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Recall that functor $p\colon \mathcal{C} \to \mathcal{D}$ of $\infty$-categories is said to be an exponentiable fibration if the following equivalent conditions hold:

  1. The pullback functor $p^*\colon \mathrm{Cat}_{/\mathcal{D}} \to \mathrm{Cat}_{/\mathcal{C}}$ admits a right adjoint $p_*\colon \mathrm{Cat}_{/\mathcal{C}} \to \mathrm{Cat}_{/\mathcal{D}}$.

  2. For any pair of composable arrows $x \xrightarrow{f} y \xrightarrow{g} z$ in $\mathcal{D}$ and any lift $\tilde{h}\colon \tilde{x} \to \tilde{y}$ of $h = g \circ f$ to $\mathcal{C}$, the $\infty$-category of factorizations of $\widetilde{h}$ in $\mathcal{C}$ which lift the given factorization $h=g\circ f$ of $h$, is weakly contractible.

The notion of exponentiable fibration is a model independent version of that of flat inner fibration studied in section B.3 of Lurie's Higher Algebra (see also Lemma 1.10 of Ayala-Francis https://arxiv.org/abs/1702.02681). For example, cartesian and cocartesian fibrations are exponentiable. The collection of exponentiable fibrations is closed under composition, so composing cartesian and cocartesian fibrations gives many examples of exponentiable fibrations which are not themselves cartesian or cocartesian.

My question is as follows:

Suppose that $\mathcal{C} \xrightarrow{p} \mathcal{D} \xrightarrow{q} \mathcal{E}$ is a composable pair of exponentiable fibrations. Does it follows that $q_*(p\colon \mathcal{C} \to \mathcal{D}) \in \mathrm{Cat}_{/\mathcal{E}}$ is exponentiable? In other words, does $q_*$ send exponentiable fibrations over $\mathcal{D}$ to exponentiable fibrations over $\mathcal{E}$ for any exponentiable fibration $q\colon \mathcal{D} \to \mathcal{E}$?

For context, note that in the above situation, if $q$ is a cocartesian fibration then $q_*$ sends cartesian fibrations to cartesian fibrations, and dually if $q$ is a cartesian fibration then $q_*$ preserves cocartesian fibrations. A positive answer to the above question is hence compatible, and somewhat suggested, by these two dual facts.

Ideally this question was considered in the literature, in which case I would be grateful for a reference, but if not, any argument or even sketch of argument could be extremely useful.

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  • $\begingroup$ The context you mention suggests a special case to consider: does pushforward along a cartesian fibration preserve cartesian fibrations? $\endgroup$
    – Tim Campion
    Commented Sep 28, 2023 at 1:39
  • $\begingroup$ Not in general. For example suppose that $q: D \to E$ is a cartesian fibration and $p: D \times X \to D$ is a constant fibration with fibre $X$. Then the pushforward $q_*(D \times X)$ is a cocartesian fibration whose fibres are $Fun(D_e,X)$ for $e$ in $E$, and this will generally only be covariantly functorial in $e$ via restriction, but not contravariantly so (unless some right Kan extensions exist). But I'm hoping it would still be true in general that $q_*$ send cartesian fibrations to exponentiable fibrations. $\endgroup$
    – Yonatan Harpaz
    Commented Sep 28, 2023 at 6:26
  • $\begingroup$ If the answer is "yes", then a quick search seems to indicate that it doesn't seem to have been anticipated in the strict 1-categorical literature... OTOH I'm having trouble even answering the question "does there exist a category where exponentiable maps are not closed under pushforward?" $\endgroup$
    – Tim Campion
    Commented Oct 21, 2023 at 4:06

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