Michael Shulman
exponentials in a 2-category

Cat is not locally cartesian closed

The 2-category CatCat is cartesian closed, in an appropriate 2-categorical sense; see also the discussion here. However, it is not locally cartesian closed. This failure is fundamental and has nothing to do with strictness or size issues; pullbacks just don’t preserve colimits. For example, let X=(012)X=(0\to 1\to 2) be the ordinal 3\mathbf{3}; then there is a pushout

(1) (12) (01) (012)\array{(1) & \to & (1\to 2)\\ \downarrow && \downarrow\\ (0\to 1) & \to & (0\to 1\to 2)}

in Cat/XCat/X which pulls back along the inclusion (02)X(0\to 2)\to X to

(2) (0) (02)\array{\emptyset & \to & (2)\\ \downarrow && \downarrow\\ (0) & \to & (0\to 2)}

which is certainly not a pushout. Note that the same counterexample applies in PosPos.

It is similarly easy to write down examples of coinserters, coinverters, and coequifiers that are not preserved by pullbacks. Coproducts are preserved by pullback (in fact, CatCat is extensive), as are quotients of 2-congruences (since CatCat is regular), but these seem to be about it for pullback-stable colimits in CatCat.

2-categories with exponentials

However, there are a number of other useful exponentiability properties that do hold when K=CatK=Cat.

  1. Fibrations and opfibrations are exponentiable. That is, if f:AXf:A\to X is an (op)fibration, then exponentials (BX) (AX)(B\to X)^{(A\to X)} exist in the slice 2-category K/XK/X. Equivalently, f *:K/XK/Af^*:K/X\to K/A has a right adjoint Π f\Pi_f. (Fibrations and opfibrations are not the only exponentiable morphisms in CatCat, but they are certainly the most important and most commonly encountered ones.)

  2. If AXA\to X is an opfibration and BXB\to X is a fibration, then the exponential (BX) (AX)(B\to X)^{(A\to X)} in K/XK/X is a fibration, and dually.

  3. For any f:YXf:Y\to X, the functor f *:Fib(X)Fib(Y)f^*:Fib(X)\to Fib(Y) has a right adjoint Ran fRan_f, and likewise for f *:Opf(X)Opf(Y)f^*:Opf(X)\to Opf(Y).

  4. For any f:YXf:Y\to X, the functors f *:DFib(X)DFib(Y)f^*:DFib(X)\to DFib(Y) and f *:DOpf(X)DOpf(Y)f^*:DOpf(X)\to DOpf(Y) have right adjoints Ran fRan_f.

  5. The 2-categories Fib(X)Fib(X) and Opf(X)Opf(X) are all cartesian closed.

  6. KK itself is cartesian closed.

  7. The 2-categories DFib(X)DFib(X) and DOpf(X)DOpf(X) are all locally cartesian closed.

Note that Ran fRan_f and Π f\Pi_f are not the same even when both exist, and likewise the exponentials in Fib(X)Fib(X) are not the same as the exponentials in K/XK/X even when both exist. The latter are better-behaved in some ways, for instance they are stable under pullback (because they are “fiberwise”).

Perhaps surprisingly, it turns out that the first of these properties is sufficient to imply all the others. The goal of the rest of this page is to prove this claim. Therefore, we define:

Definition

A 2-category KK with finite limits is said to have exponentials if all fibrations and opfibrations in KK are exponentiable.

Note that a 1-category, and in fact any (2,1)-category, has exponentials if and only if it is locally cartesian closed, since every morphism is a fibration and opfibration.

Basic observations

We begin with a couple of easy observations.

Proposition

If either fibrations or opfibrations are exponentiable in KK, then KK is cartesian closed.

Proof

Of course, KK is cartesian closed just when every morphism A1A\to 1 is exponentiable; but every such morphism is both a fibration and an opfibration.

Proposition

If AXA\to X is an opfibration and BXB\to X is a fibration, then the exponential (BX) (AX)(B\to X)^{(A\to X)} in K/XK/X (if it exists) is a fibration, and dually.

Proof

We say briefly how to construct an action morphism

X 2× X(BX) (AX)(BX) (AX).X ^{\mathbf{2}} \times_X(B\to X)^{(A\to X)} \to (B\to X)^{(A\to X)}.

