ the assignment $(S, Spec A) \mapsto Spec (\Omega^\bullet_{poly}(S) \otimes A)$ models the $\infty$-copowering in $cdgAlg_k^{op}$.

By the discussion at <a href="http://nlab.mathforge.org/nlab/show/limit+in+a+quasi-category#Tensoring">(∞,1)-copowering</a> it is sufficient to to establish an equivalence

$$
  (dgcAlg_{k}^{op})^\circ(Spec (\Omega^\bullet_{poly}(S) \otimes A), Spec B)
  \simeq
   \infty Grpd(S, (dgcAlg_{k}^{op})^\circ(Spec A, Spec B))
$$

natural in $B$. Consider a cofibrant model of $B$, which we denote by the same symbol. The we compute with 1-categorical [[end]]/[[coend]] calculus

$$
  \begin{aligned}
    sSet(S, cdgAlg_k^{op}(Spec A,Spec B))
    & \simeq
    \int^{[r] \in\Delta}
      \Delta[r] \cdot Hom_{sSet}(S \times \Delta[r],
        cdgAlg_k^{op}(Spec A, Spec B))
   \\
   & \simeq
    \int^{[r] \in\Delta}
      \Delta[r] \cdot
      \int_{[k] \in \Delta}
       Hom_{Set}(S_k \times \Delta[k,r],
        Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B))
   \\
   & \simeq
    \int^{[r] \in\Delta}
      \Delta[r] \cdot
      \int_{[k] \in \Delta}
        Hom_{cdgAlg_k^{op}}((S_k \times \Delta[k,r]) \cdot Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B))
    \\
    & \simeq
    \int^{[r] \in\Delta}
      \Delta[r] \cdot
        Hom_{cdgAlg_k^{op}}(\int^{[k] \in \Delta}
 (S_k \times \Delta[k,r]) \cdot Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B))
    \\
    & \simeq
    \int^{[r] \in\Delta}
      \Delta[r] \cdot
        Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(S \times \Delta[r]) \times Spec A, Spec B))
  \end{aligned}
  \,,
$$

where all steps are [[isomorphism]]s and the dot denotes the ordinary 1-categorical [[copower]]ing of the 1-category $cdgAlg^{op}$ over [[Set]]. In the last step we are using that the [[tensor product]] commutes with finite limits of dg-algebras. (This is where the finiteness assumption is needed).

Now we use that $\Omega^\bullet_{poly}$ preserves [[product]]s up to [[quasi-isomorphism]] (as discussed <a href="http://nlab.mathforge.org/nlab/show/differential+forms+on+simplices#Properties">here</a>)

$$
  \Omega^\bullet_{poly}(S \times \Delta[r])
  \simeq
  \Omega^\bullet_{poly}(S) \otimes \Omega_{poly}^\bullet(\Delta[r])
  \,.
$$

This being a weak equivalence between fibrant objects and since $B$ is assumed cofibrant, we have by the [above discussion](#SimplicialHomObjects) of the derived hom-functor (and using the [[factorization lemma]]) a weak equivalence

$$
  \cdots \simeq
    \int^{[r] \in\Delta}
      \Delta[r] \cdot
        Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(S) \times Spec \Omega^\bullet_{poly}\Delta[r]) \times Spec A, Spec B))
  \,.
$$

Since all this is [[natural transformation|natural]] in $B$, this proves the claim.

=--

#### Path objects
 {#PathObjectsForUnboundedCommutative}
+-- {: .num_prop}
###### Proposition

For $A \in cdgAlg_k$, a [[path object]]

$$
  A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A
$$

for $A$ is given by

$$
  P(A) := A \otimes_k \Omega^\bullet_{poly}(\Delta[1])
$$

=--

This follows along the above lines. The statement appears for instance as
([Behrend, lemma 1.19](#Behrend)).

#### Relation to $H \mathbb{Z}$-algebra spectra
 {#RelationToAInfinityAlgebras}

For every [[ring spectrum]] $R$ there is the notion of [[algebra spectra]] over $R$. Let $R := H \mathbb{Z}$ be the [[Eilenberg-MacLane spectrum]] for the [[integer]]s. Then unbounded dg-algebras (over $\mathbb{Z}$) are one model for $H \mathbb{Z}$-algebra spectra.

