nLab Conduché functor



Category theory

Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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 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:EBp\colon E\to B is a strict Conduché functor if given any morphism α\alpha in EE, and any way of factoring its image in BB, we can lift that factorization back up to a factorization of α\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 CatCat is cartesian closed, it is not locally cartesian closed.

It is easy to write down examples of colimits in CatCat that are not preserved by pullback (as they would be if pullback had a right adjoint). For instance, let 2\mathbf{2} denote the walking arrow, i.e. the ordinal 22 regarded as a category, 11 the terminal category, and 3=2 12\mathbf{3} = \mathbf{2} \sqcup_1 \mathbf{2} the ordinal 3=(abc)3 = (a \to b \to c) regarded as a category. Then the pushout square

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

in the slice category Cat/3Cat/\mathbf{3} pulls back along the inclusion 23\mathbf{2}\to \mathbf{3} of the arrow (ac)(a\to c) to the square

0 1 1 2\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 23\mathbf{2}\to\mathbf{3} fails to notice that the morphism (ac)(a\to c) acquires a new factorization in 3\mathbf{3} which it didn’t have in 2\mathbf{2}. Conduché‘s observation was that this latter failure is really the only problem that can prevent a functor from being exponentiable.


A functor p:EBp\colon E\to B is a strict Conduché functor if for any morphism α:ab\alpha\colon a\to b in EE and any factorization paβcγpbp a \overset{\beta}{\to} c \overset{\gamma}{\to} p b of pαp \alpha in BB, we have:

  1. there exists a factorization aβ˜dγ˜ba \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b of α\alpha in EE such that pβ˜=βp \tilde{\beta} = \beta and pγ˜=γp \tilde{\gamma} = \gamma, and

  2. any two such factorizations in EE are connected by a zigzag of commuting morphisms which map to id cid_c in BB.

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

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


The theorem is then that the following are equivalent:

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

For exponentiability in the weak 2-category CatCat, in the sense of pullback having a weak/pseudo 2-adjoint, we can simply weaken the condition. We say that p:EBp\colon E\to B is a (weak) Conduché functor if for any morphism α:ab\alpha\colon a\to b in EE and any factorization paβcγpbp a \overset{\beta}{\to} c \overset{\gamma}{\to} p b of pαp \alpha in BB, we have:

  1. there exists a factorization aβ˜dγ˜ba \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b of α\alpha in EE, and an isomorphism pdcp d \cong c, such that modulo this isomorphism pβ˜=βp \tilde{\beta} = \beta and pγ˜=γp \tilde{\gamma} = \gamma, and

  2. any two such factorizations in EE are connected by a zigzag of commuting morphisms which map to isomorphisms in BB.

A functor can then be shown to be a weak Conduché functor if and only if it is exponentiable in the weak sense in CatCat.

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

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:EBp\colon E\to B is essentially the same as to give a normal lax 2-functor BProfB\to Prof from BB 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 pp, we define BProfB\to Prof as follows. Each object bBb\in B is sent to the fiber category p 1(b)p^{-1}(b) of objects lying over bb and morphism lying over 1 b1_b. And each morphism f:abf\colon a\to b in BB to the profunctor H f:p 1(a)p 1(b)H_f\colon p^{-1}(a) ⇸ p^{-1}(b) for which H f(x,y)H_f(x,y) is the set of arrows xyx\to y in EE lying over ff. The lax structure maps H fH gH gfH_f \otimes H_g \to H_{g f} are given by composition in EE. The converse construction of a functor pp from a normal lax 2-functor into ProfProf is an evident generalization of the Grothendieck construction. Now we can say that:

  • pp is a fibration iff the corresponding functor BProfB\to Prof factors through a pseudo 2-functor landing in Cat opCat^{op}, via the contravariant inclusion Cat opProfCat^{op}\to Prof.
  • Similarly, pp is an opfibration iff BProfB\to Prof factors through a pseudo 2-functor landing in CatCat via the covariant inclusion CatProfCat \to Prof.
  • The functor BProfB\to Prof factors through a lax 2-functor landing in Cat opCat^{op} iff pp admits all “weakly cartesian” liftings, and dually.
  • Finally, pp is a (strict) Conduché functor iff the functor BProfB\to Prof is itself a pseudo 2-functor (though it may not land in CatCat or Cat opCat^{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 BB correspond to pseudofunctors from BB, regarded as a locally discrete bicategory, to the bicategory ProfProf. However, morphisms between Conduché functors over BB do not correspond to pseudonatural transformations between such pseudofunctors. To get the correct transformations, we must instead regard BB as a vertically discrete double category, and ProfProf as a pseudo double category with profunctors horizontally and functors vertically; then pseudo double functors BProfB\to Prof again correspond to Conduché functors into BB, and vertical double transformations between them correspond to functors between Conduché functors into BB.

More generally, the slice category Cat/BCat/B is equivalent to the hom-category Dbl normal,lax(B,Prof)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 ProfProf (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 (,1)(\infty,1)-functors into an (,1)(\infty,1)-category BB correspond to (,1)(\infty,1)-functors B(,1)ProfB\to (\infty,1)Prof.

As in the 1-categorical case, ordinary (,1)(\infty,1)-transformations between functors B(,1)ProfB\to (\infty,1)Prof do not give the correct maps between exponentiable (,1)(\infty,1)-functors over BB; we need to instead regard (,1)Prof(\infty,1)Prof as a sort of “(,1)(\infty,1)-double category”. Ayala and Francis consider only the vertically-invertible fragment of this (,1)(\infty,1)-double category, which can be represented as a functor from an \infty-groupoid to an (,1)(\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” (,1)(\infty,1)-category and is also what is represented by a non-complete Segal space. Of course, restricting to the vertically-invertible fragment of (,1)Prof(\infty,1)Prof also restricts what it classifies to the \infty-groupoid of exponentiable (,1)(\infty,1)-functors over BB rather than the whole (,1)(\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, AA is complete iff for any small MM and locally small CC, K:MCK: M\to C a functor, T:AMT : A\to M, TT has a pointwise right Kan extension along KK; in pseudo double categories, this is too strong, and we must restrict to double functors K:MCK: M\to C satisfying a Conduché condition.

A pseudo-double category is unitary if the vertical identity arrows are strict identities, so 1 Af=f=f1 B1_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:𝔸𝔹F: \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 FAF(1 A)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 1𝔸1\to\mathbb{A} is automatically strict, as 11 has no nontrivial compositions, whereas a lax functor 1𝔸1\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:𝔸𝔹F : \mathbb{A}\to\mathbb{B} as the right Kan extension of FF along ! 𝔸:𝔸1!_{\mathbb{A}}:\mathbb{A}\to 1, we probably want to consider the initial extension 1𝔹1\to \mathbb{B} along unitary lax functors rather than lax functors, so that our answer is just an object of 𝔹\mathbb{B} and a cocone rather than a vertical monad in 𝔹\mathbb{B}. Thus we will here consider unitary Kan extensions.

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

is exact. Here, 𝕀 o\mathbb{I}_o is the category of objects and horizontal morphisms of 𝕀\mathbb{I}, and 𝕀 a\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.


  • 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\mathbf{2} denotes the interval category, then any normal lax functor out of 2\mathbf{2} is necessarily pseudo, since there are no composable pairs of nonidentity arrows in 2\mathbf{2}. It follows that, as pointed out by Jean Benabou, any functor with codomain 2\mathbf{2} is a Conduché functor. Note that functors with codomain 2\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)

  • 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

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

Last revised on December 29, 2023 at 20:21:13. See the history of this page for a list of all contributions to it.