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\section*{Conduché functor}

[[!redirects Conduche functor]]

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

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

[[!include category theory - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{failure_of_local_cartesian_closedness_in_cat}{Failure of local cartesian closedness in Cat}\dotfill \pageref*{failure_of_local_cartesian_closedness_in_cat} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{conduch_functors_and_2functors_to_prof}{Conduch\'e{} functors and 2-functors to Prof}\dotfill \pageref*{conduch_functors_and_2functors_to_prof} \linebreak
\noindent\hyperlink{highercategorical_versions}{Higher-categorical versions}\dotfill \pageref*{highercategorical_versions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \textbf{Conduch\'e{} functor}, also called a \textbf{Conduch\'e{} fibration} or an \textbf{exponentiable functor}, is a [[functor]] which is an [[exponentiable morphism]] in [[Cat]]. (In accordance with [[Baez's law]], the notion was actually defined in \hyperlink{Giraud64}{Giraud 64} before \hyperlink{Conduche72}{Conduch\'e{}}.) This turns out to be equivalent to a certain ``factorization lifting'' property which includes both [[Grothendieck fibrations]] and opfibrations.

\hypertarget{failure_of_local_cartesian_closedness_in_cat}{}\subsection*{{Failure of local cartesian closedness in Cat}}\label{failure_of_local_cartesian_closedness_in_cat}

As is evident from the fact that such functors have a name, not every functor is exponentiable in [[Cat]]. In particular, although $Cat$ is [[cartesian closed category|cartesian closed]], it is not [[locally cartesian closed category|locally cartesian closed]].

It is easy to write down examples of [[colimits]] in $Cat$ that are not preserved by [[pullback]] (as they would be if pullback had a [[right adjoint]]). For instance, let $\mathbf{2}$ denote the [[walking arrow]], i.e. the [[ordinal]] $2$ regarded as a [[category]], $1$ the [[terminal category]], and $\mathbf{3} = \mathbf{2} \sqcup_1 \mathbf{2}$ the ordinal $3 = (a \to b \to c)$ regarded as a category. Then the [[pushout]] square

\begin{displaymath}
\itexarray{1 & \overset{}{\to} & \mathbf{2}\\
  \downarrow && \downarrow\\
  \mathbf{2}& \underset{}{\to} & \mathbf{3}}
\end{displaymath}
in the [[slice category]] $Cat/\mathbf{3}$ pulls back along the inclusion $\mathbf{2}\to \mathbf{3}$ of the arrow $(a\to c)$ to the square

\begin{displaymath}
\itexarray{0 & \overset{}{\to} & 1\\
  \downarrow && \downarrow\\
  1& \underset{}{\to} & \mathbf{2}}
\end{displaymath}
which is certainly not a pushout.

One way to describe the problem is that the pushout has ``created new morphisms'' that didn't exist before. But another way to describe the problem is that the inclusion $\mathbf{2}\to\mathbf{3}$ fails to notice that the morphism $(a\to c)$ acquires a new factorization in $\mathbf{3}$ which it didn't have in $\mathbf{2}$. Conduch\'e{}`s observation was that this latter failure is really the only problem that can prevent a functor from being exponentiable.

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

A [[functor]] $p\colon E\to B$ is a \textbf{strict Conduch\'e{} functor} if for any [[morphism]] $\alpha\colon a\to b$ in $E$ and any factorization $p a \overset{\beta}{\to} c \overset{\gamma}{\to} p b$ of $p \alpha$ in $B$, we have:

\begin{enumerate}%
\item there exists a factorization $a \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b$ of $\alpha$ in $E$ such that $p \tilde{\beta} = \beta$ and $p \tilde{\gamma} = \gamma$, and


\item any two such factorizations in $E$ are connected by a [[zigzag]] of commuting morphisms which map to $id_c$ in $B$.



\end{enumerate}
(Here, `commuting morphism' means a morphism $d \to d'$ in $E$ such that the pair of triangles in

\begin{displaymath}
\itexarray{
 & & d & \stackrel{\gamma}{\to} & b \\
 & ^\mathllap{\beta} \nearrow & \downarrow & \nearrow^\mathrlap{\gamma'} & \\
a & \underset{\beta'}{\to} & d' & & 
}
\end{displaymath}
[[commuting diagram|commute]].)

The theorem is then that the following are equivalent:

\begin{itemize}%
\item $p$ is a Conduch\'e{} functor.
\item $p$ is [[exponentiable morphism|exponentiable]] in the 1-category $Cat$.
\item $p$ is exponentiable in the [[strict 2-category]] $Cat$.

\end{itemize}
By ``exponentiable in the strict 2-category $Cat$'' we mean that pullback along $p$ has a strict right 2-adjoint (i.e.~a $Cat$-enriched right adjoint). Of course, this implies ordinary exponentiability in the 1-category $Cat$, while the converse follows via an argument involving [[cotensors]] with $\mathbf{2}$ in $Cat$.

