nLab exponentiable functor

Idea

An exponentiable functor, also called a Conduché functor or Conduché fibration, is a functor which is an exponentiable morphism in Cat. (In accordance with Stigler's law of eponymy, the notion was actually defined in Giraud 64 before Conduché 72.) This turns out to be equivalent to a certain “factorization lifting” property which includes both Grothendieck fibrations and opfibrations.

So, roughly speaking, a functor $p\colon E\to B$ is a strict Conduché functor if given any morphism $\alpha$ in $E$, and any way of factoring its image in $B$, we can lift that factorization back up to a factorization of $\alpha$, in a way that is unique up to isomorphism. There is also a weak version of this idea.

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, it is not 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

$$\array{1 & \overset{}{\to} & \mathbf{2}\\ \downarrow && \downarrow\\ \mathbf{2}& \underset{}{\to} & \mathbf{3}}$$

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

$$\array{0 & \overset{}{\to} & 1\\ \downarrow && \downarrow\\ 1& \underset{}{\to} & \mathbf{2}}$$

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

Definition

A functor $p\colon E\to B$ is a strict Conduché 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:

1. 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

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

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

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

commute.)

The theorem is then that the following are equivalent:

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

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 (weak) Conduché 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:

1. 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

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 $Cat$.

A Conduché functor is discrete if each factorisation is unique (equivalently, if it reflects identities). Discrete Conduché functors generalise discrete fibrations and discrete opfibrations.

Discrete Conduché functors are said to satisfy the unique lifting of factorisations. Discrete Conduché functors are therefore sometimes called ULF functors.

Properties

• ULF functors form the right class of an orthogonal factorisation system on 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 $p\colon E\to B$ is essentially the same as to give a normal lax 2-functor $B\to Prof$ from $B$ to Prof, 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 $H_f\colon p^{-1}(a) ⇸ p^{-1}(b)$ 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:

• $p$ is a fibration iff the corresponding functor $B\to Prof$ factors through a pseudo 2-functor landing in $Cat^{op}$, via the contravariant inclusion $Cat^{op}\to Prof$.
• Similarly, $p$ is an opfibration iff $B\to Prof$ factors through a pseudo 2-functor landing in $Cat$ via the covariant inclusion $Cat \to Prof$.
• The functor $B\to Prof$ factors through a lax 2-functor landing in $Cat^{op}$ iff $p$ admits all “weakly cartesian” liftings, and dually.
• Finally, $p$ is a (strict) Conduché 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.

Thus Conduché functors into $B$ correspond to pseudofunctors from $B$, regarded as a locally discrete bicategory, to the bicategory $Prof$. However, morphisms between Conduché 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 Conduché functors into $B$, and vertical double transformations between them correspond to functors between Conduché 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 Conduché functors corresponding to the pseudo double functors.

Higher-categorical versions

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

Ayala and Francis prove an analogous characterization of exponentiable (∞,1)-functors. The (∞,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.,

In a double category - completeness

Functors satisfying a Conduché-like condition are central to completeness/cocompleteness in pseudo-double categories?. In short, in 1-category theory, $A$ is complete iff for any small $M$ and locally small $C$, $K: M\to C$ a functor, $T : A\to M$, $T$ has a pointwise right Kan extension along $K$; in pseudo double categories, this is too strong, and we must restrict to double functors $K: M\to C$ satisfying a Conduché condition.

A pseudo-double category is unitary if the vertical identity arrows are strict identities, so $1_A\circ f = f = f\circ 1_B$ on-the-nose. We here assume all double categories are unitary; this is not a severe restriction in practice.

Call a lax functor $F: \mathbb{A}\to\mathbb{B}$ between pseudo-double categories unitary if it it preserves identity vertical arrows on-the-nose and the comparison map $1_{FA}\to F(1_A)$ is the identity; therefore it is only really lax from the point of view of composition. A unitary lax functor $1\to\mathbb{A}$ is automatically strict, as $1$ has no nontrivial compositions, whereas a lax functor $1\to\mathbb{A}$ is an object endowed with a vertical monad.

Note that if we are working in a 2-category of lax functors, and we want to define the limit of a functor $F : \mathbb{A}\to\mathbb{B}$ as the right Kan extension of $F$ along $!_{\mathbb{A}}:\mathbb{A}\to 1$, we probably want to consider the initial extension $1\to \mathbb{B}$ along unitary lax functors rather than lax functors, so that our answer is just an object of $\mathbb{B}$ and a cocone rather than a vertical monad in $\mathbb{B}$. Thus we will here consider unitary Kan extensions.

The correct definition of “complete” for a pseudo-double category $\mathbb{A}$ is that it admits pointwise unitary lax right Kan extensions of any lax double functor $S: \mathbb{I}\to\mathbb{A}$ ($\mathbb{I}$ small) along all lax double functor along a lax double functor $R: \mathbb{I}\to\mathbb{J}$, where $R$ satisfies the “right Conduché condition: that the laxity 2-cell for vertical composition in $R$,

is exact. Here, $\mathbb{I}_o$ is the category of objects and horizontal morphisms of $\mathbb{I}$, and $\mathbb{I}_a$ is the category of arrows and 2-cells.

The situation for cocompleteness is dual, replacing lax by colax everywhere.

For more on this see Grandis and Paré, Theorem 5.2.

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 $\mathbf{2}$ 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, any functor with codomain $\mathbf{2}$ is a Conduché 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.

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

References

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

• F. Conduché, Au sujet de l’existence d’adjoints à droite aux foncteurs ‘image reciproque’ dans la catégorie des catégories , C. R. Acad. Sci. Paris 275 Série A (1972) pp.891-894. (gallica)

• Marco Grandis, Robert Paré Lax Kan extensions for double categories (on weak double categories, part IV) , Cahiers de topologie et géométrie différentielle catégoriques, tome 48, no 3 (2007), p. 163-199

Some of 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 (e.g. see Lemma 6.1 for a proof that Conduché implies exponentiability):

• 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

Discrete Conduché functors are considered in

• M. Bunge, S. Niefield, Exponentiability and single universes , JPAA 148 (2000) pp.217-250.

• Peter Johnstone, A Note on Discrete Conduché Fibrations , TAC 5 no.1 (1999) pp.1-11. (pdf)

An analogue of Conduché functors for ∞-categories, classified by maps into an ∞-category version of Prof, is studied in

• David Ayala and John Francis, Fibrations of ∞-categories, arxiv

A discussion of ULF functors (including the fact they form a factorisation system) is contained in:

• Pullback-homomorphisms, n-Category Café (2010)

• Tom Leinster, Notions of Möbius inversion, Bulletin of the Belgian Mathematical Society-Simon Stevin 19.5 (2012): 909-933.