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\begin{document}

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\section*{internal category in homotopy type theory}

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

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

[[!include category theory - contents]]

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{RefetencesGeneral}{General}\dotfill \pageref*{RefetencesGeneral} \linebreak
\noindent\hyperlink{ReferencesCoinduction}{By coinduction}\dotfill \pageref*{ReferencesCoinduction} \linebreak


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

One may consider [[internal categories]] in [[homotopy type theory]]. Under the [[relation between type theory and category theory|interpretation]] of [[HoTT]] in an [[(infinity,1)-topos]], this corresponds to the concept of a [[category object in an (infinity,1)-category]]. The general idea is presented there at \emph{\href{category+object+in+an+%28infinity%2C1%29-category#HomotopyTypeTheoryFormulation}{Homotopy Type Theory Formulation}}.

For internal [[1-categories]] in HoTT (as opposed to more general internal [[(infinity,1)-categories]]) a comprehensive discussion was given in (\hyperlink{AhrensKapulkinShulman13}{Ahrens-Kapulkin-Shulman-13}).

In some literature, the ``Rezk-completeness'' condition on such categories is omitted from the definition, and categories that satisfy it are called \emph{saturated} or \emph{univalent}.

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[type-theoretic definition of category]]


\item Particularly useful in the context of such categories are [[displayed categories]].



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

\hypertarget{RefetencesGeneral}{}\subsubsection*{{General}}\label{RefetencesGeneral}

The relation between [[complete Segal space|Segal completeness]] (now often ``Rezk completeness'') for internal categories in HoTT and the [[univalence axiom]] had been pointed out in

\begin{itemize}%
\item [[Urs Schreiber]], \emph{\href{https://golem.ph.utexas.edu/category/2012/05/segalcompleteness_and_univalen.html}{Segal completenes and Univalence}}, 2012

\end{itemize}
A comprehensive discussion for [[1-categories]] is in

\begin{itemize}%
\item [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], \emph{Univalent categories and the Rezk completion}, Mathematical Structures in Computer Science, Volume 25, Issue 5 (\emph{From type theory and homotopy theory to Univalent Foundations of Mathematics}), June 2015, pp. 1010-1039 (\href{https://arxiv.org/abs/1303.0584}{arXiv:1303.0584})

\end{itemize}
Exposition of this includes

\begin{itemize}%
\item [[Mike Shulman]], \emph{\href{https://golem.ph.utexas.edu/category/2013/03/category_theory_in_homotopy_ty.html}{Category Theory in Homotopy Type Theory}}, 2013

\end{itemize}
Discussion of implementation of this in [[Coq]] includes

\begin{itemize}%
\item [[Jason Gross]], [[Adam Chlipala]], [[David Spivak]], \emph{Experience Implementing a Performant Category-Theory Library in Coq} (\href{http://arxiv.org/abs/1401.7694}{arXiv:1401.7694})

\end{itemize}
See also

\begin{itemize}%
\item [[homotopytypetheory:HomePage|HoTT wiki]], \emph{[[homotopytypetheory:category]]}

\end{itemize}
Generalization to [[(n,1)-categories]] is discussed in

\begin{itemize}%
\item [[Paolo Capriotti]], [[Nicolai Kraus]], \emph{Univalent Higher Categories via Complete Semi-Segal Types} (\href{https://arxiv.org/abs/1707.03693}{arXiv:1707.03693})

\end{itemize}
and, by different means, in

\begin{itemize}%
\item [[Emily Riehl]], [[Michael Shulman]], \emph{A type theory for synthetic $\infty$-categories} (\href{https://arxiv.org/abs/1705.07442}{arXiv:1705.07442})


\item [[Emily Riehl]], \emph{The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories}, talk at \href{http://www.math.ias.edu/vvmc2018}{Vladimir Voevodsky Memorial Conference 2018} (\href{http://www.math.jhu.edu/~eriehl/Voevodsky.pdf}{pdf})



\end{itemize}
Formalization of [[bicategories]]:

\begin{itemize}%
\item [[Benedikt Ahrens]], Dan Frumin, Marco Maggesi, Niels van der Weide, \emph{Bicategories in Univalent Foundations} (\href{https://arxiv.org/abs/1903.01152}{arXiv:1903.01152})

\end{itemize}
\hypertarget{ReferencesCoinduction}{}\subsubsection*{{By coinduction}}\label{ReferencesCoinduction}

A formalization in [[HoTT]]-[[Agda]] of general [[(n,r)-categories]] for $n,r \in \mathbb{N} \sqcup \{\infty\}$, defined as [[coinductive types]] of infinity-graphs, with operations defined by induction-coinduction, is implemented in

\begin{itemize}%
\item Darin Morrison, \emph{\href{https://github.com/5HT/agda-nr-cats}{agda-nr-cats}}, \emph{\href{https://github.com/freebroccolo/agda-infinity-categories/}{agda-infinity-categories}}

\end{itemize}
$\backslash$linebreak

[[!redirects internal categories in homotopy type theory]]

[[!redirects internal category in HoTT]] [[!redirects internal categories in HoTT]]

[[!redirects internal category theory in homotopy type theory]] [[!redirects internal category theory in HoTT]]

[[!redirects saturated category]] [[!redirects univalent category]] [[!redirects saturated categories]] [[!redirects univalent categories]]



\end{document}