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

There is a [[Quillen equivalence]] between the standard [[model category]] structure for $H \mathbb{Z}$-[[algebra spectra]] and the model structure on unbounded differential graded algebras.

=--

See [[algebra spectrum]] for details.

#### Relation to $\mathbb{E}_\infty$-algebras
 {#RelationToEInfinityAlgebras}

Commutative dg-algebras
over a field $k$ of characteristic 0 constitute a presentation of
[[E-infinity algebras]] over $k$ ([Lurie, prop. A.7.1.4.11]).

## Related concepts

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

* [[model structure on differential graded-commutative superalgebras]]

* [[model structure on chain complexes]]

* [[model structure on dg-modules]]

* [[model structure on dg-operads]]

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

  * **model structure on dg-algebras**

  * [[model structure on dg-categories]]

* [[model structure on dg-coalgebras]]

## References

### On connective dgc-algebras

The cofibrantly generated model structure on [[differential graded-commutative algebras]] is originally due to

* {#BousfieldGugenheim76} [[Aldridge Bousfield]], [[Victor Gugenheim]], _[[On PL deRham theory and rational homotopy type]]_, Memoirs of the AMS 179 (1976) ([ams:memo-8-179](https://bookstore.ams.org/memo-8-179))

Textbook account:

* {#GelfandManin96} [[Sergei Gelfand]], [[Yuri Manin]], Chapter V of: _[[Methods of homological algebra]]_,  transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 ([doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5))

Review:

* {#Hess06} [[Kathryn Hess]], p. 6 of: _Rational homotopy theory: a brief introduction_, contribution to _[Summer School on Interactions between Homotopy Theory and Algebra](https://jdc.math.uwo.ca/summerschool/)_, University of Chicago, July 26-August 6, 2004, Chicago ([arXiv:math.AT/0604626](http://arxiv.org/abs/math.AT/0604626)), chapter in Luchezar Lavramov, [[Dan Christensen]], [[William Dwyer]], [[Michael Mandell]], [[Brooke Shipley]] (eds.), _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics 436, AMS 2007 ([doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

* {#Moerman15} [[Joshua Moerman]], Section II of: _Rational Homotopy Theory_, 2015 ([pdf](https://www.ru.nl/publish/pages/813282/rational_homotopy_theory.pdf), [[MoermanRationalHomotopyTheory.pdf:file]])

The approach in [Hess 06](#Hess06) makes use of the general discussion in section 3 of

* [[Paul Goerss]], [[Kirsten Schemmerhorn]], _Model categories and simplicial methods_, Notes from lectures given at the University of Chicago, August 2004, in: _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics 436, AMS 2007([arXiv:math.AT/0609537](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436))

that obtains the model structure from the [[model structure on chain complexes]].

See also

* {#Baues88} Baues, Chapter I.8 of: _Algebraic Homotopy_ ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/baues4.pdf)

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

Generalization to [[equivariant rational homotopy theory]]:

* [[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))

### On non-commutative dg-algebras

For general **non-commutative** (or rather: not necessarily graded-commutative) dg-algebras a model structure is given in

* {#Jardine97} [[J. F. Jardine]], _[[JardineModelDG.pdf:file]]_, Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications, Vol. 17, AMS (1997), 55-58.

This is also the structure used in

* J. L Castiglioni, G. Cortiñas, _Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence_, J. Pure Applied Algebra **191** (2004) 119-142 [[arXiv:math/0306289](https://arxiv.org/abs/math/0306289), [doi:10.1016/j.jpaa.2003.11.009](https://doi.org/10.1016/j.jpaa.2003.11.009)]

where aspects of its relation to the [[model structure on cosimplicial rings]] is discussed. (See [[monoidal Dold-Kan correspondence]] for
more on this).

