nLab
Conduche functor

Idea

A Conduché functor, or exponentiable functor, is a functor which is an exponentiable morphism in Cat. (In accordance with Baez's law, the notion was actually defined by Giraud before Conduché.) This turns out to be equivalent to a certain “factorization lifting” property which includes both Grothendieck fibrations and opfibrations.

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 $\mathrm{Cat}$. In particular, although $\mathrm{Cat}$ is cartesian closed, it is not locally cartesian closed.

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

$$\begin{array}{ccc}1& \stackrel{}{\to }& 2\\ \downarrow & & \downarrow \\ 2& \underset{}{\to }& 3\end{array}$$\array{1 & \overset{}{\to} & \mathbf{2}\\ \downarrow && \downarrow\\ \mathbf{2}& \underset{}{\to} & \mathbf{3}}

in $\mathrm{Cat}/3$ pulls back along the inclusion $2\to 3$ of the arrow $(a\to c)$ to the square

$$\begin{array}{ccc}0& \stackrel{}{\to }& 1\\ \downarrow & & \downarrow \\ 1& \underset{}{\to }& 2\end{array}$$\array{0 & \overset{}{\to} & 1\\ \downarrow && \downarrow\\ 1& \underset{}{\to} & \mathbf{2}}

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 $2\to 3$ fails to notice that the morphism $(a\to c)$ acquires a new factorization in $3$ which it didn’t have in $2$. Conduché’s observation was that this latter failure is really the only problem that can prevent a functor from being exponentiable.

Definition

A functor $p:E\to B$ is a strict Conduché functor if for any morphism $\alpha :a\to b$ in $E$ and any factorization $pa\stackrel{\beta }{\to }c\stackrel{\gamma }{\to }pb$ of $p\alpha$ in $B$, we have:

1. there exists a factorization $a\stackrel{\tilde{\beta }}{\to }d\stackrel{\tilde{\gamma }}{\to }b$ of $\alpha$ in $E$ such that $p\tilde{\beta }=\beta$ and $p\tilde{\gamma }=\gamma$, and

2. any two such factorizations in $E$ are connected by a zigzag of commuting morphisms which map to ${\mathrm{id}}_c$ in $B$.

(Here, ‘commuting morphism’ means a morphism $d\to d\prime$ in $E$ such that the pair of triangles in

$$\begin{array}{ccccc}& & d& \stackrel{\gamma }{\to }& b\\ & {}^{\beta }\nearrow & \downarrow & {\nearrow }^{\gamma \prime }& \\ a& \underset{\beta \prime }{\to }& d\prime & & \end{array}$$\array{ & & d & \stackrel{\gamma}{\to} & b \\ & ^\mathllap{\beta} \nearrow & \downarrow & \nearrow^\mathrlap{\gamma'} & \\ a & \underset{\beta'}{\to} & d' & & }

commute.)

The theorem is then that the following are equivalent:

• $p$ is a Conduché functor.
• $p$ is exponentiable in the 1-category $\mathrm{Cat}$.
• $p$ is exponentiable in the strict 2-category $\mathrm{Cat}$.

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

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

1. there exists a factorization $a\stackrel{\tilde{\beta }}{\to }d\stackrel{\tilde{\gamma }}{\to }b$ of $\alpha$ in $E$, and an isomorphism $pd\cong c$, such that modulo this isomorphism $p\tilde{\beta }=\beta$ and $p\tilde{\gamma }=\gamma$, and

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

A functor can then be shown to be a weak Conduché functor if and only if it is exponentiable in the weak sense in $\mathrm{Cat}$.

Conduché functors and 2-functors to Prof

The Conduché criterion can be reformulated in a more conceptual way by analogy with Grothendieck fibrations. We first observe that to give a functor $p:E\to B$ is essentially the same as to give a normal lax 2-functor $B\to \mathrm{Prof}$ from $B$ to the 2-category of profunctors. Specifically, iven a functor $p$, we define $B\to \mathrm{Prof}$ 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 $f:a\to b$ in $B$ to the profunctor $H_f:p^{-1}(a)⇸p^{-1}(b)$ for which $H_f(x,y)$ is the set of arrows $x\to y$ in $E$ lying over $f$. The lax structure maps $H_f\otimes H_g\to H_{gf}$ are given by composition in $E$. The converse construction of a functor $p$ from a normal lax 2-functor into $\mathrm{Prof}$ is an evident generalization of the Grothendieck construction. Now we can say that:

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

Non-strict Conduché functors and Street fibrations may be equivalently characterized by an “up-to-iso” version of the above construction using essential fibers.

Examples

• The above considerations show that any Grothendieck fibration or opfibration is a (strict) Conduché functor, while any Street fibration or opfibration is a non-strict Conduché functor.

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

• As with exponentiable morphisms in any category, Conduché functors are closed under composition.

References

• F. Conduché, 'Au sujet de l’existence d’adjoints à droite aux foncteurs “image réciproque” dans la catégorie des catégories', C.R. Acad. Sci. Paris 275 (1972), A891–894.

• J. Giraud, 'Méthode de la descente', Bull. Math. Soc. Memoire 2 (1964)

The definitions and proofs of the above theorems, along with the 2-categorical generalization (Conduché considered only the 1-categorical case) can also be found in

• Peter Johnstone, “Fibrations and partial products in a 2-category”, Appl. Categ. Structures 1 (1993), 141–179

A description of the characterization in terms of lax normal functors can be found in

• Ross Street, 'Powerful functors', pdf