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\begin{document}

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\section*{model structure on dg-algebras}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{projective_model_structure_on_nonnegatively_graded_cochain_dgcalgebras}{Projective model structure on Non-negatively graded cochain dgc-algebras}\dotfill \pageref*{projective_model_structure_on_nonnegatively_graded_cochain_dgcalgebras} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{SullivanAlgebras}{Cofibrations and Sullivan algebras}\dotfill \pageref*{SullivanAlgebras} \linebreak
\noindent\hyperlink{HomComplexes}{Simplicial hom-complexes}\dotfill \pageref*{HomComplexes} \linebreak
\noindent\hyperlink{relation_to_cosimplicial_commutative_algbras}{Relation to cosimplicial commutative algbras}\dotfill \pageref*{relation_to_cosimplicial_commutative_algbras} \linebreak
\noindent\hyperlink{CommVsNoncomm}{Commutative vs. non-commutative dg-algebras}\dotfill \pageref*{CommVsNoncomm} \linebreak
\noindent\hyperlink{Unbounded}{Unbounded dg-algebras}\dotfill \pageref*{Unbounded} \linebreak
\noindent\hyperlink{GradingsAndConventions}{Gradings and conventions}\dotfill \pageref*{GradingsAndConventions} \linebreak
\noindent\hyperlink{definition_8}{Definition}\dotfill \pageref*{definition_8} \linebreak
\noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \linebreak
\noindent\hyperlink{properness}{Properness}\dotfill \pageref*{properness} \linebreak
\noindent\hyperlink{derived_tensor_product}{Derived tensor product}\dotfill \pageref*{derived_tensor_product} \linebreak
\noindent\hyperlink{SimplicialHomObjects}{Derived hom-functor}\dotfill \pageref*{SimplicialHomObjects} \linebreak
\noindent\hyperlink{DerivedCopowering}{Derived copowering over $sSet$}\dotfill \pageref*{DerivedCopowering} \linebreak
\noindent\hyperlink{DerivedPowering}{Derived powering over $sSet$}\dotfill \pageref*{DerivedPowering} \linebreak
\noindent\hyperlink{PathObjectsForUnboundedCommutative}{Path objects}\dotfill \pageref*{PathObjectsForUnboundedCommutative} \linebreak
\noindent\hyperlink{RelationToAInfinityAlgebras}{Relation to $H \mathbb{Z}$-algebra spectra}\dotfill \pageref*{RelationToAInfinityAlgebras} \linebreak
\noindent\hyperlink{RelationToEInfinityAlgebras}{Relation to $\mathbb{E}_\infty$-algebras}\dotfill \pageref*{RelationToEInfinityAlgebras} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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]].

\hypertarget{general}{}\subsection*{{General}}\label{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.

\hypertarget{projective_model_structure_on_nonnegatively_graded_cochain_dgcalgebras}{}\subsection*{{Projective model structure on Non-negatively graded cochain dgc-algebras}}\label{projective_model_structure_on_nonnegatively_graded_cochain_dgcalgebras}

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

\hypertarget{definition}{}\subsubsection*{{Definition}}\label{definition}

\begin{defn}
\label{dgcCochainAlgebrasInNonNegativeDegrees}\hypertarget{dgcCochainAlgebrasInNonNegativeDegrees}{}
For $k$ a [[field]] of [[characteristic zero]], write

\begin{displaymath}
dgcAlg^{\geq 0}_{k}
\end{displaymath}
for the [[category]] of [[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]].

\end{defn}
\begin{defn}
\label{dgcCochainAlgebraInNonNegDegreeOfFiniteType}\hypertarget{dgcCochainAlgebraInNonNegDegreeOfFiniteType}{}
\textbf{(finite type)}

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

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

\begin{itemize}%
\item \emph{weak equivalences} are those [[homomorphisms]] of dg-algebras whose underlying [[chain map]] is [[quasi-isomorphism]];


\item \emph{fibrations} are those [[homomorphisms]] which are degreewise [[surjections]];



\end{itemize}
\end{defn}
\begin{prop}
\label{IndeedProjectiveModelStructureOnCdgAlg}\hypertarget{IndeedProjectiveModelStructureOnCdgAlg}{}
The category $(dgcAlg^{\geq 0}_k)_{proj}$ from def. \ref{ProjectiveModelStructureOnCdgAlg} is a [[model category]], to be called the \emph{projective model structure}.