Of course, by adjointness it suffices to construct a morphism

A× XX 2× X(BX) (AX)BA\times_X X ^{\mathbf{2}} \times_X (B\to X)^{(A\to X)} \to B

over XX. Now the left-hand side is a limit of the diagram

(BX) (AX) X 2 X A X.\array{ &&&& (B\to X)^{(A\to X)}\\ &&&& \downarrow\\ && X ^{\mathbf{2}} &\to & X\\ && \downarrow\\ A & \to & X.}

We first map it via a diagonal to the limit of

(BX) (AX) X 2 X A X X 2\array{ &&&& (B\to X)^{(A\to X)}\\ &&&& \downarrow\\ && X ^{\mathbf{2}} &\to & X\\ && \downarrow && \uparrow\\ A & \to & X & \leftarrow & X ^{\mathbf{2}}}

then use the covariant action of XX on AA to map to the limit of

(BX) (AX) A X X X 2\array{ && (B\to X)^{(A\to X)}\\ && \downarrow\\ A &\to & X\\ && \uparrow\\ X & \leftarrow & X ^{\mathbf{2}}}

then the evaluation A× X(BX) (AX)BA\times_X (B\to X)^{(A\to X)} \to B to get to the limit of

B X X X 2\array{ && B\\ && \downarrow\\ && X\\ && \uparrow\\ X & \leftarrow & X ^{\mathbf{2}}}

and finally the contravariant action of XX on BB to get to

B X.\array{ B\\ \downarrow \\ X.}

It is then straightforward to check that this action makes (BX) (AX)(B\to X)^{(A\to X)} into a fibration.

Comonadicity

The key observation for many of the proofs below is the following.

Lemma

If KK has exponentials, then for any XX, Fib(X)Fib(X) and Opf(X)Opf(X) are comonadic, as well as monadic, over K/XK/X.

Proof

The monad on K/XK/X whose category of algebras is Fib(X)Fib(X) takes AXA\to X to X 2× XAX ^{\mathbf{2}} \times_X A, or equivalently Σ st *A\Sigma_s t^*A where (s,t):X 2X(s,t):X ^{\mathbf{2}} \;\rightrightarrows\; X are the two projections. But tt is an opfibration, so Σ st *\Sigma_s t^* has a right adjoint Π ts *\Pi_t s^*. And it is a standard result, valid for 2-categories as for 1-categories, that when the underlying functor of a monad has a right adjoint, its right adjoint becomes a comonad whose category of coalgebras is equivalent to the category of algebras for the original monad.

Corollary

If KK has exponentials, then Fib(X)Fib(X) and Opf(X)Opf(X) inherit any colimits possessed by KK.

Proof

The category of coalgebras for any comonad inherits colimits from the base category, and slice (2-)categories always inherit colimits.

For our main applications of comonadicity, we require the following observation.

Proposition

Given a commutative square

D 1 G¯ D 2 U 1 U 2 C 1 G C 2\array{D_1 & \overset{\overline{G}}{\to} & D_2\\ ^{U_1}\downarrow && \downarrow ^{U_2}\\ C_1 & \underset{G}{\to} & C_2}

of functors between 2-categories, if U 1U_1 and U 2U_2 are monadic, D 1D_1 has reflexive codescent objects, and GG has a left adjoint, then G¯\overline{G} also has a left adjoint. Dually, if U 1U_1 and U 2U_2 are comonadic, D 1D_1 has reflexive descent objects, and GG has a right adjoint, then G¯\overline{G} also has a right adjoint.

Proof

The 1-categorical version of this, referring to reflexive (co)equalizers, is well-known; see for instance

  • P. T. Johnstone, Adjoint lifting theorems for categories of algebras.

The idea is the same as that in Beck’s (co)monadicity theorem: we express any object of D 2D_2 as a reflexive coequalizer of free algebras, then apply the left adjoint of GG to obtain a reflexive pair of free algebras in D 1D_1 and take its coequalizer. The 2-categorical version is analogous, using the ideas of a 2-categorical monadicity theorem as found, for example, in

  • Claudio Hermida, Descent on 2-fibrations and strongly 2-regular 2-categories.

The dual is, of course, obvious.

Slicing and cartesian closure

Using comonadicity, we can show that certain exponentials are stable under slicing. First we observe:

Lemma

If YXY\to X is a fibration, then Fib(X)/YFib(X)/Y is monadic over K/YK/Y. If additionally KK has exponentials, then Fib(X)/YFib(X)/Y is also comonadic over K/YK/Y.

