The 2-category $Cat$ 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 $X=(0\to 1\to 2)$ be the ordinal $\mathbf{3}$; then there is a pushout
in $Cat/X$ which pulls back along the inclusion $(0\to 2)\to X$ to
which is certainly not a pushout. Note that the same counterexample applies in $Pos$.
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, $Cat$ is extensive), as are quotients of 2-congruences (since $Cat$ is regular), but these seem to be about it for pullback-stable colimits in $Cat$.
However, there are a number of other useful exponentiability properties that do hold when $K=Cat$.
Fibrations and opfibrations are exponentiable. That is, if $f:A\to X$ is an (op)fibration, then exponentials $(B\to X)^{(A\to X)}$ exist in the slice 2-category $K/X$. Equivalently, $f^*:K/X\to K/A$ has a right adjoint $\Pi_f$. (Fibrations and opfibrations are not the only exponentiable morphisms in $Cat$, but they are certainly the most important and most commonly encountered ones.)
If $A\to X$ is an opfibration and $B\to X$ is a fibration, then the exponential $(B\to X)^{(A\to X)}$ in $K/X$ is a fibration, and dually.
For any $f:Y\to X$, the functor $f^*:Fib(X)\to Fib(Y)$ has a right adjoint $Ran_f$, and likewise for $f^*:Opf(X)\to Opf(Y)$.
For any $f:Y\to X$, the functors $f^*:DFib(X)\to DFib(Y)$ and $f^*:DOpf(X)\to DOpf(Y)$ have right adjoints $Ran_f$.
The 2-categories $Fib(X)$ and $Opf(X)$ are all cartesian closed.
$K$ itself is cartesian closed.
The 2-categories $DFib(X)$ and $DOpf(X)$ are all locally cartesian closed.
Note that $Ran_f$ and $\Pi_f$ are not the same even when both exist, and likewise the exponentials in $Fib(X)$ are not the same as the exponentials in $K/X$ 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:
A 2-category $K$ with finite limits is said to have exponentials if all fibrations and opfibrations in $K$ 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.
We begin with a couple of easy observations.
If either fibrations or opfibrations are exponentiable in $K$, then $K$ is cartesian closed.
Of course, $K$ is cartesian closed just when every morphism $A\to 1$ is exponentiable; but every such morphism is both a fibration and an opfibration.
If $A\to X$ is an opfibration and $B\to X$ is a fibration, then the exponential $(B\to X)^{(A\to X)}$ in $K/X$ (if it exists) is a fibration, and dually.
We say briefly how to construct an action morphism
Of course, by adjointness it suffices to construct a morphism
over $X$. Now the left-hand side is a limit of the diagram
We first map it via a diagonal to the limit of
then use the covariant action of $X$ on $A$ to map to the limit of
then the evaluation $A\times_X (B\to X)^{(A\to X)} \to B$ to get to the limit of
and finally the contravariant action of $X$ on $B$ to get to
It is then straightforward to check that this action makes $(B\to X)^{(A\to X)}$ into a fibration.
The key observation for many of the proofs below is the following.
If $K$ has exponentials, then for any $X$, $Fib(X)$ and $Opf(X)$ are comonadic, as well as monadic, over $K/X$.
The monad on $K/X$ whose category of algebras is $Fib(X)$ takes $A\to X$ to $X ^{\mathbf{2}} \times_X A$, or equivalently $\Sigma_s t^*A$ where $(s,t):X ^{\mathbf{2}} \;\rightrightarrows\; X$ are the two projections. But $t$ is an opfibration, so $\Sigma_s t^*$ has a right adjoint $\Pi_t s^*$. 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.
If $K$ has exponentials, then $Fib(X)$ and $Opf(X)$ inherit any colimits possessed by $K$.
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.
Given a commutative square
of functors between 2-categories, if $U_1$ and $U_2$ are monadic, $D_1$ has reflexive codescent objects, and $G$ has a left adjoint, then $\overline{G}$ also has a left adjoint. Dually, if $U_1$ and $U_2$ are comonadic, $D_1$ has reflexive descent objects, and $G$ has a right adjoint, then $\overline{G}$ also has a right adjoint.