\end{prop}
(\hyperlink{BousfieldGugenheim76}{Bousfield-Gugenheim 76, theorem 4.3})

\begin{remark}
\label{}\hypertarget{}{}
\textbf{(category of fibrant objects)}

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

\end{remark}
The nature of the cofibrations is discussed \hyperlink{SullivanAlgebras}{below}.

\hypertarget{properties}{}\subsubsection*{{Properties}}\label{properties}

\hypertarget{SullivanAlgebras}{}\paragraph*{{Cofibrations and Sullivan algebras}}\label{SullivanAlgebras}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(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]]

\begin{displaymath}
S(n) \coloneqq (\wedge^\bullet k[n], 0)
\end{displaymath}
and for $n \geq 1$ the [[semifree dgc-algebras]]

\begin{displaymath}
D(n) 
   \coloneqq
   \left\lbrace
     \itexarray{
         0 & (n = 0)
         \\
        (\wedge^\bullet (k[n] \oplus k[n-1]), 0) & (n \gt 0)
     }
   \right.
\end{displaymath}
for which the differential sends the generator of $k[n-1]$ to that of $k[n]$

Write

\begin{displaymath}
i_n \colon S(n) \to D(n)
\end{displaymath}
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

\begin{displaymath}
j_n \colon k[0] \to D(n)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{(generating cofibrations)}

The sets

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

\begin{displaymath}
J = \{j_n \}_{n \gt 1}
\end{displaymath}
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]].

\end{prop}
review includes (\hyperlink{Hess06}{Hess 06, p. 6})

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 $(k,0) = (\wedge^\bullet 0, 0)$.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(Sullivan algebras)}

A \textbf{[[relative Sullivan algebra]]} is a [[morphism]] of dg-algebras that is an inclusion

\begin{displaymath}
(A,d) \to (A \otimes_k \wedge^\bullet V, d')
\end{displaymath}
for $(A,d)$ some dg-algebra and for $V$ some graded vector space, such that

\begin{itemize}%
\item there is a [[well ordered set]] $J$


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


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

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


\end{itemize}
This is called a \textbf{minimal} [[relative Sullivan algebra]] if in addition the condition

\begin{displaymath}
(\alpha \lt \beta) \Rightarrow (deg v_\alpha  \leq deg v_\beta)
\end{displaymath}
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 \textbf{Sullivan algebra}. And a \textbf{minimal Sullivan algebra} if $(k,0) \to (\wedge^\bullet V, d)$ is a minimal relative Sullivan algebra.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
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.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
\textbf{($L_\infty$-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

\begin{displaymath}
\omega(D(v_1 \vee v_2 \vee \cdots v_n)) =
  - (d \omega) (v_1, v_2, \cdots,  v_n)
\end{displaymath}
for all $\omega \in V$ and all $v_i \in V^*$.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{(cofibrations are relative Sullivan algebras)}

The cofibration in $(dgcAlg^{\geq 0}_{k})_{proj}$ are precisely the [[retract]]s 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 $(\wedge^\bullet V, d)$

\end{prop}
\hypertarget{HomComplexes}{}\paragraph*{{Simplicial hom-complexes}}\label{HomComplexes}

We discuss [[simplicial set|simplicial]] [[mapping spaces]] between [[dgc-algebras]]. These \emph{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]].

\begin{defn}
\label{MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees}\hypertarget{MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees}{}
\textbf{(simplicial mapping spaces)}

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

\begin{displaymath}
Maps(A,B)
  \in 
  sSet
\end{displaymath}
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]] $\Omega_{poly}^\bullet(\Delta^n)$.

\begin{displaymath}
Maps(A,B)_n
   \;\coloneqq\;
  Hom_{dgcAlg^{\geq 0}_k}
  \left(
    A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B 
  \right)
\end{displaymath}
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 \emph{simplicial [[mapping space]]} from $A$ to $B$. This construction naturally extends to a [[functor]]

\begin{displaymath}
Maps(-,-)
  \;\colon\;
  (dgcAlg^{\geq 0}_k)^{op}
  \times
  dgcAlg^{\geq 0}_k
    \longrightarrow
   dgcAlg^{\geq 0}_k
\end{displaymath}
from the [[product category]] of the [[opposite category]] of [[dgc-algebras]] with the category itself.