Proof

The first statement is an instance of a general fact: if TT is a monad on a (2-)category CC and YY is a TT-algebra, then there is an induced monad T YT_Y on C/YC/Y whose (2-)category of algebras is TAlg/YT Alg/Y, defined by taking AYA\to Y to the composite TATYYT A \to T Y \to Y.

The second statement is also an instance of a general fact: if such a TT has a right adjoint GG, then T YT_Y also has a right adjoint G YG_Y defined to take AYA\to Y to the pullback

G YA GA Y GY.\array{G_Y A & \to & G A\\ \downarrow && \downarrow\\ Y & \to & G Y.}

Here the lower map YGYY\to G Y is the adjunct of the algebra structure map TYYT Y\to Y.

Proposition

If KK has exponentials, then a morphism in Fib(X)Fib(X) or Opf(X)Opf(X) is exponentiable if its underlying morphism in KK is so. In particular, if KK has exponentials, then for any XX, fibrations are exponentiable in Fib(X)Fib(X) and opfibrations are exponentiable in Opf(X)Opf(X).

Proof

Suppose that f:ZYf:Z\to Y is a morphism in Fib(X)Fib(X) and that ff is exponentiable in KK. Then we have a commutative square

Fib(X)/Y f * Fib(X)/Z K/Y f * K/Z\array{Fib(X)/Y & \overset{f^*}{\to} & Fib(X)/Z\\ \downarrow && \downarrow\\ K/Y& \underset{f^*}{\to} & K/Z}

in which the vertical functors are comonadic by Lemma 2, and the bottom functor f *f^* has a right adjoint since ff is exponentiable in KK. Therefore, by Proposition 3, the top functor f *f^* has a right adjoint as well. The second statement follows because the underlying morphism in KK of any fibration in Fib(X)Fib(X) is a fibration in KK, and dually (see the theorems on iterated fibrations).

Corollary

If KK has exponentials and a duality involution, then Fib(X)Fib(X) and Opf(X)Opf(X) also have exponentials for any XX.

Proof

After Proposition 4, it remains to show that opfibrations are exponentiable in Fib(X)Fib(X) (and dually, fibrations are exponentiable in Opf(X)Opf(X)). Note that an opfibration in Fib(X)Fib(X) will not, in general, be an opfibration in KK. But with a duality involution we have Fib(X)Opf(X o)Fib(X)\simeq Opf(X^o), and opfibrations are exponentiable in Opf(X o)Opf(X^o), hence also in Fib(X)Fib(X).

We say that a 2-category KK has local exponentials if KK has exponentials and each 2-category Fib(X)Fib(X) and Opf(X)Opf(X) also has local exponentials. Of course, the recursion in this definition is not well-founded, but we can reformulate it in explicit terms to say that

  1. KK has exponentials,
  2. Each Opf(X)Opf(X) and Fib(X)Fib(X) has exponentials,
  3. Each Opf Fib(X)(AX)Opf_{Fib(X)}(A\to X) and Fib Opf(X)(AX)Fib_{Opf(X)}(A\to X) has exponentials,
  4. and so on.

Since duality involutions are stable under fibrational slicing, Corollary 2 implies that if KK has exponentials and a duality involution, then it has local exponentials. It would be nice to have a finite list of axioms that implies local exponentials without invoking a duality involution (since not all Grothendieck 2-toposes have dualities).

Corollary

If KK has exponentials, then the 2-categories Fib(X)Fib(X) and Opf(X)Opf(X) are all cartesian closed.

Proof

Combine Propositions 4 and 1.

Corollary

If KK has exponentials, then the categories DFib(X)DFib(X) and DOpf(X)DOpf(X) are all locally cartesian closed.

Proof

Since right adjoints preserve discrete objects, Corollary 3 implies that DFib(X)DFib(X) and DOpf(X)DOpf(X) are cartesian closed for any XX. Now, given a discrete fibration AXA\to X, we have Fib K(A)Fib Fib(X)(AX)Fib_K(A) \simeq Fib_{Fib(X)}(A\to X) by the theorem on iterated fibrations, and so DFib K(A)DFib Fib(X)(AX)DFib K(X)/(AX)DFib_K(A) \simeq DFib_{Fib(X)}(A\to X) \simeq DFib_K(X)/(A\to X). Thus, since DFib K(A)DFib_K(A) is cartesian closed, so is DFib K(X)/(AX)DFib_K(X)/(A\to X), and thus DFib K(X)DFib_K(X) is locally cartesian closed.