### On unbounded dg-algebras

Discussion of the model structure on unbounded dg-algebras over a field of characteristic 0 is in

* {#ToenVezzosi} [[Bertrand Toën]], [[Gabriele Vezzosi]], _HAG II, geometric stacks and applicatons_ ([arXiv:math/0404373v4](http://arxiv.org/abs/math/0404373v4))

A general discussion of algebras over an operad in unbounded chain complexes is 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)))

A survey of some useful facts with an eye towards [[dg-geometry]] is in

* {#Behrend} [[Kai Behrend]], _Differential graded schemes I: prefect resolving algebras_ ([arXiv:0212225](http://arxiv.org/abs/math/0212225))

Discussion of cofibrations in unbounded dg-algebras are in

* {#Keller} [[Bernhard Keller]], _$A_\infty$-algebras, modules and functor categories_ ([pdf](http://www.math.jussieu.fr/~keller/publ/ainffun.pdf))

### More

The derived copowering of unbounded commutative dg-algebras over $sSet$ is discussed (somewhat implicitly) in

* {#GinotTradlerZeinalian} [[Grégory Ginot]], Thomas Tradler, Mahmoud Zeinalian, _Derived higher Hochschild homology, topological chiral homology and factorization algebras_, ([arxiv/1011.6483](http://arxiv.org/abs/1011.6483))

The _commutative_ product on the dg-algebra of the higher order Hochschild complex is discussed in

* [[Grégory Ginot]], Thomas Tradler, Mahmoud Zeinalian, _A Chen model for mapping spaces and the surface product_ ([pdf](http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.2231v1.pdf))
{#GinotTradlerZeinalianChenModel}

The relation to [[E-infinity algebras]] is discussed in

* {#KrizMay95} [[Igor Kriz]] and [[Peter May]], _Operads, algebras, modules and motives_ , Astérisque No 233 (1995)

* {#Lurie} [[Jacob Lurie]], section 7.1 of _Higher algebra_ ([pdf](http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf))

The relation between commutative and non-commutative dgas is further discussed in

* {#Amrani14} [[Ilias Amrani]], _Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view_ ([arXiv:1401.7285](http://arxiv.org/abs/1401.7285))

* {#Amrani14b} [[Ilias Amrani]], _Rational homotopy theory of function spaces and Hochschild cohomology_ ([arXiv:1406.6269](http://arxiv.org/abs/1406.6269))

For more see also at _[[model structure on dg-algebras over an operad]]_.

Discussion of [[homotopy limits]] and [[homotopy colimits]] of dg-algebras is in

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

[[!redirects model structure on dg-rings]]
[[!redirects model structure on dg-algebra]]
[[!redirects Sullival algebra]]
[[!redirects model structures on dg-algebras]]

[[!redirects model structure on differential graded-commutative algebras]]
[[!redirects model structures on differential graded-commutative algebras]]

[[!redirects projective model structure on differential graded-commutative algebras]]
[[!redirects projective model structures on differential graded-commutative algebras]]

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

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

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

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


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

#Contents#
* table of contents
{:toc}

## Idea

This entry discusses [[model category]] structures on categories of [[algebras over an operad]] in the [[category of chain complexes]] (unbounded).

This is a special case of the general discussion at [[model structure on algebras over an operad]], but the category of (unbounded) chain complexes warrents some special attention.

For the operad of ordinary (commutative) algebras see also [[model structure on dg-algebras]].

## Definition
 {#Definition}

Let $k$ be a commutative ring.
Write $Ch_\bullet(k)$ for the [[category of chain complexes|category of unbounded chain complexes]] of $k$-[[module]]s.

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

An [[operad]] $P$ over $Ch_\bullet(k)$ is called **$\Sigma$-split** if (...)

A quasi-isomorphism between such operads $P_1 \to P_2$ is said to be compatible with $\Sigma$-splitting if (...)

=--

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

If $k$ contains the ring of [[rational number]]s, $\mathbb{Q} \hookrightarrow k$, then every $Ch_\bullet(k)$-operad is $\Sigma$-split and every quasi-isomorphism of operads is compatible with $\Sigma$-splitting.

The [[associative operad]] $Assoc_k$ is $\Sigma$-split for all $k$.

=--

This is ([Hinich, example 4.2.5](#Hinich)).

