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\section*{Segal space}

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

\hypertarget{internal_categories}{}\paragraph*{{Internal $(\infty,1)$-Categories}}\label{internal_categories}

[[!include internal infinity-categories 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{InSetByNervesOfCategories}{In $Set$}\dotfill \pageref*{InSetByNervesOfCategories} \linebreak
\noindent\hyperlink{ConstructionFromACategory}{Construction in $1Grpd$ from a category}\dotfill \pageref*{ConstructionFromACategory} \linebreak
\noindent\hyperlink{ExamplesInIGrpd}{In $1Grpd$}\dotfill \pageref*{ExamplesInIGrpd} \linebreak
\noindent\hyperlink{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{Segal space} is a [[precategory object in an (∞,1)-category|pre-category object]] in [[∞Grpd]].

A genuine [[category object in an (∞,1)-category|category object]] in [[∞Grpd]] is a \emph{[[complete Segal space]]}. This is a way of speaking of [[(∞,1)-categories]].

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

A \textbf{Segal space} $X_\bullet$ is a [[simplicial topological space]] or [[bisimplicial set]] $X_\bullet : \Delta^{op} \to Top$ which satisfies the [[Segal conditions]]:

for all $m,n \in \mathbb{N}$ the square

\begin{displaymath}
\itexarray{
    X_{m+n} &\stackrel{
      p^*_{0,\cdots, m}
    }{\to}& X_m
    \\
    {}^{\mathllap{p^*_{m, \cdots, m+n}}}\downarrow 
    && \downarrow^{\mathrlap{p^*_m}}
    \\
    X_n &\stackrel{p^*_0}{\to}& X_0
  }
\end{displaymath}
is a [[homotopy pullback]] square.

A Segal space for which $X_0$ is a [[discrete space]] is called a \emph{[[Segal category]]}. See there for more dicussion.

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

\hypertarget{InSetByNervesOfCategories}{}\subsubsection*{{In $Set$}}\label{InSetByNervesOfCategories}

For $\mathcal{C}$ a ([[small category|small]]) [[category]] we may regard its ordinary [[nerve]] [[simplicial set]] $N(\mathcal{C}) \in Set^{\Delta^{op}}$ as a Segal space, under the canonical inclusion $Set \hookrightarrow \infty Grpd$,

\begin{displaymath}
N(\mathcal{C}) \in Set^{\Delta^{op}} \hookrightarrow \infty Grpd^{\Delta^{op}}
  \,.
\end{displaymath}
In fact, the classical ``nerve theorem'' about the [[Segal conditions]] says that a simplicial set is the nerve of a category precisely if it is a Segal space.

Notice that $Equiv(N(\mathcal{C})_1) \hookrightarrow N(\mathcal{C})_1$ is precisely the [[subset]] of [[isomorphisms]] in all [[morphisms]] of $\mathcal{C}$.

Therefore under this identification, $N(\mathcal{C})$ is a [[complete Segal space]] precisely if $\mathcal{C}$ is a [[gaunt category]], hence precisely if the only isomorphisms in $\mathcal{C}$ are the [[identities]].

In particular if $\mathcal{C}$ is a [[(0,1)-category]], hence a [[preordered set]], then $N(\mathcal{C})$ is complete Segal precisely if $\mathcal{C}$ is in fact an [[partially ordered set]].

\hypertarget{ConstructionFromACategory}{}\subsubsection*{{Construction in $1Grpd$ from a category}}\label{ConstructionFromACategory}

Let $\mathcal{C}$ be an ordinary category. We discuss how Segal spaces are associated with this.

Let $\mathcal{K}$ be a [[groupoid]] and $p \colon \mathcal{K} \to \mathcal{C}$ a [[functor]] which is [[essentially surjective functor|essentially surjective]].

