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Let $\mathcal D$ be a category. A familiar bit of category theory says that Grothendieck fibrations $\mathcal C \to \mathcal D$ are equivalent to pseudofunctors $F: \mathcal D^\mathrm{op} \to \mathsf{Cat}$.

Let's say that a $\mathcal D$-fibered Grothendieck opfibration is a functor $\mathcal B \to \mathcal C$ satsifying the following conditions:

  1. The composite $\mathcal B \to \mathcal C \to \mathcal D$ is a Grothendieck fibration.

  2. $\mathcal B \to \mathcal C$ preserves $\mathcal D$-cartesian morphisms.

  3. For every $D \in \mathcal D$, the fiber functor $\mathcal B_D \to \mathcal C_D$ is a Grothendieck opfibration.

  4. For every $f: D' \to D$ in $\mathcal D$ and $\mathcal B_D$-cocartesian morphism $g: X \to Y$ in $\mathcal C_D$, the reindexed morphism $f^\ast(g): f^\ast(X) \to f^\ast(Y)$ in $\mathcal C_{D'}$ is $\mathcal B_{D'}$-cocartesian.

Questions:

(0. Is there a more standard name for $\mathcal D$-fibered Grothendieck opfibrations?)

  1. What is the analog of the fibration - pseudofunctor correspondence in this setting?

For Question 1, the idea is that if $\mathcal C \to \mathcal D$ is a Grothendieck fibration, then a $\mathcal D$-fibered Grothendieck opfibration $\mathcal B \to \mathcal C$ should straighten to assign to every $D \in \mathcal D$ a pseudofunctor $G_D: \mathcal C_D \to \mathsf{Cat}$ which is "suitably functorial in $D$". But I'm wondering if somebody has written up the details.

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  • $\begingroup$ I would guess that this falls under the heading of "indexed category theory", which I dabbled in only for a short while. I would guess this is something like an "indexed opfibration"? $\endgroup$ – Harry Gindi Jan 23 at 23:11
  • $\begingroup$ If you add the additional condition that B—>C preserves Cartesian morphisms then this has been studied-see, for example, Jay Shah’s thesis on indexed colimits (you have to put opposites everywhere though- their convention is to be cocartesian over the base). In this case, if you take ‘vertical opposites’ of B and C over D to get B’ —>C’ cartesian. The straightening of this to a functor (C’)^{op}—>Cat is what you’re after I think. $\endgroup$ – Dylan Wilson Jan 24 at 15:02
  • $\begingroup$ (By vertical opposite I mean the construction explained in Barwick-Glasman-Nardin: arxiv.org/pdf/1409.2165.pdf. Also, since you op twice in the end, the category (C’)^{op} is their C^{\vee}.) $\endgroup$ – Dylan Wilson Jan 24 at 15:09
  • $\begingroup$ @DylanWilson I think with that extra condition what you have is an internal opfibration in the 2-category of fibrations over $D$, right? $\endgroup$ – Mike Shulman Jan 24 at 16:53
  • $\begingroup$ @DylanWilson Wow, I definitely should add that condition. I'm thinking there should be some description not involving the vertical opposite, but I suppose that's a good way to think about it. $\endgroup$ – Tim Campion Jan 24 at 19:08

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