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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".)

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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

  1. each category $\Phi(b)$ is accessible,
  2. each functor $\Phi(\beta) : \Phi(b') \to \Phi(b)$ is accessible,
  3. the category $B$ is $\kappa$-accessible, and
  4. 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 well-known 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

  1. each category $\Phi(b)$ is locally presentable,
  2. each functor $\Phi(\beta) : \Phi(b') \to \Phi(b)$ has a left adjoint (i.e. the fibration is also an opfibration),
  3. the category $B$ is locally $\kappa$-presentable, and
  4. $\Phi$ 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.

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  • $\begingroup$ Great, that's exactly the result I had hoped would be true! $\endgroup$ Jul 13 '12 at 2:22

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