Suppose I have a functor $\mathcal{E} \to \mathcal{B}$ that is both a Grothendieck fibration and an opfibration, $\mathcal{B}$ is presentable, and the fibres $\mathcal{E}_{b}$ for $b \in \mathcal{B}$ are all presentable. Are there any known conditions on this setup under which the category $\mathcal{E}$ will be presentable? (By "presentable" I mean the same thing as "locally presentable".)
1 Answer
In section 5.3 of Accessible categories by Makkai and Pare, they prove that if $\Phi : B^{\mathrm{op}} \to \mathrm{Cat}$ is a pseudofunctor such that
 each category $\Phi(b)$ is accessible,
 each functor $\Phi(\beta) : \Phi(b') \to \Phi(b)$ is accessible,
 the category $B$ is $\kappa$accessible, and
 the pseudofunctor $\Phi$ preserves $\kappa$filtered (weak 2)colimits,
then the total category of the Grothendieck construction of $\Phi$ is again accessible. An accessible category is locally presentable iff it is complete iff it is cocomplete, and it is wellknown that the total category of a fibration is complete if the base and fibers are complete and the restriction functors are continuous. Moreover, an accessible functor between locally presentable categories is continuous iff it has a left adjoint, and any adjoint between locally presentable categories is accessible. Thus, the total category of $\Phi$ is locally presentable if
 each category $\Phi(b)$ is locally presentable,
 each functor $\Phi(\beta) : \Phi(b') \to \Phi(b)$ has a left adjoint (i.e. the fibration is also an opfibration),
 the category $B$ is locally $\kappa$presentable, and
 $\Phi$ preserves $\kappa$filtered (weak 2)colimits.
So it sounds like to your hypotheses you need only to add that $\Phi$ preserves $\kappa$filtered (weak 2)colimits, for some $\kappa$ such that $B$ is locally $\kappa$presentable.

$\begingroup$ Great, that's exactly the result I had hoped would be true! $\endgroup$ Jul 13, 2012 at 2:22