For exponentiability in the weak [[2-category]] $Cat$, in the sense of pullback having a weak/pseudo 2-adjoint, we can simply weaken the condition. We say that $p\colon E\to B$ is a \textbf{(weak) Conduch\'e{} functor} if for any morphism $\alpha\colon a\to b$ in $E$ and any factorization $p a \overset{\beta}{\to} c \overset{\gamma}{\to} p b$ of $p \alpha$ in $B$, we have:

\begin{enumerate}%
\item there exists a factorization $a \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b$ of $\alpha$ in $E$, and an isomorphism $p d \cong c$, such that modulo this isomorphism $p \tilde{\beta} = \beta$ and $p \tilde{\gamma} = \gamma$, and


\item any two such factorizations in $E$ are connected by a zigzag of commuting morphisms which map to isomorphisms in $B$.



\end{enumerate}
A functor can then be shown to be a weak Conduch\'e{} functor if and only if it is exponentiable in the weak sense in $Cat$.

\hypertarget{conduch_functors_and_2functors_to_prof}{}\subsection*{{Conduch\'e{} functors and 2-functors to Prof}}\label{conduch_functors_and_2functors_to_prof}

The Conduch\'e{} criterion can be reformulated in a more conceptual way by analogy with [[Grothendieck fibrations]]. We first observe that to give a functor $p\colon E\to B$ is essentially the same as to give a normal [[lax 2-functor]] $B\to Prof$ from $B$ to the 2-category of [[profunctors]]. The latter is also known as a [[displayed category]]; see there for more on this correspondence.

Specifically, given a functor $p$, we define $B\to Prof$ as follows. Each object $b\in B$ is sent to the fiber category $p^{-1}(b)$ of objects lying over $b$ and morphism lying over $1_b$. And each morphism $f\colon a\to b$ in $B$ to the profunctor $H_f\colon p^{-1}(a) &#8696; p^{-1}(b)$ for which $H_f(x,y)$ is the set of arrows $x\to y$ in $E$ lying over $f$. The lax structure maps $H_f \otimes H_g \to H_{g f}$ are given by composition in $E$. The converse construction of a functor $p$ from a normal lax 2-functor into $Prof$ is an evident generalization of the [[Grothendieck construction]]. Now we can say that:

\begin{itemize}%
\item $p$ is a fibration iff the corresponding functor $B\to Prof$ factors through a pseudo 2-functor landing in $Cat^{op}$, via the contravariant inclusion $Cat^{op}\to Prof$.
\item Similarly, $p$ is an opfibration iff $B\to Prof$ factors through a pseudo 2-functor landing in $Cat$ via the covariant inclusion $Cat \to Prof$.
\item The functor $B\to Prof$ factors through a \emph{lax} 2-functor landing in $Cat^{op}$ iff $p$ admits all ``weakly cartesian'' liftings, and dually.
\item Finally, $p$ is a (strict) Conduch\'e{} functor iff the functor $B\to Prof$ is itself a pseudo 2-functor (though it may not land in $Cat$ or $Cat^{op}$). This can be seen by comparing the definition of the tensor product of profunctors with the explicit description in terms of unique factorizations above.

\end{itemize}
Thus Conduch\'e{} functors into $B$ correspond to pseudofunctors from $B$, regarded as a locally discrete bicategory, to the bicategory $Prof$. However, morphisms between Conduch\'e{} functors over $B$ do not correspond to pseudonatural transformations between such pseudofunctors. To get the correct transformations, we must instead regard $B$ as a vertically discrete [[double category]], and $Prof$ as a pseudo double category with profunctors horizontally and functors vertically; then pseudo double functors $B\to Prof$ again correspond to Conduch\'e{} functors into $B$, and \emph{vertical} double transformations between them correspond to functors between Conduch\'e{} functors into $B$.

More generally, the slice category $Cat/B$ is equivalent to the hom-category $Dbl_{normal,lax}(B,Prof)$, with its full subcategory consisting of Conduch\'e{} functors corresponding to the pseudo double functors.

\hypertarget{highercategorical_versions}{}\subsection*{{Higher-categorical versions}}\label{highercategorical_versions}

Non-strict Conduch\'e{} functors and [[Street fibrations]] may be equivalently characterized by an ``up-to-iso'' version of the above constructions using [[essential fibers]].

\hyperlink{AyalaFrancis}{Ayala and Francis} prove an analogous characterization of exponentiable [[(∞,1)-functors]]. The [[(∞,1)-category theory|(∞,1)-categorical context]] eliminates the ``level-shifting'' in the characterization via $Prof$ (i.e. the presence of a [[bicategory]] [[Prof]] when discussing only exponentiable 1-functors). Thus, there is an [[(∞,1)-category]] [[(∞,1)Prof]] such that exponentiable $(\infty,1)$-functors into an $(\infty,1)$-category $B$ correspond to $(\infty,1)$-functors $B\to (\infty,1)Prof$.