The 1-categorical version of this, referring to reflexive (co)equalizers, is well-known; see for instance
The idea is the same as that in Beck’s (co)monadicity theorem: we express any object of $D_2$ as a reflexive coequalizer of free algebras, then apply the left adjoint of $G$ to obtain a reflexive pair of free algebras in $D_1$ and take its coequalizer. The 2-categorical version is analogous, using the ideas of a 2-categorical monadicity theorem as found, for example, in
The dual is, of course, obvious.
Using comonadicity, we can show that certain exponentials are stable under slicing. First we observe:
If $Y\to X$ is a fibration, then $Fib(X)/Y$ is monadic over $K/Y$. If additionally $K$ has exponentials, then $Fib(X)/Y$ is also comonadic over $K/Y$.
The first statement is an instance of a general fact: if $T$ is a monad on a (2-)category $C$ and $Y$ is a $T$-algebra, then there is an induced monad $T_Y$ on $C/Y$ whose (2-)category of algebras is $T Alg/Y$, defined by taking $A\to Y$ to the composite $T A \to T Y \to Y$.
The second statement is also an instance of a general fact: if such a $T$ has a right adjoint $G$, then $T_Y$ also has a right adjoint $G_Y$ defined to take $A\to Y$ to the pullback
Here the lower map $Y\to G Y$ is the adjunct of the algebra structure map $T Y\to Y$.
If $K$ has exponentials, then a morphism in $Fib(X)$ or $Opf(X)$ is exponentiable if its underlying morphism in $K$ is so. In particular, if $K$ has exponentials, then for any $X$, fibrations are exponentiable in $Fib(X)$ and opfibrations are exponentiable in $Opf(X)$.
Suppose that $f:Z\to Y$ is a morphism in $Fib(X)$ and that $f$ is exponentiable in $K$. Then we have a commutative square
in which the vertical functors are comonadic by Lemma 2, and the bottom functor $f^*$ has a right adjoint since $f$ is exponentiable in $K$. Therefore, by Proposition 3, the top functor $f^*$ has a right adjoint as well. The second statement follows because the underlying morphism in $K$ of any fibration in $Fib(X)$ is a fibration in $K$, and dually (see the theorems on iterated fibrations).
If $K$ has exponentials and a duality involution, then $Fib(X)$ and $Opf(X)$ also have exponentials for any $X$.
After Proposition 4, it remains to show that opfibrations are exponentiable in $Fib(X)$ (and dually, fibrations are exponentiable in $Opf(X)$). Note that an opfibration in $Fib(X)$ will not, in general, be an opfibration in $K$. But with a duality involution we have $Fib(X)\simeq Opf(X^o)$, and opfibrations are exponentiable in $Opf(X^o)$, hence also in $Fib(X)$.
We say that a 2-category $K$ has local exponentials if $K$ has exponentials and each 2-category $Fib(X)$ and $Opf(X)$ 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
Since duality involutions are stable under fibrational slicing, Corollary 2 implies that if $K$ 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).
If $K$ has exponentials, then the 2-categories $Fib(X)$ and $Opf(X)$ are all cartesian closed.
If $K$ has exponentials, then the categories $DFib(X)$ and $DOpf(X)$ are all locally cartesian closed.
Since right adjoints preserve discrete objects, Corollary 3 implies that $DFib(X)$ and $DOpf(X)$ are cartesian closed for any $X$. Now, given a discrete fibration $A\to X$, we have $Fib_K(A) \simeq Fib_{Fib(X)}(A\to X)$ by the theorem on iterated fibrations, and so $DFib_K(A) \simeq DFib_{Fib(X)}(A\to X) \simeq DFib_K(X)/(A\to X)$. Thus, since $DFib_K(A)$ is cartesian closed, so is $DFib_K(X)/(A\to X)$, and thus $DFib_K(X)$ is locally cartesian closed.
We now turn to the existence of left and right adjoints to pullback functors. So far what we know is
Note that even if $f$ is a fibration, so that $\Sigma_f$ maps $Fib(Y)$ to $Fib(X)$, it will not in general be a left adjoint to $f^*:Fib(X)\to Fib(Y)$. However, if $f$ is a discrete fibration, then $\Sigma_f:DFib(Y)\to DFib(X)$ is left adjoint to $f^*:DFib(X)\to DFib(Y)$, since $DFib(X)$ is a full sub-2-category of $K/X$.