Observe that

\begin{displaymath}
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)  
  \,,
\end{displaymath}
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

\begin{displaymath}
\circ^{sSet}_{A,B,C}
  \;\colon\;
  Maps(A,B) \times Maps(B,C)
    \longrightarrow
  Maps(A,C)
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{BousfieldGugenheim76}{Bousfield-Gugenheim 76, 5.1})

\begin{remark}
\label{}\hypertarget{}{}
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:

\begin{displaymath}
Maps(A,B)_0 \simeq Hom_{dgcAlg^{\geq 0}_k}(A,B)
\end{displaymath}
and under this identification the two notions of [[composition]] agree.

\end{remark}
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:

\begin{prop}
\label{PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets}\hypertarget{PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets}{}
\textbf{(powering over finite simplicial sets)}

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

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

\begin{itemize}%
\item $S$ is a [[finite set|finite]] [[simplicial set]];


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



\end{itemize}
\end{prop}
(\hyperlink{BousfieldGugenheim76}{Bousfield-Gugenheim 76, lemma 5.2})

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(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})

\begin{displaymath}
p^i
  \;\colon\;
  Maps(W,X)
   \longrightarrow
  Maps(V,X)
   \underset{Maps(V,Y)}{\times}
  Maps(W,Y)
\end{displaymath}
is

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


\item 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}.



\end{enumerate}
\end{prop}
(\hyperlink{BousfieldGugenheim76}{Bousfield-Gugenheim 76, prop. 5.3})

\begin{remark}
\label{}\hypertarget{}{}
Prop. \ref{PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets} \emph{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.

\end{remark}
\hypertarget{relation_to_cosimplicial_commutative_algbras}{}\paragraph*{{Relation to cosimplicial commutative algbras}}\label{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]] $(cAlg_k^{\Delta})_{proj}$.

\hypertarget{CommVsNoncomm}{}\paragraph*{{Commutative vs. non-commutative dg-algebras}}\label{CommVsNoncomm}

\begin{quote}%
this needs harmonization


\end{quote}
\begin{prop}
\label{}\hypertarget{}{}
The [[forgetful functor]]

\begin{displaymath}
F dgcAlg_k \to dgAlg_k
\end{displaymath}
from (graded-)commutative [[dg-algebra]]s to dg-algebras is the [[right adjoint]] part of a [[Quillen adjunction]]

\begin{displaymath}
Ab
  \colon
  dgAlg 
   \stackrel{\leftarrow}{\to}
  CdgAlg
  :
  F
\end{displaymath}
\end{prop}
\begin{quote}%
boundedness?


\end{quote}
\begin{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 $A/[A,A]$.

\end{proof}
\begin{theorem}
\label{}\hypertarget{}{}
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

\begin{itemize}%
\item a morphism of [[E-k-operad|E-∞ algebras]] (where $A_c$ has the obvious [[E-k-operad|E-∞ algebras]] structure)


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



\end{itemize}
\end{theorem}
This is in (\hyperlink{KrizMay95}{Kriz-May 95, II.1.5}).

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.

\begin{remark}
\label{HomotopyFaithfulnessOfforgettingCommutativity}\hypertarget{HomotopyFaithfulnessOfforgettingCommutativity}{}
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 (\hyperlink{Amrani14}{Amrani 14}).

\end{remark}
\hypertarget{Unbounded}{}\subsection*{{Unbounded dg-algebras}}\label{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]].

\hypertarget{GradingsAndConventions}{}\subsubsection*{{Gradings and conventions}}\label{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 \emph{[[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

\begin{displaymath}
\left(
    \itexarray{
        \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(
  \itexarray{
    \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(
    \itexarray{
     \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)
  \,.
\end{displaymath}
On the other hand, a representable $X$ has itself a \emph{[[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

