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\begin{document}

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\section*{model structure on cosimplicial rings}

\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{the_model_structure}{The model structure}\dotfill \pageref*{the_model_structure} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{simplicial_model_category_structure}{Simplicial model category structure}\dotfill \pageref*{simplicial_model_category_structure} \linebreak
\noindent\hyperlink{relation_to_the_model_structure_on_cochain_dgcalgebras}{Relation to the model structure on cochain dgc-algebras}\dotfill \pageref*{relation_to_the_model_structure_on_cochain_dgcalgebras} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A [[model category]] structure on [[cosimplicial objects]] in unital, [[commutative algebras]] over some [[field]] $k$.

Under the [[monoidal Dold-Kan correspondence]] this is [[Quillen equivalence|Quillen equivalent]] to the [[model structure on dg-algebras|model structure on commutative non-negative cochain dg-algebras]].

\hypertarget{the_model_structure}{}\subsection*{{The model structure}}\label{the_model_structure}

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

\begin{defn}
\label{CosimplicalCommutativeAlgebras}\hypertarget{CosimplicalCommutativeAlgebras}{}
Write $cAlg_k^\Delta$ for the [[category]] of [[cosimplicial objects]] in the [[category]] of unital, [[commutative algebras]] over $k$.

\end{defn}
\begin{remark}
\label{ComparisonFunctors}\hypertarget{ComparisonFunctors}{}
Sending $k$-[[associative algebras|algebras]] to their underlying $k$-[[modules]] yields a [[forgetful functor]]

\begin{displaymath}
U \colon cAlg_k^\Delta \longrightarrow k Mod^\Delta
\end{displaymath}
from cosimplicial $k$-allgebras (def. \ref{CosimplicalCommutativeAlgebras}) to [[cosimplicial objects]] in $k$-[[vector spaces]].

Moreover, the [[Dold-Kan correspondence]] provides the [[normalized cochain complex]] functor

\begin{displaymath}
N \colon k Mod_k^\Delta \to Ch^{\geq 0}(k)
\end{displaymath}
from [[cosimplicial objects|cosimplicial]] $k$-[[vector spaces]] to [[cochain complexes]] (i.e. with [[differential]] of degree +1) in non-negative degrees.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
Say that morphism $f \colon A \to B$ in $cAlg_k^{\Delta}$ (def. \ref{CosimplicalCommutativeAlgebras}) is

\begin{enumerate}%
\item a \emph{weak equivalence} if its image $N(U(f)) \colon N(U(A)) \to N(U(B))$ under the comparison functors from remark \ref{ComparisonFunctors} is a [[quasi-isomorphism]] in $Ch^{\geq 0}(k)$;

\end{enumerate}
1.a \emph{fibration} if $f$ is an [[epimorphism]] (i.e. degreewise a [[surjection]]).

Then

\begin{enumerate}%
\item this defines a [[model category]] structure, to be called the \emph{projective model structure} on comsimplicial commutative $k$-algebras. $(cAlg_k^\Delta)_{poj}$.


\item this is a [[cofibrantly generated model category]]


\item and a [[simplicial model category]].



\end{enumerate}
\end{prop}
e.g. \hyperlink{Toen00}{To\"e{}n 00, theorem 2.1.2}

\begin{proof}
The first two statements follow by observing that $(cAlg_k^{\Delta})_{proj} is$the [[transferred model structure]] along the [[forgetful functor]] $U \circ N$ from remark \ref{ComparisonFunctors} of the [[projective model structure on chain complexes]], by \href{transferred+model+structure#SufficientConditions}{this prop.}.

The third statement is the content of prop. \ref{SimplicialModelStructure} below.

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{simplicial_model_category_structure}{}\subsubsection*{{Simplicial model category structure}}\label{simplicial_model_category_structure}

There is also the structure of an [[sSet]]-[[enriched category]] on $cAlg_k^\Delta$ (def. \ref{cAlgDelta})

\begin{defn}
\label{cAlgDelta}\hypertarget{cAlgDelta}{}
For $X$ a [[simplicial set]] and $A \in Alg_k$ let $A^X  \in Alg_k^\Delta$ be the corresponding $A$-valued [[cochains on simplicial sets]]

\begin{displaymath}
A^X \;\colon\; [n]  \mapsto (A_n)^{X_n} = \underset{X_n}{\prod} A_n
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If we write $C(X) \coloneqq Hom_{Set}(X_\bullet,k)$ for the cosimplicial algebra of [[cochains on simplicial sets]] then for $X$ degreewise finite this may be written as

\begin{displaymath}
A^X = A \otimes C(X)
\end{displaymath}
where the tensor product is the degreewise tensor product of $k$-algebras.