We now consider how to construct left adjoints for non-discrete fibrations.
If $K$ has exponentials and reflexive codescent objects, then each pullback functor $f^*:Fib(Y)\to Fib(X)$ and $f^*:Opf(Y)\to Opf(X)$ has a left adjoint $Lan_f$.
Because of the existence of $\Sigma_f$, by Proposition 3 it suffices to show that $Fib(X)$ and $Opf(X)$ 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 $FinCat$, 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).
We now consider the existence of right adjoints to pullback functors between fibrational slices.
If $K$ has exponentials, then $f^*:Opf(X)\to Opf(Y)$ and $f^*:Fib(X)\to Fib(Y)$ have right adjoints $Ran_f$ for any $f$.
If $f$ is exponentiable in $K$, this follows directly from Lemma 1 and Proposition 3. In particular, this applies when $f$ is a fibration or opfibration.
Now let $f:Y\to X$ be any morphism, and consider the comma square:
Suppose first that all the left adjoints $Lan_g$ exist for opfibrational slices and satisfy the Beck-Chevalley condition for comma squares. Then this comma square gives us an equivalence $f^*\simeq Lan_t s^*$, so that $f^*:Opf(X)\to Opf(Y)$ has the right adjoint $Ran_f := Ran_s t^*$.
We now show that regardless of the overall existence of left adjoints, the particular value $Lan_t s^*(A)$ exists for any $A\in Opf(X)$ and is given by $f^*A$. This clearly suffices to prove the result.
According to the proof of Proposition 5, $Lan_t s^*(A)$ should be given by the codescent object of
where
Now we have a map
the first map is a Beck-Chevalley morphism, and the second comes from the fact that $f^*A$ is an opfibration. We claim that $q$ is actually a codescent object of the above diagram, and in fact it is a split codescent object. Recall from
that a split codescent object is a diagram
with isomorphisms $c\tau \cong \sigma q$, $\pi_{12} \mu \cong \tau c$, $\pi_{02}\mu \cong \tau d$, $1\cong\pi_{01}\mu$, $1\cong d\tau$, and $1\cong q\sigma$ satisfying certain axioms. In this situation, $q$ is necessarily the codescent object of the diagram composed of $d,c,\pi_{01},\pi_{02},\pi_{12}$.
Tracing through the definitions above, we see that our $T$ is the limit of the span $A\to X \leftarrow Y$ weighted by
Likewise, $S$ is the limit of $A\to X \leftarrow Y$ weighted by
and $R$ is the limit by an evident more complicated weight. Finally, $f^*A$ itself is the limit of the same span weighted by
The morphism $q$ is induced by the action of $X$ on the opfibration $A$, applied twice. The morphism $d$ is induced by the inclusion of the second arrow in a composable pair and the action of $X$ on $A$ applied once. The morphism $c$ is induced by the composition of a composable pair. We define the splitting $\sigma,\tau,\mu$ 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.
If $K$ has exponentials, then $f^*:DOpf(X)\to DOpf(Y)$ and $f^*:DFib(X)\to DFib(Y)$ have right adjoints $Ran_f$ for any $f$.
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.
One further observation:
If $K$ has exponentials, then each category $DFib(X)$ is enriched over the cartesian closed category $disc(K)$.
We define the hom-object $\mathbf{DFib(X)}(A,B)$ to be $Ran_X B^A$, where $B^A$ is the exponential in $DFib(X)$ and
is the right adjoint to $X^*$. A composition map is obtained by adjointness in the usual way. We also have
so the underlying ordinary category of this $disc(K)$-enriched category is the ordinary category $DFib(X)$.
In fact, more is true: I believe $DFib(X)$ can be made into a locally internal category, or equivalently a locally small fibration, over $disc(K)$. Its fiber over a discrete object $Z\in K$ is the category $DFib(X\times Z)$. This follows by localizing the previous lemma in the slices $Fib(Z)\simeq K/Z$; one also has to check that a suitable Beck-Chevalley condition is satisfied by the right adjoints $Ran_f$.