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\section*{simplicially enriched category}

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

\hypertarget{enriched_category_theory}{}\paragraph*{{Enriched category theory}}\label{enriched_category_theory}

[[!include enriched category theory contents]]

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

[[!include homotopy - contents]]

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory 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{as_models_for_categories}{As models for $(\infty,1)$-categories}\dotfill \pageref*{as_models_for_categories} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{simplicial_localization}{Simplicial localization}\dotfill \pageref*{simplicial_localization} \linebreak
\noindent\hyperlink{model_structure}{Model structure}\dotfill \pageref*{model_structure} \linebreak
\noindent\hyperlink{homotopy_kan_extension}{Homotopy Kan extension}\dotfill \pageref*{homotopy_kan_extension} \linebreak
\noindent\hyperlink{presentation_of_topos_theory}{Presentation of $(\infty,1)$-topos theory}\dotfill \pageref*{presentation_of_topos_theory} \linebreak
\noindent\hyperlink{as_models_for_categories_2}{As models for $(\infty,2)$-categories}\dotfill \pageref*{as_models_for_categories_2} \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{simplicially enriched category} is a [[category]] with a [[simplicial set]] of [[morphism]]s between any two objects.

One may think of the 1-cells in a hom-simplicial set as a [[2-morphism]], the 2-cells as a [[3-morphism]] and generally a $(k-1)$-cell as a [[k-morphism]]. Therefore simplicially enriched categories may serves as models for [[∞-categories]]. Precisely which notion of $\infty$-category depends on which extra [[stuff, structure, property|structure and property]] one imposes.

For instance

\begin{itemize}%
\item requiring the hom-simplicial sets to be [[Kan complex]]es makes simplicially enriched categories a model for [[(∞,1)-categories]];


\item similary, equipping the $sSet$-enriched category with the structure of a $sSet_{Quillen}$-[[enriched model category]] -- a [[simplicial model category]] -- makes it a model for an $(\infty,1)$-category.

This is discussed in more detail at [[relation between quasi-categories and simplicial categories]].


\item on the other hand, eqipping the $sSet$-enriched category with the structure of an $sSet_{Joyal}$-enriched model category over the Joyal-[[model structure for quasi-categories]] makes it a model for an [[(∞,2)-category]].



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

\begin{udefn}
A \emph{simplicially enriched category} is a [[enriched category|category enriched over]] the [[cartesian monoidal category]] [[sSet]] of [[simplicial sets]].

\end{udefn}
\begin{uremark}
These $sSet$-enriched categories are sometimes, somewhat imprecisely, called just \emph{[[simplicial categories]]}.

\end{uremark}
There is a related notion of [[simplicial groupoid]] with the added requirement that all arrows in the categories concerned are [[isomorphisms]].

\hypertarget{as_models_for_categories}{}\subsection*{{As models for $(\infty,1)$-categories}}\label{as_models_for_categories}

Since simplicial sets that are [[Kan complex]]es are an incarnation of [[∞-groupoid]]s, an $sSet$-category all whose [[hom-object]]s happen to be [[Kan complex]]es may be regarded as a category enriched in [[∞-groupoid]]s. By the logic of [[(n,r)-category]] theory this should be a model for an [[(∞,1)-category]].

Treating simplicial categories this way as models for $(\infty,1)$-categories is one of the central tools in [[homotopy coherent category theory]].

Indeed, there is a [[model structure on simplicial categories]] whose fibrant objects are [[Kan complex|Kan-complex]]-enriched categories, and which is one model for the [[(∞,1)-category of (∞,1)-categories]].

By a web of [[Quillen equivalence]]s this is related to the other incarnations of $(\infty,1)$-categories. Notably to [[quasi-categories]] and [[complete Segal space]]s. For more on this see

\begin{itemize}%
\item [[relation between quasi-categories and simplicial categories]].

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

\hypertarget{simplicial_localization}{}\paragraph*{{Simplicial localization}}\label{simplicial_localization}

To every [[category with weak equivalences]] $(C,W)$ is associated its [[simplicial localization]] $L_W C$, which is an $sSet$-category with the property that its [[homotopy category of an (∞,1)-category]] coincides with the [[homotopy category]] $Ho_W(C)$.

