nLab
Conduche functor

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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 CatCat. 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 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.

Definition

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{\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:

  • pp is a Conduché functor.
  • pp is exponentiable in the 1-category CatCat.
  • pp is exponentiable in the strict 2-category CatCat.

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.

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 the 2-category of profunctors. Specifically, iven 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.

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\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.

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

Revised on December 5, 2011 11:07:54 by Urs Schreiber (82.113.98.199)