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\begin{document}

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\section*{simplicial object}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

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

[[!include homotopy - 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{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{category_of_simplicial_objects}{Category of simplicial objects}\dotfill \pageref*{category_of_simplicial_objects} \linebreak
\noindent\hyperlink{simplicial_enrichment}{Simplicial enrichment}\dotfill \pageref*{simplicial_enrichment} \linebreak
\noindent\hyperlink{geometric_realization}{Geometric realization}\dotfill \pageref*{geometric_realization} \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 object} $X$ in a [[category]] $C$ is an \emph{[[simplicial set]]} [[internalization|internal]] to $C$: a collection $\{X_n\}_{n \in \mathbb{N}}$ of [[objects]] in $C$ that behave as if $X_n$ were an object of $n$-dimensional [[simplices]] [[internalization|internal to]] $C$ equipped with maps between these space that assign faces and degenerate simplices.

For instance, and there is a longer list further down this page, a simplicial object in $Grps$ is a collection $\{G_n\}_{n\in \mathbb{N}}$ of groups, together with face and degeneracy \emph{homomorphisms} between them. This is just a [[simplicial group]]. We equally well have other important instances of the same idea, when we replace $Grps$ by other categories, or higher categories.

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

A \textbf{simplicial object} in a [[category]] $C$ is a [[functor]] $\Delta^{op} \to C$, where $\Delta$ is the [[simplex category|simplicial indexing category]].

More generally, a simplicial object in an [[(∞,1)-category]] is an [[(∞,1)-functor]] $\Delta^{op} \to C$.

A \textbf{[[cosimplicial object]]} in $C$ is similarly a functor out of the [[opposite category]], $\Delta \to C$.

Accordingly, simplicial and cosimplicial objects in $C$ themselves form a [[category]] in an obvious way, namely the [[functor category]] $[\Delta^{op},C]$ and $[\Delta,C]$, respectively.

\textbf{Remark}

A \textbf{simplicial object} $X$ in $C$ is often specified by the objects, $X_n$, which are the images under $X$, of the objects $[n]$ of $\Delta$, together with a description of the face and degeneracy morphisms, $d_i$ and $s_j$, which must satisfy the [[simplicial identities]].

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{itemize}%
\item A simplicial object in [[Set]] is a [[simplicial set]].


\item A simplicial object in a category of [[presheaves]] is a [[simplicial presheaf]].


\item A simplicial object in [[Top]] is a [[simplicial topological space]].


\item A simplicial object in [[Diff]] is a [[simplicial manifold]].


\item A simplicial object in the category [[Grp]] of [[groups]] is a [[simplicial group]]. See also [[Dold-Kan correspondence]].


\item A simplicial object in the category of [[topological group]]s is a [[simplicial topological group]].


\item A simplicial object in [[Lie algebra]]s is a [[simplicial Lie algebra]].


\item A simplicial object in [[Ring]] is a [[simplicial ring]].


\item A cosimplicial object in the category of [[rings]] ([[algebras]]) is a [[cosimplicial ring]] ([[cosimplicial algebra]]).


\item A simplicial object in a category of simplicial objects is a [[bisimplicial object]].


\item A cosimplicial object in [[sSet]] is a [[cosimplicial simplicial set]] (equivalently a simplicial object in cosimplicial sets).


\item The [[bar construction]] produces a simplicial object from a [[monad]] and an algebra over that monad.



\end{itemize}
\hypertarget{category_of_simplicial_objects}{}\subsection*{{Category of simplicial objects}}\label{category_of_simplicial_objects}

For $D$ a [[category]], we write $D^{\Delta^{op}}$ for the [[functor category]] from $\Delta^{op}$ to $D$: its category of simplicial objects.

