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\begin{document}

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\section*{displayed category}

\hypertarget{displayed_categories}{}\section*{{Displayed categories}}\label{displayed_categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{correspondence_to_slices}{Correspondence to slices}\dotfill \pageref*{correspondence_to_slices} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{displayed category} over a [[category]] $C$ is the ``classifying map'' of a category over $C$. That is, it is equivalent to the data of a category $D$ and a functor $F:D\to C$, but organized differently as a ``functor'' associating to each object or morphism of $C$ the [[fiber]] over it. The operation (which is an [[equivalence]]) taking a displayed category to the corresponding functor $F:D\to C$ is a generalization of the [[Grothendieck construction]].

Displayed categories are particularly useful in [[type theory]] (especially [[internal categories in homotopy type theory]]) and to preserve the [[principle of equivalence]], since they allow a more ``category-theoretic'' formulation of various notions (such as [[Grothendieck fibrations]] and strict [[creation of limits]]) that, if stated in terms of a functor $F:D\to C$, would involve equality of objects.

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

A \textbf{displayed category} over a category $C$ is a [[lax functor]] from $C$, regarded as a bicategory with only identity 2-cells, to the [[bicategory]] [[Span]].

Better, it is a lax [[double functor]] from $C$, regarded as a [[double category]] ``horizontally'' with only identity vertical arrows and 2-cells, to the (pseudo) double category $Span$ with sets as objects, functions as vertical arrows, and spans as horizontal arrows. Although this produces an equivalent notion, it is better because a \textbf{displayed functor} is then a [[vertical transformation]] between such lax double functors.

A displayed category may equivalently be described as a \emph{normal} lax functor from $C$ to [[Prof]] (either the bicategory or the double category, as appropriate), meaning one that strictly preserves identities. Formally, this is because $Prof = Mod(Span)$, where $Mod(-)$ denotes the double category of horizontal monads and modules, and $Mod$ is a [[right adjoint]] to the inclusion of [[virtual double categories]] and normal (lax) functors into all (lax) functors; see \hyperlink{CS}{(CS, Prop. 5.14)}.

Equivalently, it is a [[double profunctor]] between $C$ and the terminal double category $1$, i.e., a double presheaf on $C$.

\hypertarget{correspondence_to_slices}{}\subsection*{{Correspondence to slices}}\label{correspondence_to_slices}

The category over $C$ corresponding to a displayed category $D:C\to Span$ is the pullback

\begin{displaymath}
\itexarray{ D & \to & Span_* \\ \downarrow & & \downarrow \\ C & \to & Span }
\end{displaymath}
Where $Span_*=Span(Set_*)$ is the bicategory (or double category) of [[pointed sets]] and pointed spans. This is a strict pullback, which exists in the 2-category of bicategories (or double categories) and lax functors because the projection $Span_* \to Span$ is not just lax but strong. Equivalently, it is the pullback

\begin{displaymath}
\itexarray{ D & \to & Prof_* \\ \downarrow & & \downarrow \\ C & \to & Prof }
\end{displaymath}
where $Prof_*$ consists of pointed categories and pointed profunctors, a pointed profunctor $H:(A,a_0)&#8696;(B,b_0)$ being equipped with an element of $H(a_0,b_0)$.

This construction induces an [[equivalence of categories]] $Disp(C) \to Cat/C$, which restricts to the following equivalences:

\begin{itemize}%
\item A displayed category factors through the inclusion $Set \hookrightarrow Span$ (or equivalently $Set \hookrightarrow Prof$) if and only if $F:D\to C$ is a [[discrete opfibration]]. Similarly, it factors through $Set^{op}$ if and only if $F$ is a discrete fibration.


\item A factorization of a displayed category $C\to Prof$ through the inclusion $Cat^{op} \hookrightarrow Prof$ (as co[[representable profunctors]]) is equivalent to giving $F:D\to C$ the structure of a [[cloven fibration|cloven]] [[prefibred category]], i.e. equipped with a choice of weakly cartesian liftings. The factorization is a [[pseudofunctor]] precisely when $F:D\to C$ is a [[Grothendieck fibration]]; in this case we see the usual [[Grothendieck construction]] of a pseudofunctor.


\item Similarly, factorizations through $Cat\hookrightarrow Prof$ corresponds to cloven Grothendieck (pre)opfibrations.


\item An arbitrary displayed category $C\to Prof$ is a pseudofunctor if and only if $F:D\to C$ is a [[Conduché functor]], i.e. an exponentiable morphism in $Cat$.



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

The correspondence between categories over $C$ and normal lax functors $C\to Prof$ was observed by

\begin{itemize}%
\item Jean B\'e{}nabou, \emph{Distributors at work}, Notes by Thomas Streicher from lectures given at TU Darmstadt, 2000, \href{http://www.mathematik.tu-darmstadt.de/~streicher/FIBR/DiWo.pdf}{http\char58\char47\char47www\char46mathematik\char46tu\char45darmstadt\char46de\char47\char126streicher\char47FIBR\char47DiWo\char46pdf}

\end{itemize}
The term ``displayed category'', and the applications to type theory, are due to:

\begin{itemize}%
\item [[Benedikt Ahrens]], [[Peter LeFanu Lumsdaine]], \emph{Displayed Categories}, \href{https://arxiv.org/abs/1705.04296}{arxiv}

\end{itemize}
An unpacking of the definition as lax functors into [[Span]] is in 2.2 of

\begin{itemize}%
\item [[Duško Pavlović]], [[Samson Abramsky]], \emph{Specifying Interaction Categories}, \href{https://link.springer.com/chapter/10.1007/BFb0026986}{LNCS}

\end{itemize}
Also cited above:

\begin{itemize}%
\item [[Geoff Cruttwell]], [[Mike Shulman]], \emph{A unified framework for generalized multicategories} \href{http://tac.mta.ca/tac/volumes/24/21/24-21abs.html}{TAC}

\end{itemize}
[[!redirects displayed categories]] [[!redirects displayed functor]] [[!redirects displayed functors]]



\end{document}