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

Let $P$ be a $\Sigma$-split operad, def. \ref{SigmaSplitOperad}, in $Ch_\bullet(k)$. Then the category $Alg_{Ch_\bullet(k)}(P)$ of [[algebras over an operad|algebras over the operad]] admits a [[model category]] structure whose

* weak equivalences are the underlying [[quasi-isomorphisms]]

* whose fibrations are the degreewise surjections

in $Ch_\bullet(k)$.

The [[free-forgetful adjunction]]

$$
  Alg_{Ch_\bullet(k)}(P)
  \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  Ch_\bullet(k)
$$

is therefore a [[Quillen adjunction]] (see at _[[transferred model structure]]_).

=--

This appears as ([Hinich, theorem 4.1.1](#Hinich)).

## Properties

### Invariance under equivalence and rectification

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

If $P_1 \to P_2$ is a [[quasi-isomorphism]] of $\Sigma$-split operads compatible with splittings, then there is an induced [[Quillen equivalence]]

$$
  Alg(P_1) \stackrel{\overset{}{\leftarrow}}{\underset{}{\rightarrow}}
  Alg(P_2)
$$

between the corresponding model structures on their algebras, as above.

=--

This is ([Hinich, theorem 4.7.4](#Hinich)).

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

Theorem \ref{ResolutionInvariance} in particular provides
rectification results for [[homotopy algebras]]: if
$P$ is some operad and $\tilde P \stackrel{\simeq}{\to} P$ a
[[cofibrant resolution]] in the suitable [[model structure on operads]], then the theorem says that $P$-[[homotopy algebras]] have the same homotopy theory as the plain $P$-algebras.

Famous examples include the [[Quillen equivalence]] between the [[model structure on dg-Lie algebras]] and the [[model structure for L-infinity algebras]].

=--

### Simplicial enrichment {#SimplicialEnrichment}

We discuss how the [above model structure](#ModelStructure) on $Alg_{Ch_\bullet(k)}(P)$ is _almost_ enhanced to a [[simplicial model category]] structure.

First we recall the standard definition of polynomial [[differential forms on simplices]]:

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

For $n \in \mathbb{N}$ let $\Omega_{poly}^\bullet(\Delta^n)$ be the commutative [[dg-algebra]] of **[[polynomial]] [[differential form]]s** on the $n$-[[simplex]]:

as a graded algebra it is

$$
  \Omega_{poly}^{\bullet}(\Delta^n)
  :=
  k[t_0, \cdots, t_n, d t_0, \cdots, d t_n]/(\sum t_i -1, \sum d t_i)
$$

with the differential the usual de Rham differential under the embedding $\Omega^\bullet_{poly}(\Delta^n) \hookrightarrow \Omega^\bullet(\Delta^n)$.

For $f : [k] \to [l]$ a [[morphism]] in the [[simplex category]] let

$$
  \Omega^\bullet_{poly}(f) : \Omega^\bullet_{poly}(\Delta^l) \to \Omega^\bullet_{poly}(\Delta^k)
$$

be the morphism of dg-algebras given on generators by

$$
  \Omega^\bullet_{poly}(f) : t_i \mapsto \sum_{f(j) = i} t_j
  \,.
$$

This yields a [[simplicial object|simplicial]] commutative dg-algebra

$$
  \Omega^\bullet_{poly}(\Delta^{(-)}) : \Delta^{op} \to cdgAlg_k
$$

or equivalently a [[cosimplicial object]] in the [[opposite category]] $cdgAlg_k^{op}$.

By the general definition of [[differential forms on presheaves]] this extends by left [[Kan extension]] to a functor

$$
  \Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}
$$

given by

$$
  \Omega^\bullet_{poly}(S) = \int^{[k]\in \Delta} S_k \cdot \Omega^\bullet_{poly}(\Delta^k)
  \,,
$$

where on the right be have the [[coend]] over the [[copowering]] of $cdgAlg_k^{op}$ over [[Set]].

=--

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

For $P$ a dg-operad as above, define [[sSet]]-[[hom-object]]s between objects $A,B \in Alg(P)$ by

$$
  Alg_P(A,B) := ([n] \mapsto Hom_{Alg_P}(A, B \otimes \Omega^\bullet_{poly}(\Delta^n))
  \in sSet
  \,.
$$

=--

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

These simplicial hom-objects satisfy the dual of the [[pushout-product axiom]] (see [[enriched model category]]).