\hypertarget{simplicial_enrichment}{}\subsubsection*{{Simplicial enrichment}}\label{simplicial_enrichment}

\begin{defn}
\label{SimplicialEnrichment}\hypertarget{SimplicialEnrichment}{}
Let $D$ be a [[nLab:category]] with all [[nLab:limit]]s and [[nLab:colimit]]s. This implies that it is [[nLab:copower|tensored]] over [[nLab:Set]]

\begin{displaymath}
\cdot : D \times Set \to D
  \,.
\end{displaymath}
This induces a functor

\begin{displaymath}
\cdot^{\Delta^{op}} : D^{\Delta^{op}} \times sSet \to D^{\Delta^{op}}
\end{displaymath}
which we shall also write just ``$\cdot$''.

For $X,Y \in D^{\Delta^{op}}$ write

\begin{displaymath}
D^{\Delta^{op}}(X,Y) := Hom_{D^{\Delta^{op}}}(X \cdot \Delta[\bullet], Y)
  \in sSet
\end{displaymath}
and for $X,Y,Z \in D^{\Delta^{op}}$ let

\begin{displaymath}
D^{\Delta^{op}}(X,Y)
    \times  
  D^{\Delta^{op}}(Y,Z)
  \to 
  D^{\Delta^{op}}(X,Z)
\end{displaymath}
be given in degree $n$ by

\begin{displaymath}
(X \cdot \Delta[n] \to Y, Y \cdot \Delta[n] \to Z)
  \mapsto
  ( X \cdot \Delta[n] \to X \cdot \Delta[n]\times \Delta[n] \to Y \cdot \Delta[n]  \to Z)
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
With the above definitions $D^{\Delta^{op}}$ becomes an [[nLab:sSet]]-[[nLab:enriched category]] which is both [[nLab:copower|tensored]] as well as [[nLab:power|cotensored]] over $sSet$.

\end{prop}
\begin{defn}
\label{}\hypertarget{}{}
We may regard the category of cosimplicial objects $D^{\Delta}$ as an $sSet$-enriched category using the above enrichment by identifying

\begin{displaymath}
D^{\Delta} \simeq ({D^{op}}^{\Delta^{op}})^{op}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{geometric_realization}{}\subsubsection*{{Geometric realization}}\label{geometric_realization}

If $D$ is already a [[simplicially enriched category]] in its own right, with [[powers]] and [[copowers]], we can define the [[geometric realization]] of a simplicial object $X\in D^{\Delta^{op}}$ as a [[coend]]:

\begin{displaymath}
|X| = \int^{[n]\in\Delta} \Delta[n] \odot X_n
\end{displaymath}
where $\odot$ denotes the [[copower]] for the simplicial enrichment of $D$. This is [[left adjoint]] to the ``total singular object'' functor $D \to D^{\Delta^{op}}$ sending $Y$ to the simplicial object $n\mapsto Y^{\Delta[n]}$, the [[power]] for the simplicial enrichment.

Perhaps surprisingly, this adjunction is even a \emph{simplicially enriched} adjunction when $D^{\Delta^{op}}$ has its simplicial structure from Definition \ref{SimplicialEnrichment}, even though the latter makes no reference to the given simplicial enrichment of $D$. A proof can be found in \hyperlink{RSS01}{RSS01, Proposition 5.4}.

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

\begin{itemize}%
\item [[simplex]], [[simplex category]]


\item \textbf{simplicial object}

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


\item [[simplicial diagram]]


\item [[simplicial object in an (∞,1)-category]]



\end{itemize}

\item [[semi-simplicial object]]

\begin{itemize}%
\item [[semisimplicial set]]

\end{itemize}

\item [[globular set]], [[cubical set]]


\item [[nerve]], [[nerve and realization]]


\item [[Segal condition]]



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

\begin{itemize}%
\item [[Peter May]], \emph{Simplicial objects in algebraic topology} , University of Chicago Press, 1967, (\href{http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu}{djvu})


\item [[Charles Rezk]], [[Stefan Schwede]], and [[Brooke Shipley]], \emph{Simplicial structures on model categories and functors}, \href{https://arxiv.org/abs/math/0101162}{arxiv}



\end{itemize}
[[!redirects simplicial objects]]

[[!redirects category of simplicial objects]]

[[!redirects categories of simplicial objects]]

category: simplicial object



\end{document}