\begin{displaymath}
\mathcal{O}(X)
   \;\;\;\;\;
  =
   \;\;\;\;\;
  \left(
   \itexarray{
     \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(
    \itexarray{
        \vdots
        \\
        \uparrow
        \\
        0
        \\
        \uparrow
        \\
        0
        \\
        \uparrow
        \\
       \mathcal{O}(X)_0
       \\
       \uparrow
       \\
       \mathcal{O}(X)_{-1}
       \\
       \uparrow
       \\
       \mathcal{O}(X)_{-2}
       \\
       \uparrow
       \\
       \vdots
    }
  \right)
  \,.
\end{displaymath}
Taking this together, for $X_\bullet$ a general [[∞-stack]], its function algebra is generally an \emph{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:

\begin{displaymath}
\mathcal{O}(X_\bullet)
   \;\;\;\;\;
   =
   \;\;\;\;\;
  \left(
     \itexarray{
        \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)
  \,.
\end{displaymath}
\hypertarget{definition_8}{}\subsubsection*{{Definition}}\label{definition_8}

\begin{theorem}
\label{}\hypertarget{}{}
For $k$ a [[field]] of [[characteristic]] 0 let

\begin{displaymath}
cdgAlg = CMon(Ch_\bullet(k))
\end{displaymath}
be the category of undounded commutative dg-algebras. With fibrations the degreewise surjections and weak equivalences the [[quasi-isomorphism]]s this is a

\begin{itemize}%
\item [[model category]]

\end{itemize}
which is

\begin{itemize}%
\item [[proper model category|proper]];


\item [[combinatorial model category|combinatorial]].



\end{itemize}
\end{theorem}
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 (\hyperlink{ToenVezzosi}{To\"e{}nVezzosi}):

\begin{itemize}%
\item in lemma 2.3.1.1 they state that $cdgAlg_+$ constitutes the first two items in a triple which they call an \emph{HA context} .


\item this implies their assumption 1.1.0.4 which asserts properness and combinatoriality



\end{itemize}
Discussion of cofibrations in $dgAlg_{proj}$ is in (\hyperlink{Keller}{Keller}).

\hypertarget{properties_2}{}\subsubsection*{{Properties}}\label{properties_2}

\hypertarget{properness}{}\paragraph*{{Properness}}\label{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 \href{http://mathoverflow.net/q/204414/381}{here}.

\hypertarget{derived_tensor_product}{}\paragraph*{{Derived tensor product}}\label{derived_tensor_product}

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

\begin{prop}
\label{}\hypertarget{}{}
For cofibrant $A \in cdgAlg_k$, the functor

\begin{displaymath}
A\otimes_k (-) : k Mod \to A Mod
\end{displaymath}
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

\begin{displaymath}
A \coprod_k^{L} B \stackrel{}{\to} A \otimes_k^L B
\end{displaymath}
is an isomorphism in $Ho(k Mod)$.

\end{prop}
This follows by the above with (\hyperlink{ToenVezzosi}{To\"e{}nVezzosi, assumption 1.1.0.4, and page 8}).

\hypertarget{SimplicialHomObjects}{}\paragraph*{{Derived hom-functor}}\label{SimplicialHomObjects}

The model structure on unbounded dg-algebras is \emph{almost} a [[simplicial model category]]. See the section \emph{\href{model+structure+on+dg-algebras+over+an+operad#SimplicialEnrichment}{simplicial enrichment}} at \emph{[[model structure on dg-algebras over an operad]]} for details.

\begin{defn}
\label{}\hypertarget{}{}
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]]

\begin{displaymath}
cdgAlg_k(A,B) : ([n] \mapsto Hom_{cdgAlg_k}(A, B \otimes_k \Omega^\bullet_{poly}(\Delta[n]))
  \,.
\end{displaymath}
This extends to a functor

\begin{displaymath}
cdgAlg_k(-,-) : cdgAlg_k^{op} \times cdgAlg_k \to sSet
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
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

\begin{displaymath}
(i^*, p_*) : cdgalg_k(A,B) \to 
    cdgAlg_k(A,X)
      \times_{cdgAlg_k(A,Y)} cdgAlg_k(B,Y)
\end{displaymath}
is a [[Kan fibration]], which is acyclic if $i$ or $p$ is.