Then let $X_1 \coloneqq (p/p)\in Grpd$ be the ``lax fiber product'' of $p$ with itself, or rather the [[comma object]] of $p$ with itself, hence the [[comma category]], sitting in the universal square

\begin{displaymath}
\itexarray{
    X_1 &\stackrel{\partial_1}{\to}& \mathcal{K}
    \\
    {}^{\mathllap{\partial_0}}\downarrow &\swArrow& \downarrow^{\mathrlap{p}}
    \\
    \mathcal{K} &\underset{p}{\to}& \mathcal{C}
  }
  \,.
\end{displaymath}
Next let $X_2 = (p/p/p)$ be the ``3-fold comma category'', hence the comma category in

\begin{displaymath}
\itexarray{
    X_2 &\stackrel{\partial_{0}}{\to}& \mathcal{K}
    \\
    {}^{\mathllap{}}\downarrow &\swArrow& \downarrow^{\mathrlap{p}}
    \\
    X_1 &\underset{\partial_1}{\to}& \mathcal{C}  
  }
\end{displaymath}
and so forth: $X_n \coloneqq p^{/^{n+1}}$.

This way an [[object]] of $X_n$ is an $(n+1)$-[[tuple]] of objects $(x_0,x_1, \cdots, x_n) \in \mathcal{K}$ together with a sequence of $n$ composable morphisms $p(x_0) \to p(x_1) \to \cdots \to p(x_n)$, and a [[morphism]] is an $(n+1)$-tuple of morphisms $(f_0,f_1, \cdots, f_n) \in \mathcal{K}$ and a [[pasting]] [[commuting diagram]]

\begin{displaymath}
\itexarray{
    p(x_0) &\to& p(x_1) &\to&  \cdots &\to& p(x_1) 
    \\
    {}^{\mathllap{p(f_0)}}\downarrow && \downarrow^{\mathrlap{p(f_1)}} && \cdots   && \downarrow^{\mathrlap{p(f_1)}}
    \\
    p(x'_0) &\to& p(x_1) &\to& \cdots &\to& p(x'_1) 
  }
\end{displaymath}
in $\mathcal{C}$.

By direct inspection, the maps $X_n \to X^{\partial \Delta^n}$ obtained this way are [[isofibrations]], hence [[fibrations]] in the [[canonical model structure]] on [[Grpd]] and so the [[homotopy pullbacks]] that enter the [[Segal conditions]] for $X_\bullet$ are given by ordinary [[fiber products]]. These clearly satisfy the Segal conditions, hence

\begin{displaymath}
X_\bullet \in Grpd^{\Delta} \hookrightarrow \infty Grpd^{\Delta^{op}}
\end{displaymath}
constructed this way is a Segal space.

Two special case of the functor $p$ are important:

\begin{itemize}%
\item if $\mathcal{K} \simeq core(\mathcal{C})$ is the [[core]] of $\mathcal{C}$ and $p$ is the canonical core inclusion, one finds that $Equiv(X_1) \hookrightarrow X_1$ by the above construction is $Equiv(X_1) = Core(\mathcal{C})^{\Delta^1}$, the [[arrow category]] of the [[core]] of $\mathcal{C}$. This is [[equivalence of categories|equivalent]] to $\mathcal{C}$ by, for instance, the source or restriction map. Hence for $p$ the core inclusion, the above construction gives the [[complete Segal space]] corresponding to the category $\mathcal{C}$.


\item if $p \colon \pi_0(\mathcal{C}) \to \mathcal{C}$ is a choice of basepoints in each [[isomorphism class]] of $\mathcal{C}$, then $X_\bullet$ is the [[Segal category]] incarnation of the category $\mathcal{C}$.



\end{itemize}
\hypertarget{ExamplesInIGrpd}{}\subsubsection*{{In $1Grpd$}}\label{ExamplesInIGrpd}

We consider the situation of \emph{\hyperlink{ConstructionFromACategory}{From a category}}, but now conversely, starting with a Segal space in groupoids and then extracting a category from it.

Consider a Segal space that is degreewise just a [[1-groupoid]], hence a simplicial object in the inclusion

\begin{displaymath}
X_\bullet \in 1Grpd^{\Delta^{op}} \hookrightarrow \infty Grpd^{\Delta^{op}}
  \,.
\end{displaymath}
Choosing this to be [[Reedy model structure|Reedy]] [[fibrant object|fibrant]], the map $(\partial_0,\partial_1) \colon X_1 \to X_0 \times X_0$ is an [[isofibration]].

