# Pseudofunctor correspondence for fibered opfibrations?

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.

• 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"? – Harry Gindi Jan 23 at 23:11
• 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. – Dylan Wilson Jan 24 at 15:02
• (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}.) – Dylan Wilson Jan 24 at 15:09
• @DylanWilson I think with that extra condition what you have is an internal opfibration in the 2-category of fibrations over $D$, right? – Mike Shulman Jan 24 at 16:53
• @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. – Tim Campion Jan 24 at 19:08