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

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

\begin{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 \hyperlink{BousfieldGugenheim76}{Bousfield-Gugenheim 76}, 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

\begin{displaymath}
\itexarray{
    && B \otimes \Omega^\bullet_{poly}(\Delta[n])
    \\
    & {}^{\mathllap{\sigma}}\nearrow & \downarrow
    \\
    A &\to& B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k)
  }
  \,.
\end{displaymath}
If $A$ is cofibrant, then such a lift does always exist.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For $A \in cdgAlg$ cofibrant, $cdgAlg_k(A,B)$ is the correct [[derived hom-space]]

\begin{displaymath}
cdgAlg_k(A,B)
  \simeq
  \mathbb{R}Hom(A,B)   
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the assumption that $A$ is cofibrant and according to the facts discussed at [[derived hom-space]], we need to show that

\begin{displaymath}
s B : [n] \mapsto B\otimes_k \Omega^\bullet_{poly}(\Delta[n])
\end{displaymath}
is a [[simplicial resolution|resolution]], or \emph{simplicial frame} for $B$. (Notice that every object is fibrant in $cdgAlg_k$).

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

\begin{displaymath}
const B \to s B
\end{displaymath}
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

\begin{displaymath}
(match s B)_k = B \otimes \Omega^\bullet_{poly}(\partial \Delta[k])
  \,.
\end{displaymath}
Therefore $s B$ is Reedy fibrant if in each degree the morphism

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

\end{proof}
\hypertarget{DerivedCopowering}{}\paragraph*{{Derived copowering over $sSet$}}\label{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.

\begin{prop}
\label{}\hypertarget{}{}
In $CAlg_k$ the [[coproduct]] is given by the [[tensor product]] over $k$:

\begin{displaymath}
\left(
  \itexarray{
    A 
      &\stackrel{i_A}{\to}& 
    A \coprod B 
      &\stackrel{i_B}{\leftarrow}& 
    B
   }
  \right)
  \simeq
  \left(
    \itexarray{
      A 
        &\stackrel{Id_A \otimes_k e_B}{\to}& 
      A \otimes_k B 
        & \stackrel{e_A \otimes Id_B}{\leftarrow}&
      B
   }
  \right)
\end{displaymath}
\end{prop}
\begin{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

\begin{displaymath}
\itexarray{
    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
  }
\end{displaymath}
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

\begin{displaymath}
(a,b) = (a,e_B) \cdot (e_A, b)
\end{displaymath}
this already uniquely determines $(f,g)$ to be given on elements by the map

\begin{displaymath}
(a,b) \mapsto f(a) \cdot g(b)
  \,.
\end{displaymath}
That this is indeed an $k$-algebra homomorphism follows from the fact that $f$ and $g$ are

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
For these derivations it is crucial that we are working with commutative algebras.

\end{remark}
\begin{cor}
\label{}\hypertarget{}{}
We have that the [[copower]]ing of $A$ with the map of sets from two points to the single point

\begin{displaymath}
(* \coprod * \to *) \cdot A
  \simeq
  (
    A \otimes_k A
     \stackrel{\mu}{\to}
    A
  )
\end{displaymath}
is the product morphism on $A$. And that the tensoring with the map from the empty set to the point

\begin{displaymath}
(\emptyset \to *)\cdot A
  \simeq
  (k \stackrel{e_A}{\to} A)
\end{displaymath}
is the unit morphism on $A$. Generally, for $f : S \to T$ any map of sets we have that the tensoring

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

\end{cor}
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

\begin{displaymath}
S \cdot A \in cdgAlg_k^{\Delta^{op}}
\end{displaymath}
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

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

\begin{displaymath}
C^\bullet(S \cdot A) \in CMon(Ch^\bullet_-(Ch^\bullet(k)))
  \,.
\end{displaymath}
Since the [[total complex]] functor $Tot : Ch^\bullet(Ch^\bullet(k)) \to Ch^\bullet(k)$ is itself symmetric lax monoidal (\ldots{}), this finally yields

\begin{displaymath}
Tot C^\bullet(S \cdot A) \in CMon(Ch^\bullet(k)) \simeq
    cdgAlg_k
\end{displaymath}
\begin{defn}
\label{}\hypertarget{}{}
Define the functor

\begin{displaymath}
CC : sSet \times cdgAlg \to cdgAlg
\end{displaymath}
by

\begin{displaymath}
CC(S,A) := Tot C^\bullet(S \cdot A) 
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
We have

\begin{displaymath}
CC(Y,A)^n := \bigoplus_{k \geq 0} (A^{\otimes_k |Y_k| })_{n+k}
\end{displaymath}
\end{remark}
This appears essentially (\ldots{}) as (\hyperlink{GinotTradlerZeinalian}{GinotTradlerZeinalian, def 3.1.1}).