Left Kan extensions

We now turn to the existence of left and right adjoints to pullback functors. So far what we know is

  • f *:K/YK/Xf^*:K/Y\to K/X always has a left adjoint Σ f\Sigma_f given by composition with ff.
  • If KK has exponentials and ff is a fibration or opfibration, then (by definition) f *:K/YK/Xf^*:K/Y\to K/X has a right adjoint Π f\Pi_f.
  • If KK has a comprehensive factorization (such as when it is countably-coherent), then f *:DFib(Y)DFib(X)f^*:DFib(Y)\to DFib(X) and f *:DOpf(Y)DOpf(X)f^*:DOpf(Y)\to DOpf(X) have left adjoints Lan fLan_f, which satisfy the Beck-Chevalley condition for comma squares.

Note that even if ff is a fibration, so that Σ f\Sigma_f maps Fib(Y)Fib(Y) to Fib(X)Fib(X), it will not in general be a left adjoint to f *:Fib(X)Fib(Y)f^*:Fib(X)\to Fib(Y). However, if ff is a discrete fibration, then Σ f:DFib(Y)DFib(X)\Sigma_f:DFib(Y)\to DFib(X) is left adjoint to f *:DFib(X)DFib(Y)f^*:DFib(X)\to DFib(Y), since DFib(X)DFib(X) is a full sub-2-category of K/XK/X.

We now consider how to construct left adjoints for non-discrete fibrations.

Proposition

If KK has exponentials and reflexive codescent objects, then each pullback functor f *:Fib(Y)Fib(X)f^*:Fib(Y)\to Fib(X) and f *:Opf(Y)Opf(X)f^*:Opf(Y)\to Opf(X) has a left adjoint Lan fLan_f.

Proof

Because of the existence of Σ f\Sigma_f, by Proposition 3 it suffices to show that Fib(X)Fib(X) and Opf(X)Opf(X) always have reflexive codescent objects; but this follows from Corollary 1.

The basic first-order structure we are interested in doesn’t imply the existence of such codescent objects, and a Heyting 2-pretopos, such as FinCatFinCat, need not have them. But they can be constructed with some infinitary structure; see colimits in an n-pretopos.

I do not know whether these adjunctions always satisfy the Beck-Chevalley condition for comma squares (although they do in one important case; see below).

Right Kan extensions

We now consider the existence of right adjoints to pullback functors between fibrational slices.

Proposition

If KK has exponentials, then f *:Opf(X)Opf(Y)f^*:Opf(X)\to Opf(Y) and f *:Fib(X)Fib(Y)f^*:Fib(X)\to Fib(Y) have right adjoints Ran fRan_f for any ff.

Proof

If ff is exponentiable in KK, this follows directly from Lemma 1 and Proposition 3. In particular, this applies when ff is a fibration or opfibration.

Now let f:YXf:Y\to X be any morphism, and consider the comma square:

Comma Square ( 1 / f ) (1/f) X X Y Y X X t t s s 1 1 f f

Suppose first that all the left adjoints Lan gLan_g exist for opfibrational slices and satisfy the Beck-Chevalley condition for comma squares. Then this comma square gives us an equivalence f *Lan ts *f^*\simeq Lan_t s^*, so that f *:Opf(X)Opf(Y)f^*:Opf(X)\to Opf(Y) has the right adjoint Ran f:=Ran st *Ran_f := Ran_s t^*.

We now show that regardless of the overall existence of left adjoints, the particular value Lan ts *(A)Lan_t s^*(A) exists for any AOpf(X)A\in Opf(X) and is given by f *Af^*A. This clearly suffices to prove the result.

According to the proof of Proposition 5, Lan ts *(A)Lan_t s^*(A) should be given by the codescent object of

Descent object R R S S T T

where

T= Σ ts *A× YY 2 S= Σ t(s *A× (1/f)(1/f) 2)× YY 2 R= Σ t(s *A× (1/f)(1/f) 2× (1/f)(1/f) 2)× YY 2. \begin{aligned} T =& \Sigma_t s^* A \times_Y Y ^{\mathbf{2}}\\ S =& \Sigma_t \left(s^* A\times_{(1/f)} (1/f)^{\mathbf{2}}\right) \times_Y Y ^{\mathbf{2}}\\ R =& \Sigma_t \left(s^* A\times_{(1/f)} (1/f)^{\mathbf{2}} \times_{(1/f)} (1/f)^{\mathbf{2}}\right) \times_Y Y ^{\mathbf{2}}. \end{aligned}