As in the 1-categorical case, ordinary $(\infty,1)$-transformations between functors $B\to (\infty,1)Prof$ do not give the correct maps between exponentiable $(\infty,1)$-functors over $B$; we need to instead regard $(\infty,1)Prof$ as a sort of ``$(\infty,1)$-double category''. Ayala and Francis consider only the vertically-invertible fragment of this $(\infty,1)$-double category, which can be represented as a functor from an $\infty$-groupoid to an $(\infty,1)$-category (a sort of [[proarrow equipment]] with all 2-cells and all 1-cells in the domain invertible); this is what they call a ``flagged'' $(\infty,1)$-category and is also what is represented by a non-complete [[Segal space]]. Of course, restricting to the vertically-invertible fragment of $(\infty,1)Prof$ also restricts what it classifies to the $\infty$-groupoid of exponentiable $(\infty,1)$-functors over $B$ rather than the whole $(\infty,1)$-category thereof.

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

\begin{itemize}%
\item The above considerations show that any Grothendieck fibration \emph{or} opfibration is a (strict) Conduch\'e{} functor, while any [[Street fibration]] or opfibration is a non-strict Conduch\'e{} functor.


\item If $\mathbf{2}$ denotes the [[interval category]], then any normal lax functor out of $\mathbf{2}$ is necessarily pseudo, since there are no composable pairs of nonidentity arrows in $\mathbf{2}$. It follows that, as pointed out by Jean Benabou, \emph{any} functor with codomain $\mathbf{2}$ is a Conduch\'e{} functor. Note that functors with codomain $\mathbf{2}$ can also be identified with [[profunctors]], the two fiber categories being the source and target of the corresponding profunctor.


\item As with [[exponentiable morphisms]] in any category, Conduch\'e{} functors are closed under composition.



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

\begin{itemize}%
\item [[J. Giraud]], \emph{M\'e{}thode de la descente}, Bull. Math. Soc. M\'e{}moire \textbf{2} (1964). (\href{http://www.numdam.org/numdam-bin/item?id=MSMF_1964__2__R3_0}{numdam})


\item [[F. Conduché]], \emph{Au sujet de l'existence d'adjoints \`a{} droite aux foncteurs `image reciproque' dans la cat\'e{}gorie des cat\'e{}gories} , C. R. Acad. Sci. Paris \textbf{275} S\'e{}rie A (1972) pp.891-894. (\href{http://gallica.bnf.fr/ark:/12148/bpt6k56191352/f19.image#}{gallica})



\end{itemize}
The definitions and proofs of the above theorems, along with the 2-categorical generalization (Conduch\'e{} considered only the 1-categorical case) can also be found in

\begin{itemize}%
\item [[Peter Johnstone]], ``Fibrations and partial products in a 2-category'', \emph{Appl. Categ. Structures} 1 (1993), 141--179

\end{itemize}
A description of the characterization in terms of lax normal functors can be found in

\begin{itemize}%
\item [[Ross Street]], 'Powerful functors', \href{http://www.math.mq.edu.au/~street/Pow.fun.pdf}{pdf}

\end{itemize}
Discrete Conduch\'e{} functors are considered in

\begin{itemize}%
\item [[Marta Bunge|M. Bunge]], [[Susan Niefield|S. Niefield]], \emph{Exponentiability and single universes} , JPAA \textbf{148} (2000) pp.217-250.


\item [[Peter Johnstone]], \emph{A Note on Discrete Conduch\'e{} Fibrations} , TAC \textbf{5} no.1 (1999) pp.1-11. (\href{http://www.tac.mta.ca/tac/volumes/1999/n1/n1.pdf}{pdf})



\end{itemize}
An analogue of Conduch\'e{} functors for [[∞-categories]], classified by maps into an ∞-category version of [[Prof]], is studied in

\begin{itemize}%
\item [[David Ayala]] and [[John Francis]], \emph{Fibrations of ∞-categories}, \href{https://arxiv.org/abs/1702.02681}{arxiv}

\end{itemize}
[[!redirects Conduché fibration]] [[!redirects Conduche fibration]] [[!redirects Conduche functor]] [[!redirects Conduche functors]] [[!redirects Conduché functor]] [[!redirects Conduché functors]] [[!redirects exponentiable functor]] [[!redirects exponentiable functors]] [[!redirects strict Conduche functors]] [[!redirects strict Conduche functors]] [[!redirects strict Conduché functor]] [[!redirects strict Conduché functors]] [[!redirects strictly exponentiable functor]] [[!redirects strictly exponentiable functors]] [[!redirects powerful functor]] [[!redirects powerful functors]] [[!redirects strictly powerful functor]] [[!redirects strictly powerful functors]]

[[!redirects Conduché ∞-fibration]] [[!redirects Conduche ∞-fibration]] [[!redirects Conduche ∞-functor]] [[!redirects Conduche ∞-functors]] [[!redirects Conduché ∞-functor]] [[!redirects Conduché ∞-functors]] [[!redirects exponentiable ∞-functor]] [[!redirects exponentiable ∞-functors]] [[!redirects Conduché infinity-fibration]] [[!redirects Conduche infinity-fibration]] [[!redirects Conduche infinity-functor]] [[!redirects Conduche infinity-functors]] [[!redirects Conduché infinity-functor]] [[!redirects Conduché infinity-functors]] [[!redirects exponentiable infinity-functor]] [[!redirects exponentiable infinity-functors]]



\end{document}