=--

This is ([Hinich, lemma 4.8.4](#Hinich)).

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

For $S$ a degreewise finite simplicial set, we have a [[natural isomorphism]]

$$
  Hom_{Alg_P}(A, B \otimes \Omega^\bullet_{poly}(S))
  \simeq
  Hom_{sSet}(S, Alg_P(A,B))
  \,.
$$

=--

This is ([Hinich, lemma 4.8.3](#Hinich)).

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

The [[homotopy category]] of $Alg_P$ is given by

$$
  Ho(Alg_P)(A,B) \simeq \pi_0 Alg_P(Q A,B)
  \,,
$$

where $Q A$ is a cofibrant [[resolution]] of $A$.

=--

This appears as ([Hinich, section 4.8.10](#Hinich)).

## Examples

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

## 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

The model structure on dg-algebras over an operad is discussed in

* [[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}

based on results in

* A.K. Bousfield, V.K.A.M. Gugenheim, _On PL de Rham theory and rational homotopy type_ , Memoirs AMS, t.8, 179(1976)


model structure on dg-categories

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

#Contents#
* table of contents
{:toc}

## Idea

The general notion [[model structure on enriched categories]] gives in particular a model structure on [[dg-categories]], called the _Dwyer-Kan model structure_, and analogous to the usual [[model structure on sSet-categories]] which models [[(infinity,1)-categories]].

There are interesting [[left Bousfield localizations]] of this model structure, called the _quasi-equiconic_ and _Morita_ model structures.  Here the [[fibrant objects]] are the [[pretriangulated dg-categories]], resp. [[idempotent complete category|idempotent complete]] [[pretriangulated dg-categories]].  In characteristic zero, the Morita model structure is known to present the [[(infinity,1)-category]] of linear [[stable (infinity,1)-categories]] ([Cohn 13](#Cohn13)).

## Statement

### With Dwyer-Kan equivalences

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

Let $k$ be a [[commutative ring]]. Write $dgCat_k$ for the [[category]] of [[small category|small]] [[dg-categories]] over $k$.

There is the structure of a [[cofibrantly generated model category]] on $dgCat_k$ where a dg-functor $F : A \to B$ is

* a weak equivalence if

  1. for all objects $x,y \in A$ the component $F_{x,y} : A(x,y) \to B(F(x), F(y))$ is a [[quasi-isomorphism]] of [[chain complexes]];

  1. the induced functor on [[homotopy categories]] $H^0(F)$ (obtained by taking degree 0 [[chain homology]] in each hom-object)  is an [[equivalence of categories]].

* a fibration if

  1. for all objects  $x,y \in A$ the component $F_{x,y}$ is a degreewise surjection of chain complexes;

  1. for each [[isomorphism]] $F(x) \to Z$ in $H^0(B)$ there is a lift to an isomorphism in $H^0(A)$.

=--

This is due to ([Tabuada](#Tabuada)).

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

The definition is entirely analogous to the [[model structure on sSet-categories]].   Both are special cases of the [[model structure on enriched categories]].

=--

### With Morita equivalences
  {#WithMoritaEquivalences}

There is another model category structure with more weak equivalences, the _Morita equivalences_ ([Tabuada 05](#Tabuada05)).  This is in fact the [[left Bousfield localization]] of the above model structure with respect to the Morita equivalences, i.e. functors $F: C \to D$ whose induced [[restriction of scalars]] functor $\mathbf Lf^* : \mathbf D(D) \to \mathbf D(C)$ is an [[equivalence of categories]].

The [[fibrant objects]] with respect to this model structure are the [[dg-categories]] A for which the canonical inclusion $H^0(A) \hookrightarrow \mathbf D(A)$ has its [[essential image]] stable under [[cones]], [[suspensions]], and [[direct sums]].  Hence the [[homotopy category]] with respect to this model structure is identified with the [[full subcategory]] of Ho(DGCat), the homotopy category of the Dwyer-Kan model structure, spanned by dg-categories of this form.