\hypertarget{model_structure}{}\paragraph*{{Model structure}}\label{model_structure}

There is a [[model structure on sSet-categories]] that presents the [[(∞,1)-category]] [[(∞,1)Cat]].

\hypertarget{homotopy_kan_extension}{}\paragraph*{{Homotopy Kan extension}}\label{homotopy_kan_extension}

The notion of [[homotopy Kan extension]] and hence in particular that of [[homotopy limit]] and [[homotopy colimit]] has a direct formulation in terms of [[Kan complex|Kan-complex]]-enriched categories. See [[homotopy Kan extension]] for more.

\hypertarget{presentation_of_topos_theory}{}\paragraph*{{Presentation of $(\infty,1)$-topos theory}}\label{presentation_of_topos_theory}

All of [[(∞,1)-topos theory]] can be modeled in terms of $sSet$-categories. (\hyperlink{Toenvezzosie}{To\"e{}nVezzosi}). There is a notion of [[sSet-site]] $C$ that models the notion of [[(∞,1)-site]] and a [[model structure on sSet-enriched presheaves]] on $sSet$-sites that is a [[presentable (∞,1)-category|presentation]] for the [[∞-stack]] [[(∞,1)-topos]]es on $C$.

\hypertarget{as_models_for_categories_2}{}\subsection*{{As models for $(\infty,2)$-categories}}\label{as_models_for_categories_2}

See [[(∞,2)-category]].

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

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

\begin{itemize}%
\item \textbf{simplicially enriched category}


\item [[simplicial object in Cat]]



\end{itemize}

\item [[simplicial groupoid]]


\item [[relation between quasi-categories and simplicial categories]]


\item [[topologically enriched category]]



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

In the context of [[model category]] theory, simplicially enriched categories ([[simplicial model categories]]) appear in

\begin{itemize}%
\item [[Dan Quillen]], chapter II, section 1 of \emph{Homotopical algebra}, Lecture Notes in Mathematics \textbf{43}, Springer-Verlag 1967, iv+156 pp.

\end{itemize}
The original references on homotopy theory in terms of $sSet$-categories are

\begin{itemize}%
\item [[William Dwyer]], [[Dan Kan]], \emph{Simplicial localization of categories}, J. Pure and Appl. Algebra 17 (1980), 267-284.


\item [[William Dwyer]], [[Dan Kan]], \emph{Equivalences between homotopy theories of diagrams} , in Algebraic topology and algebraic K-theory, Annals of Math. Studies 113, Princeton University Press, Princeton, 1987, 180-205.



\end{itemize}
Simplicially enriched categories as models for $(\infty,1)$-categories are discussed in some detail in section A.3 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
as well as in section 2 of

\begin{itemize}%
\item [[Bertrand Toën]], [[Gabriele Vezzosi]], \emph{Topos theory} (\href{http://arxiv.org/abs/math/0207028}{arXiv:0207028})

\end{itemize}
Homotopy coherent category theory on $sSet$-categories is discussed in

\begin{itemize}%
\item [[Jean-Marc Cordier]], and [[Timothy Porter]], \emph{Homotopy coherent category theory} (\href{http://www.ams.org/journals/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf}{pdf})

\end{itemize}
which describes resolutions of the simplicial functor categories between two simplicial categories and

\begin{itemize}%
\item [[Michael Batanin]], \emph{Homotopy coherent category theory and $A_\infty$-structures in monoidal categories} (\href{http://dodo.pdmi.ras.ru/~topology/books/batanin.pdf}{pdf})

\end{itemize}
which shows that these resolved functor categories are in fact $sSet$-[[A-∞ categories]].

[[!redirects simplicially enriched category]] [[!redirects simplicially enriched categories]] [[!redirects sSet-enriched category]] [[!redirects sSet-enriched categories]] [[!redirects SSet-enriched category]] [[!redirects SSet-enriched categories]] [[!redirects SimpSet-enriched category]] [[!redirects SimpSet-enriched categories]] [[!redirects sSet-category]] [[!redirects sSet-categories]] [[!redirects SSet-category]] [[!redirects SSet-categories]] [[!redirects SimpSet-category]] [[!redirects SimpSet-categories]]



\end{document}