We may write an object $K \in X_1$ as a horizontal morphism

\begin{displaymath}
\partial_1(K) \stackrel{K}{\to} \partial_2(K)
\end{displaymath}
and a morphism $\lambda \colon L \to K$ in $X_1$ as a vertical [[double category]] arrow:

\begin{displaymath}
\itexarray{
     \partial_1(L) & \stackrel{L}{\to} & \partial_0(L)
     \\
     {}^{\mathllap{\partial_1(\lambda)}}\downarrow &\Downarrow^{\mathrlap{\lambda}}& \downarrow^{\mathrlap{\partial_0(\lambda)}}
     \\
     \partial_1(K) &\stackrel{K}{\to}& \partial_0(K)
  }
  \,.
\end{displaymath}
Then the fact that $(\partial_1,\partial_0)$ is an [[isofibration]] means that for every ``niche''

\begin{displaymath}
\itexarray{
     y_0 & & y_1
     \\
     {}^{\mathllap{f_0}}\downarrow 
     && 
    \downarrow^{\mathrlap{f_1}}
     \\
     x_0 &\stackrel{K}{\to}& x_1
  }
  \,,
\end{displaymath}
namely for every pair of morphisms $f_0, f_1$ in $X_0$ and lift of its codomain to an object $K \in X_1$, there is a ``niche filler''

\begin{displaymath}
\itexarray{
     y_0 & \stackrel{L}{\to} & Y_1
     \\
     {}^{\mathllap{f_0}}\downarrow 
     &\Downarrow^{\mathrlap{\lambda}}& 
    \downarrow^{\mathrlap{f_1}}
     \\
     x_0 &\stackrel{K}{\to}& x_1
  }
  \,,
\end{displaymath}
namely a lift of the whole pair $(f_0,f_1)$ to a morphism $\lambda$ in $X_1$, and this is necessarily universal in that any other such lift uniquely factors through this one (because $X_1$ is a groupoid).

Comparison with the definition of a [[2-category equipped with proarrows]] in the incarnation \emph{\href{2-category+equipped+with+proarrows#DefinitionAsDoubleCategory}{as a double category}} shows that this is the beginning of the construction of a [[pseudo double category]] whose vertical category is $X_0$ and whose weak horizontal composition is that induced by the Segal maps.

Assume next that $X_3 \to X^{\partial \Delta^2}$ is a [[1-monomorphism]], as are all the higher $X^n \to X^{\partial \Delta^n }$, for $n \geq 3$, hence that $X_\bullet$ is [[coskeleton|2-coskeletal]] as a simplicial object. This means that the horizontal composition in this pseudo double category has unique composites, hence that the horizontal category is an ordinary category. If then furthermore the composite $Equiv(X_1) \to X_1 \stackrel{\partial_0}{\to} X_0$ is an equivalence, hence is the Segal space is a [[complete Segal space]] this means that $X_\bullet$ arises from this horizontal category by the construction \hyperlink{ConstructionFromACategory}{above}.

\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

\begin{itemize}%
\item [[reduced Segal space]], [[Segal category]],


\item [[semi-Segal space]]


\item [[complete Segal space]], [[model structure for complete Segal spaces]]


\item [[higher Segal space]]



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

See generally the references at \emph{[[complete Segal space]]}.

The ``[[Segal conditions]]'' are first discussed in

\begin{itemize}%
\item [[Graeme Segal]], \emph{Classifying spaces and spectral sequences}, Inst. Hautes \'E{}tudes Sci. Publ. Math., vol. 34, pp. 105--112 (1968),

\end{itemize}
where it is attributed to [[Alexander Grothendieck]].

The term ``Segal space'' is due to

\begin{itemize}%
\item [[Charles Rezk]], \ldots{}

\end{itemize}
The \emph{invertible} case of Segal spaces, hence models for [[groupoid objects in an (infinity,1)-category]] are discussed in section 3 of

\begin{itemize}%
\item [[Julia Bergner]], \emph{Adding inverses to diagrams II: Invertible homotopy theories are spaces}, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (\href{http://www.intlpress.com/hha/v10/n2/a9/}{web}, \href{http://arxiv.org/abs/0710.2254}{arXiv:0710.2254})

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



\end{document}