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\begin{document}

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\section*{Segal category}

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

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

[[!include quasi-category theory contents]]

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

[[!include enriched 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{model_category_structure}{Model category structure}\dotfill \pageref*{model_category_structure} \linebreak
\noindent\hyperlink{operadic_version}{Operadic version}\dotfill \pageref*{operadic_version} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{inclusion_of_ordinary_categories}{Inclusion of ordinary categories}\dotfill \pageref*{inclusion_of_ordinary_categories} \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}

The notion of \emph{Segal category} is one of the models for that of \emph{[[(∞,1)-category]]}, given by regarding an $(\infty,1)$-category as an [[∞Grpd]]-[[enriched (∞,1)-category]].

So the notion can be understood as modelling the notion of an [[sSet]]-[[enriched category|enrichment]] \emph{up to [[coherence|coherent]] [[homotopy]]}, i.e. a \emph{weak} enrichment. As such it is closely related to the notion of [[complete Segal space]], which models the notion of an [[internal category in an (∞,1)-category|internal category]] in [[sSet]].

Indeed, Segal categories may be considered with enrichment not just over [[sSet]], but over other suitable [[model categories]]. In particular, an iterated enrichment over itself gives rise to the notion of \emph{[[Segal n-category]]} which is a model for \emph{[[(∞,n)-categories]]}.

Since the major difference between ([[small category|small]]) $\mathcal{V}$-[[enriched categories]] and $\mathcal{V}$-[[internal categories]] is that in the first case the [[objects]] (as opposed to all the [[hom objects]]) form an ordinary [[set]], while in the second these form an object of $\mathcal{V}$, too, accordingly a the definition of \emph{Segal category} is like that of \emph{([[complete Segal space|complete]]) [[Segal space]]}, only that the simplicial set of objects is required to be an ordinary set (a [[discrete object|discrete]] simplicial set).

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

\begin{def}
\label{}\hypertarget{}{}
A \textbf{Segal category} is

\begin{itemize}%
\item a [[simplicial object|simplicial]] [[simplicial set]] ($\simeq$ [[bisimplicial set]]) $X \in [\Delta^{op}, sSet]$,

where we call $X_0$ the \emph{simplicial set of [[objects]]}, $X_1$ the \emph{simplicial set of [[morphisms]]}; and $X_k$ for $k \geq 2$ the \emph{simplicial set of sequences of composable morphisms of length $k$};


\item such that $X_0$ is a [[discrete object|discrete]] (= constant) simplicial set;


\item and such that the [[Segal maps]]

\begin{displaymath}
X^{Sp[k] \hookrightarrow \Delta[k]}
   :
   X_k 
     \stackrel{\simeq}{\to}    
   X_1 \times_{X_0} \cdots \times_{X_0} X_1 \;\;(k factors)
\end{displaymath}
induced by the [[spine]] inclusions $Sp[k] \hookrightarrow \Delta[k]$ are [[model structure on simplicial sets|weak equivalences of simplicial sets]] for $k \geq 2$.



\end{itemize}
\end{def}
\begin{remark}
\label{}\hypertarget{}{}
There is \emph{no} condition that a Segal category be [[fibrant object|fibrant]] with respect to the [[Reedy model structure]] on bisimplicial sets.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
For $X$ a Segal category, the [[fiber product]] simplicial set $X_1 \times_{X_0} X_1$ is manifestly the space of pairs of composable 1-[[morphism]]s in $X$, and the [[weak equivalence]]

\begin{displaymath}
(d_0,d_2) 
    :  
  X_2 \stackrel{\simeq}{\to} X_1 \times_{X_0} X_1
\end{displaymath}
given by the above definition together with the remaining face map $d_1 : X_2 \to X_1$ constitutes an [[∞-anafunctor]]

\begin{displaymath}
\circ : X_1 \times_{X_0} X_1 &#8696; X_1
\end{displaymath}
given by the [[span]]

\begin{displaymath}
\itexarray{
    X_2 &\stackrel{d_1}{\to}& X_1
    \\
    {}^{\mathllap{\simeq}}\downarrow
    \\
    X_1 \times_{X_0} X_1
  }
  \,.
\end{displaymath}
This encodes the [[composition]] operation in the Segal category $X$.