Now we have a map

q:Σ ts *A× YY 2f *A× YY 2f *A;q:\Sigma_t s^* A \times_Y Y ^{\mathbf{2}} \to f^* A \times_Y Y ^{\mathbf{2}} \to f^*A;

the first map is a Beck-Chevalley morphism, and the second comes from the fact that f *Af^*A is an opfibration. We claim that qq is actually a codescent object of the above diagram, and in fact it is a split codescent object. Recall from

  • Claudio Hermida, Descent on 2-fibrations and strongly 2-regular 2-categories.

that a split codescent object is a diagram

Descent object R R π 01 \pi_{01} π 12 \pi_{12} π 02 \pi_{02} μ \mu S S c c d d τ \tau T T q q σ \sigma Q Q

with isomorphisms cτσqc\tau \cong \sigma q, π 12μτc\pi_{12} \mu \cong \tau c, π 02μτd\pi_{02}\mu \cong \tau d, 1π 01μ1\cong\pi_{01}\mu, 1dτ1\cong d\tau, and 1qσ1\cong q\sigma satisfying certain axioms. In this situation, qq is necessarily the codescent object of the diagram composed of d,c,π 01,π 02,π 12d,c,\pi_{01},\pi_{02},\pi_{12}.

Tracing through the definitions above, we see that our TT is the limit of the span AXYA\to X \leftarrow Y weighted by

(0)(012)(12).(0) \;\to\; (0\to 1\to 2) \;\leftarrow\; (1\to 2).

Likewise, SS is the limit of AXYA\to X \leftarrow Y weighted by

(0)(0 1 2 3 4)(234)(0) \;\to\; \left(\array{0&\to&1\\ \downarrow && \downarrow \\ 2 &\to & 3& \to & 4}\right) \;\leftarrow\; (2 \to 3\to 4)

and RR is the limit by an evident more complicated weight. Finally, f *Af^*A itself is the limit of the same span weighted by

()()().(\cdot) \;\to\; (\cdot) \;\leftarrow\; (\cdot).

The morphism qq is induced by the action of XX on the opfibration AA, applied twice. The morphism dd is induced by the inclusion of the second arrow in a composable pair and the action of XX on AA applied once. The morphism cc is induced by the composition of a composable pair. We define the splitting σ,τ,μ\sigma,\tau,\mu in terms of the limit weights by adding in identity morphisms in appropriate places. It is then straightforward to check that the axioms for a split codescent object are satisfied.

Corollary

If KK has exponentials, then f *:DOpf(X)DOpf(Y)f^*:DOpf(X)\to DOpf(Y) and f *:DFib(X)DFib(Y)f^*:DFib(X)\to DFib(Y) have right adjoints Ran fRan_f for any ff.

Proof

Right adjoints preserve discrete objects.

This completes the proof that exponentiability of fibrations and opfibrations implies all the other notable exponentiability properties we might want to require of a 2-category.

Enrichment

One further observation:

Proposition

If KK has exponentials, then each category DFib(X)DFib(X) is enriched over the cartesian closed category disc(K)disc(K).

Proof

We define the hom-object DFib(X)(A,B)\mathbf{DFib(X)}(A,B) to be Ran XB ARan_X B^A, where B AB^A is the exponential in DFib(X)DFib(X) and

Ran X:DFib(X)DFib(1)disc(K)Ran_X:DFib(X) \to DFib(1)\simeq disc(K)

is the right adjoint to X *X^*. A composition map is obtained by adjointness in the usual way. We also have

K(1,DFib(X)(A,B)) K(1,Ran XB A) DFib(X)(1 X,B A) DFib(X)(A,B)\begin{aligned} K(1,\mathbf{DFib(X)}(A,B)) \cong& K(1, Ran_X B^A)\\ \cong& DFib(X)(1_X, B^A)\\ \cong& DFib(X)(A,B) \end{aligned}

so the underlying ordinary category of this disc(K)disc(K)-enriched category is the ordinary category DFib(X)DFib(X).

In fact, more is true: I believe DFib(X)DFib(X) can be made into a locally internal category, or equivalently a locally small fibration, over disc(K)disc(K). Its fiber over a discrete object ZKZ\in K is the category DFib(X×Z)DFib(X\times Z). This follows by localizing the previous lemma in the slices Fib(Z)K/ZFib(Z)\simeq K/Z; one also has to check that a suitable Beck-Chevalley condition is satisfied by the right adjoints Ran fRan_f.

Revised on February 27, 2009 14:59:16 by Jacques Distler? (128.83.114.63)