This model structure is a presentation of the [[(∞,1)-category]] of [[stable (∞,1)-categories]] ([Cohn 13](#Cohn13)).

The [[pretriangulated dg-category|pretriangulated envelope]] of [[Bondal]]-[[Kapranov]] is a [[fibrant replacement]] functor for the Morita model structure.  The [[DG quotient]] of [[Drinfeld]] is a model for the [[homotopy cofibre]] with respect to the Morita model structure.

### Quasi-equiconic model structure

Here the [[fibrant objects]] are the [[pretriangulated dg-categories]]. (...)

## 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
 {#References}

The model structure on dg-categories is due to

* [[Gonçalo Tabuada]], _Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories_ C. R. Acad. Sci. Paris Sér. I Math. 340 (1) (2005), 15–19.
{#Tabuada}

It is reproduced as theorem 4.1 in

* [[Bernhard Keller]], _On differential graded categories_ ([pdf](http://atlas.mat.ub.es/grgta/articles/Keller.pdf))

A summary of the various model structures on dg-categories:

* {#CisinskiTabuada11} [[Denis-Charles Cisinski]], [[Goncalo Tabuada]], Section 2 of: _Non-connective K-theory via universal invariants_, Compositio Math. **147** (2011) 1281-1320 [[arXiv:0903.3717](http://arxiv.org/abs/0903.3717), [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/Non-connective-K-theory.pdf)]

Discussion of [[internal homs]] of dg-categories in terms of refined [[Fourier-Mukai transforms]] is in

* [[Bertrand Toën]], _The homotopy theory of dg-categories and derived Morita theory_, Invent. Math. 167 (2007), 615–667

Discussion of [[internal homs]] of dg-categories using (just) the structure of a [[category of fibrant objects]] is in

* Alberto Canonaco, Paolo Stellari, _Internal Homs via extensions of dg functors_ ([arXiv:1312.5619](http://arxiv.org/abs/1312.5619))

The derived internal Hom in the homotopy category of DG-categories is equivalent to the dg-category of A_infty-functors.

* [[Giovanni Faonte]], _A-infinity functors and homotopy theory of DG-categories_, [arXiv](http://arxiv.org/abs/1412.1255).

A proof that the [[internal hom]] of Ho(DGCat) constructed by Toën is in fact the right [[derived functor]] of the internal hom of DGCat is in

* Beatriz Rodriguez Gonzalez, _A derivability criterion based on the existence of adjunctions_, 2012, [arXiv:1202.3359](http://arxiv.org/abs/1202.3359).

There is also

* David Rosoff, _Mapping spaces of $A_\infty$-algebras_ ([pdf](http://www.math.washington.edu/~rosoff/ainfs.pdf))

The model structure with Morita equivalences as weak equivalences is discussed in

* {#Tabuada05} [[Goncalo Tabuada]], _Invariants additifs de dg-catgories_. Internat. Math. Res. Notices 53 (2005), 33093339.

That the Morita model structure on dg-categories presents the homotopy theory of $k$-linear [[stable (infinity,1)-categories]] was shown in

* {#Cohn13} [[Lee Cohn]], _Differential Graded Categories are k-linear Stable Infinity Categories_ ([arXiv:1308.2587](http://arxiv.org/abs/1308.2587))

See also

* Piergiorgio Panero, Boris Shoikhet, _A Quillen model structure on the category of Kontsevich-Soibelman weakly unital dg categories_ ([arXiv:1907.07970](https://arxiv.org/abs/1907.07970))

* Piergiorgio Panero, Boris Shoikhet, _A closed model structure on the category of weakly unital dg categories, II_ ([https://arxiv.org/abs/2007.12716](https://arxiv.org/abs/2007.12716))

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

[[!redirects Dwyer-Kan model structure on dg-categories]]
[[!redirects canonical model structure on dg-categories]]
[[!redirects quasi-equiconic model structure on dg-categories]]
[[!redirects Morita model structure on dg-categories]]

[[!redirects (infinity,1)-category of dg-categories]]
[[!redirects infinity-category of dg-categories]]