Accordingly, the analogous spans out of $X_k$ for $k \geq 3$ encode the [[associativity]] of this composition as well as all its [[coherence|coherences]].

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

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

The category of [[bisimplicial sets]] carries a [[model category]] structure whose [[fibrant objects]] are the [[Reedy model structure|Reedy fibrant]] Segal categories. This \emph{[[model structure for Segal categories]]} is a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-categories]].

\hypertarget{operadic_version}{}\subsubsection*{{Operadic version}}\label{operadic_version}

The [[operad|operadic]] generalization of Segal category is that of \emph{[[Segal operad]]}. Segal categories are precisely those Segal operads whose only [[inhabited set|inhabited]] operations-spaces are those of unary operations.

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

\hypertarget{inclusion_of_ordinary_categories}{}\subsubsection*{{Inclusion of ordinary categories}}\label{inclusion_of_ordinary_categories}

Let $C$ be an ordinary [[small category]] and write $N(C) \in sSet$ for its [[nerve]]. Regard this as a bisimplicial set under the inclusion $sSet \simeq [\Delta^{op}, Set] \hookrightarrow [\Delta^{op}, sSet]$.

Then $N(C)$ is a Segal category. Each simplicial set $N(C)_k$ is discrete, for all $k \in \mathbb{N}$, and all the morphisms

\begin{displaymath}
N(C)_k \to Mor(C) \times_{Obj(C)} \cdots \times_{Obj(C)} Mor(C)
\end{displaymath}
are in fact [[isomorphisms]] / [[bijections]] of sets. This property of the [[nerve]] of an ordinary category goes by the name \emph{Segal condition} and is what gave Segal categories its name.

One may also form the $n$-fold [[comma object]]-fiber product of a choice of base points $\pi_0(C) \to C$ with itself. This yields a Segal category incarnation of $C$ where in degree 1 we have the [[groupoid]] [[core]] of the [[arrow category]] of $C$. For more on this see at \emph{\href{Segal+space#ConstructionFromACategory}{Segal space -- Examples - From a category}}.

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

\begin{itemize}%
\item [[Gamma space]]


\item [[Segal operad]]


\item [[table - models for (infinity,1)-operads]]



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

The idea of Segal categories goes back (implicitly) to

\begin{itemize}%
\item [[Graeme Segal]], \emph{Categories and cohomology theories}, Topology 13 (1974), 293-312.

\end{itemize}
They were named \emph{Segal categories} in

\begin{itemize}%
\item [[William Dwyer]], [[Daniel Kan]], [[Jeff Smith]], \emph{Homotopy-commutative diagrams and their realizations}. J. P. A. A. 57 (1989), 5-24.

\end{itemize}
An overview is on pages 164 to 169 of

\begin{itemize}%
\item [[André Joyal]], \emph{The theory of Quasi-Categories and its Applications}, notes from a lecture at \href{http://www.crm.cat/HigherCategories/}{Simplicial Methods in Higher Categories} (\href{http://www.crm.cat/HigherCategories/hc2.pdf}{pdf})

\end{itemize}
A discussion with emphasis on the comparison of the various [[model category]] structures is in

\begin{itemize}%
\item [[Julia Bergner]], \emph{A survey of $(\infty, 1)$-categories} (\href{http://arxiv.org/abs/math.AT/0610239}{arXiv:0610239})

\end{itemize}
The generalization to [[Segal n-categories]] is discussed in section 2 of

\begin{itemize}%
\item [[André Hirschowitz]], [[Carlos Simpson]], \emph{Descente pour les $n$-champs (Descent for $n$-stacks)} (\href{http://arxiv.org/abs/math/9807049}{arXiv:9807049})

\end{itemize}
In the more general context of [[enriched (∞,1)-categories]], this is discussed in

\begin{itemize}%
\item [[Carlos Simpson]], \emph{[[Homotopy Theory of Higher Categories]]} (\href{http://arxiv.org/abs/1001.4071}{arXiv:1001.4071})

\end{itemize}
and in section 2 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{(Infinity,2)-Categories and the Goodwillie Calculus I} (\href{http://arxiv.org/abs/0905.0462}{arXiv:0905.0462})

\end{itemize}
[[!redirects Segal categories]]

[[!redirects Segal groupoid]] [[!redirects Segal groupoids]]



\end{document}