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

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

\begin{prop}
\label{}\hypertarget{}{}
The functor

\begin{displaymath}
CC : sSet \times cdgAlg_k \to cdgAlg_k
\end{displaymath}
preserves weak equivalences in both arguments.

\end{prop}
This is essentially due to (\hyperlink{Pirashvili}{Pirashvili}). The full statement is (\hyperlink{GinotTradlerZeinalian}{GinotTradlerZeinalian, prop. 4.2.1}).

\begin{remark}
\label{}\hypertarget{}{}
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  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  for more details.

\end{remark}
\hypertarget{DerivedPowering}{}\paragraph*{{Derived powering over $sSet$}}\label{DerivedPowering}

\begin{prop}
\label{}\hypertarget{}{}
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 $\Omega^\bullet_{poly}(S)$.

\begin{displaymath}
\mathbb{R}{\lim_{\leftarrow}}_S const k \simeq \Omega^\bullet_{poly}(S) 
  \,.
\end{displaymath}
\end{prop}
\begin{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  it is sufficient to to establish an equivalence

\begin{displaymath}
(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))
\end{displaymath}
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{displaymath}
\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}
  \,,
\end{displaymath}
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 )

\begin{displaymath}
\Omega^\bullet_{poly}(S \times \Delta[r])
  \simeq
  \Omega^\bullet_{poly}(S) \otimes \Omega_{poly}^\bullet(\Delta[r])
  \,.
\end{displaymath}
This being a weak equivalence between fibrant objects and since $B$ is assumed cofibrant, we have by the \hyperlink{SimplicialHomObjects}{above discussion} of the derived hom-functor (and using the [[factorization lemma]]) a weak equivalence

\begin{displaymath}
\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))       
  \,.
\end{displaymath}
Since all this is [[natural transformation|natural]] in $B$, this proves the claim.

\end{proof}
\hypertarget{PathObjectsForUnboundedCommutative}{}\paragraph*{{Path objects}}\label{PathObjectsForUnboundedCommutative}

\begin{prop}
\label{}\hypertarget{}{}
For $A \in cdgAlg_k$, a [[path object]]

\begin{displaymath}
A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A
\end{displaymath}
for $A$ is given by

\begin{displaymath}
P(A) := A \otimes_k \Omega^\bullet_{poly}(\Delta[1])
\end{displaymath}
\end{prop}
This follows along the above lines. The statement appears for instance as (\hyperlink{Behrend}{Behrend, lemma 1.19}).

\hypertarget{RelationToAInfinityAlgebras}{}\paragraph*{{Relation to $H \mathbb{Z}$-algebra spectra}}\label{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.

\begin{prop}
\label{}\hypertarget{}{}
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.

\end{prop}
See [[algebra spectrum]] for details.

\hypertarget{RelationToEInfinityAlgebras}{}\paragraph*{{Relation to $\mathbb{E}_\infty$-algebras}}\label{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).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[model structure on differential graded-commutative superalgebras]]


\item [[model structure on chain complexes]]


\item [[model structure on dg-modules]]


\item [[model structure on dg-operads]]


\item [[model structure on dg-algebras over an operad]]

\begin{itemize}%
\item \textbf{model structure on dg-algebras}


\item [[model structure on dg-categories]]



\end{itemize}

\item [[model structure on dg-coalgebras]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The cofibrantly generated model structure on [[differential graded-commutative algebras]] is surveyed usefully for instance on p. 6 of

\begin{itemize}%
\item [[Kathryn Hess]], \emph{Rational homotopy theory: a brief introduction} (\href{http://arxiv.org/abs/math.AT/0604626}{arXiv:math.AT/0604626})

\end{itemize}
This makes use of the general discussion in section 3 of

\begin{itemize}%
\item [[Paul Goerss]], [[Kirsten Schemmerhorn]], \emph{Model categories and simplicial methods} Notes from lectures given at the University of Chicago, August 2004 (\href{http://arxiv.org/abs/math.AT/0609537}{arXiv})

\end{itemize}
that obtains the model structure from the [[model structure on chain complexes]].

