nLab Conduché functor

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category theory

Applications

Higher category theory

higher category theory

Contents

Idea

A Conduché functor, also called a Conduché fibration or an exponentiable functor, is a functor which is an exponentiable morphism in Cat. (In accordance with Baez's law, the notion was actually defined in Giraud 64 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 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

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

commute.)

The theorem is then that the following are equivalent:

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

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 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.

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.

• 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)

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

Discrete Conduché functors are considered in

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