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\begin{document}

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\section*{simplicial ring}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{connected_components}{Connected components}\dotfill \pageref*{connected_components} \linebreak
\noindent\hyperlink{higher_homotopy_groups}{Higher homotopy groups}\dotfill \pageref*{higher_homotopy_groups} \linebreak
\noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{simplicial_fields}{Simplicial fields}\dotfill \pageref*{simplicial_fields} \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 \emph{simplicial ring} is a [[simplicial object]] in the [[category]] [[Ring]] of [[rings]].

This may be understood conceptually as follows:

\begin{itemize}%
\item as ordinary rings are algebras over the ordinary [[algebraic theory]] $T$ of rings, if we regard this as an [[(∞,1)-algebraic theory]] then simplicial rings model the $(\infty,1)$-algebras over that;


\item the category [[Ring]]${}^{op}$ is naturally equipped with the structure of a [[geometry (for structured (infinity,1)-toposes)|pregeometry]]. The corresponding [[geometry (for structured (∞,1)-toposes)]] is $sRing^{op}$, the opposite of the category of simplicial rings.



\end{itemize}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

A \textbf{simplicial ring} is a [[simplicial object]] in the [[category]] [[Ring]] of [[rings]].

There is an evident notion of [[(∞,1)-category]] of [[module]]s over a simplicial ring. The corresponding [[bifibration]] $sMod \to sRing$ of modules over simplicial ring is equivalently to the [[tangent (∞,1)-category]] of the [[(∞,1)-category]] of simplicial rings.

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

\hypertarget{connected_components}{}\subsubsection*{{Connected components}}\label{connected_components}

Given a simplicial ring $A = A_{\bullet}$, its [[connected components]] (the 0th ``[[homotopy group]]'')) is an ordinary [[ring]]

\begin{displaymath}
\pi_0(A) \in Ring
  \,.
\end{displaymath}
Forming connected components is a [[functor]] from simplicial rings to plain rings, which is [[left adjoint]] to the inclusion of ordinary rings as simplicially constant simplicial rings, exhibiting a [[reflective subcategory]] inclusion

\begin{displaymath}
Ring \stackrel{\stackrel{\pi_0}{\longleftarrow}}{\hookrightarrow} Ring^{\Delta^{op}}
  \,.
\end{displaymath}
(\href{Toen14}{To\"e{}n 14, section 2.2})

So on [[formal duals]] of [[commutative ring|commutative]] (simplicial) rings this is a [[coreflection]] of [[affine schemes]] in affine [[derived schemes]]

\begin{displaymath}
CRing^{op}
    \stackrel{\hookrightarrow}{\longleftarrow}
  (CRing^{\Delta^{op}})^{op}
  \,.
\end{displaymath}
Notice that coreflective embeddings are also given for instance by the inclusion of [[manifolds]] into [[formal manifolds]]. This is one way in which formal duals of simplicial rings manifest themselves as [[infinitesimal neighbourhoods]] of formal duals of plain rings.

\hypertarget{higher_homotopy_groups}{}\subsubsection*{{Higher homotopy groups}}\label{higher_homotopy_groups}

The higher [[homotopy groups]] $\pi_n(A)$ of a simplicial ring $A_\bullet$ are naturally [[modules]] over the ring $\pi_0(A)$ of connected components, so $A_\bullet$ is weakly [[contractible]], as a [[simplicial set]], iff $\pi_0(A)=0$.

(Again this is a manifestation of the simplicial ring being just an infinitesimal thickening of its connected components.)

Notice that $\pi_0(A)$ is equivalently the [[cokernel]] of $d_0-d_1 \colon A_1 \longrightarrow A_0$. Accordingly, any chain of [[face maps]] $A_n \longrightarrow A_0$ compose with the projection to $\pi_0(A)$ is independent of the choices. These maps

\begin{displaymath}
A_n \longrightarrow \pi_0(A)
\end{displaymath}
give a [[surjective]] map from $A_\bullet$ to the constant simplicial ring $\pi_0(A)$.

(This is just the simplest piece of the [[Postnikov tower]].) If $\pi_0(A)\neq 0$ we can then compose with a surjective map to a constant simplicial field.

\hypertarget{model_category_structure}{}\subsubsection*{{Model category structure}}\label{model_category_structure}

There is a [[model category]] structure on simplicial rings that presents $\infty$-rings. See [[model structure on simplicial T-algebras]] for more.

We describe here the [[model category]] presentation of the [[(∞,1)-category]] of modules over simplicial rings.

Let $A$ be a simplicial commutative algebra. Write $A SMod$ for the [[category]] which, with $A$ regarded as a [[monoid]] in the category $SAb$ of abelian [[simplicial group]]s is just the category of $A$-[[module]]s in $SAb$. This means that

\begin{itemize}%
\item objects are abelian [[simplicial group]]s $N$ equipped with an [[action]] [[morphism]] $A \otimes N \to N$ of simplicial abelian groups;

\end{itemize}
Equip $A SMod$ with the structure of a [[model category]] by setting:

\begin{itemize}%
\item a morphism $N_1\to N_2$ of $A$-[[modules]] is a weak equivalence resp. a fibration precisely if the underlying morphism of [[simplicial set]]s is a weak equivalence, resp. fibration, in the standard [[model structure on simplicial sets]].

\end{itemize}
\textbf{Proposition} This defines a [[model category]] structure which is

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


\item [[proper model category|proper]];


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



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{simplicial_fields}{}\subsubsection*{{Simplicial fields}}\label{simplicial_fields}

All simplicial [[fields]] are simplicially constant. This is because the composite $A_0\xrightarrow{s_0^n}A_n\xrightarrow{d_0^n}A_0$ is the identity, so $d_0^n$ is surjective, but all field homomorphisms are injective, so $d_0^n$ is an isomorphism.

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

\begin{itemize}%
\item [[simplicial algebra]]


\item [[model structure on simplicial T-algebras]]



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

Introduction and survey includes

\begin{itemize}%
\item [[Bertrand Toën]], chapter 4 of \emph{Simplicial presheaves and derived algebraic geometry} , lecture at \href{http://www.crm.es/HigherCategories/}{Simplicial methods in higher categories} (\href{http://www.crm.cat/HigherCategories/hc1.pdf}{pdf})


\item [[Bertrand Toën]], \emph{Derived Algebraic Geometry} (\href{http://arxiv.org/abs/1401.1044}{arXiv:1401.1044})



\end{itemize}
See [[model structure on simplicial algebras]] for references on the model structure discussed above.

Some of the above material is taken from .

Discussion in the context of [[homotopy theory]], hence for simplicial [[ring spectra]] includes

\begin{itemize}%
\item [[Paul Goerss]], [[Michael Hopkins]], \emph{Simplicial Structured Ring Spectra} (1999) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.34.9216}{web})

\end{itemize}
[[!redirects simplicial rings]] [[!redirects simplicial commutative ring]] [[!redirects simplicial commutative rings]]



\end{document}