A standard textbook reference is section V.3 of

\begin{itemize}%
\item [[Sergei Gelfand]], [[Yuri Manin]], \emph{Methods of homological algebra}, transl. from the 1988 Russian (Nauka Publ.) original. Springer 1996. xviii+372 pp. 2nd corrected ed. 2002.

\end{itemize}
An original reference seems to be

\begin{itemize}%
\item [[Aldridge Bousfield]], V. Gugenheim, \emph{On PL deRham theory and rational homotopy type} Memoirs of the AMS 179 (1976)

\end{itemize}
For general \textbf{non-commutative} (or rather: not necessarily graded-commutative) dg-algebras a model structure is given in

\begin{itemize}%
\item [[Rick Jardine]], \emph{[[JardineModelDG.pdf:file]]} , Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications, Vol. 17, AMS (1997), 55-58.

\end{itemize}
This is also the structure used in

\begin{itemize}%
\item J.L. Castiglioni G. Corti\~n{}as, \emph{Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence} (\href{http://arxiv.org/abs/math/0306289v2}{arXiv})

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

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

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{HAG II, geometric stacks and applicatons} (\href{http://arxiv.org/abs/math/0404373v4}{arXiv:math/0404373v4})

\end{itemize}
A general discussion of algebras over an operad in unbounded chain complexes is in

\begin{itemize}%
\item [[Vladimir Hinich]], \emph{Homological algebra of homotopy algebras} Communications in algebra, 25(10). 3291-3323 (1997)(\href{http://arxiv.org/abs/q-alg/9702015}{arXiv:q-alg/9702015}, \emph{Erratum} (\href{http://arxiv.org/abs/math/0309453}{arXiv:math/0309453}))

\end{itemize}
A survey of some useful facts with an eye towards [[dg-geometry]] is in

\begin{itemize}%
\item [[Kai Behrend]], \emph{Differential graded schemes I: prefect resolving algebras} (\href{http://arxiv.org/abs/math/0212225}{arXiv:0212225})

\end{itemize}
Discussion of cofibrations in unbounded dg-algebras are in

\begin{itemize}%
\item [[Bernhard Keller]], \emph{$A_\infty$-algebras, modules and functor categories} (\href{http://www.math.jussieu.fr/~keller/publ/ainffun.pdf}{pdf})

\end{itemize}
The derived copowering of unbounded commutative dg-algebras over $sSet$ is discussed (somewhat implicitly) in

\begin{itemize}%
\item [[Grégory Ginot]], Thomas Tradler, Mahmoud Zeinalian, \emph{Derived higher Hochschild homology, topological chiral homology and factorization algebras}, (\href{http://arxiv.org/abs/1011.6483}{arxiv/1011.6483})

\end{itemize}
The \emph{commutative} product on the dg-algebra of the higher order Hochschild complex is discussed in

\begin{itemize}%
\item [[Grégory Ginot]], Thomas Tradler, Mahmoud Zeinalian, \emph{A Chen model for mapping spaces and the surface product} (\href{http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.2231v1.pdf}{pdf})

\end{itemize}
The relation to [[E-infinity algebras]] is discussed in

\begin{itemize}%
\item [[Igor Kriz]] and [[Peter May]], \emph{Operads, algebras, modules and motives} , Ast\'e{}risque No 233 (1995)


\item [[Jacob Lurie]], section 7.1 of \emph{Higher algebra} (\href{http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf}{pdf})



\end{itemize}
The relation between commutative and non-commutative dgas is further discussed in

\begin{itemize}%
\item [[Ilias Amrani]], \emph{Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view} (\href{http://arxiv.org/abs/1401.7285}{arXiv:1401.7285})


\item [[Ilias Amrani]], \emph{Rational homotopy theory of function spaces and Hochschild cohomology} (\href{http://arxiv.org/abs/1406.6269}{arXiv:1406.6269})



\end{itemize}
For more see also at \emph{[[model structure on dg-algebras over an operad]]}.

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

\begin{itemize}%
\item [[Ben Walter]], \emph{Rational Homotopy Calculus of Functors} (\href{http://arxiv.org/abs/math/0603336}{arXiv:math/0603336})

\end{itemize}
[[!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 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]]

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



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