\end{remark}
See also \hyperlink{CastiglioniCortinas03}{Castiglioni-Cortinas 03, p. 10}.

\begin{defn}
\label{}\hypertarget{}{}
For $A,B \in Alg_k^\Delta$ define the [[sSet]]-[[hom-object]] $Alg_k^\Delta(A,B)$ by

\begin{displaymath}
Alg_k^\Delta(A,B) \coloneqq Hom_{sSet}(A, B^{\Delta[\bullet]})
  = Hom_{sSet}(A, B \otimes C(\Delta[\bullet]))
  \in sSet
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
For $B \in Alg_k$ regarded as a constant cosimplicial object under the canonical embedding $Alg_k \hookrightarrow Alg_k^\Delta$ we have

\begin{displaymath}
Alg_k^\Delta(A, B^{\Delta[n]})
  =
  Alg_k^\Delta(A, B \otimes C(\Delta[n]))
  \simeq
  Alg_k(A_n,B)
  \,.
\end{displaymath}
\end{remark}
\begin{proof}
Let $f : A \to B \otimes C(\Delta[n])$ be a morphism of cosimplicial algebras and write

\begin{displaymath}
f_n : A_n \to B
\end{displaymath}
for the component of $f$ in degree $n$ with values in the copy $B = B \otimes k$ of functions $k$ on the unique non-degenerate $n$-[[simplex]] of $\Delta[n]$. The $n+1$ coface maps $C(\Delta[n])_n \leftarrow C(\Delta[n])_{n-1}$ obtained as the pullback of the $(n+1)$ face inclusions $\Delta[n-1] \to \Delta[n]$ restrict on the non-degenerate $(n-1)$-cells to the $n+1$ projections $k \leftarrow k^{n+1} : p_i$.

Accordingly, from the naturality squares for $f$

\begin{displaymath}
\itexarray{
    A_n &\stackrel{f_n}{\to}& B
    \\
    \uparrow^{\mathrlap{\delta_i}} && \uparrow^{\mathrlap{p_i}}
    \\
    A_{n-1} &\stackrel{f_{n-1}}{\to}& B^{n+1}
  }
\end{displaymath}
the bottom horizontal morphism is fixed to have components

$f_{n-1} = (f_n \circ \delta_0, \cdots, f_n \circ \delta_n)$

in the functions on the non-degenerate simplices.

By analogous reasoning this fixes all the components of $f$ in all lower degrees with values in the functions on degenerate simplices.

\end{proof}
The above [[sSet]]-[[enriched category theory|enrichment]] makes $cAlg_k^\Delta$ into a [[simplicially enriched category]] which is [[copower|tensored]] and [[power|cotensored]] over $sSet$.

And this is compatible with the model category structure:

\begin{prop}
\label{SimplicialModelStructure}\hypertarget{SimplicialModelStructure}{}
With the definitions as above, $(cAlg_k^\Delta)_{proj}$ is a [[simplicial model category]].

\end{prop}
\hyperlink{Toen00}{To\"e{}n 00, theorem 2.1.2}

\hypertarget{relation_to_the_model_structure_on_cochain_dgcalgebras}{}\subsubsection*{{Relation to the model structure on cochain dgc-algebras}}\label{relation_to_the_model_structure_on_cochain_dgcalgebras}

Under the [[monoidal Dold-Kan correspondence]] this is related to the [[model structure on dg-algebras|model structure on commutative non-negative cochain dg-algebras]].

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

Details are in

\begin{itemize}%
\item [[Bertrand Toën]], section 2.1 of \emph{Affine stacks (Champs affines)} (\href{http://arxiv.org/abs/math/0012219}{arXiv:math/0012219}) .

\end{itemize}
See also

\begin{itemize}%
\item [[Paul Goerss]], [[Rick Jardine]], \emph{[[Simplicial homotopy theory]]}

\end{itemize}
The generalization to arbitrary cosimplicial rings is proposition 9.2 of

\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/0306289}{arXiv:math/0306289})

\end{itemize}
There also aspects of relation to the [[model structure on dg-algebras]] is discussed. (See [[monoidal Dold-Kan correspondence]] for more on this).

[[!redirects model structure on cosimplicial algebras]]



\end{document}