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• CommentRowNumber1.
• CommentAuthorHurkyl
• CommentTimeMar 8th 2021

Added direct descriptions of the various universal fibrations.

• CommentRowNumber2.
• CommentAuthorHurkyl
• CommentTimeMar 8th 2021

Figured the proof of the description for the universal (co)cartesian fibration would be better duplicated here

• CommentRowNumber3.
• CommentAuthorHurkyl
• CommentTimeMar 14th 2021

Described the hom-spaces in the universal cocartesian fibration, and that it can be constructed using a lax coslice.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 14th 2021
• (edited Mar 14th 2021)

tried to give the references to online resources more canonical and more informative formatting: how about doing pointers to Kerdodon like this (with an eye towards their own request here on how to cite them):

• CommentRowNumber5.
• CommentAuthorHurkyl
• CommentTimeMar 24th 2021

I think there isn’t a consistent naming scheme whether the universal cocartesian fibration uses lax points or oplax points, so I’ve remarked on that ambiguity.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeDec 29th 2021

added pointer to:

• CommentRowNumber7.
• CommentAuthorHurkyl
• CommentTimeMar 27th 2022
• (edited Mar 28th 2022)

Here’s a subtle issue in my construction of the universal cocartesian fibration that I’m wondering if there is an easy solution to.

Consider two functors $\infty Cat \to \widehat{\infty Cat}$ from small $\infty$-categories to large $\infty$-categories given as follows:

• The covariant functor $P(-)$, the “$\infty$-category of presheaves”, where functorality comes from the universal property $Fun^L(P(C), E) \simeq Fun(C, E)$ for $\infty$-categories $E$ with small colimits.
• The contravariant functor $R = Fun((-)^{op}, \infty Gpd)$

Lurie defines $P$ pointwise by setting $P(C) = Fun(C^{op}, \infty Gpd)$, so the content of the first bullet point is to make it into a functor.

These two functors are closely related; for any functor $f : C \to D$, Proposition 5.2.6.3 of Higher Topos Theory proves that $P(f) \dashv R(f)$.

Let $\overline{el}(R) \to \infty Cat$ be the cartesian fibration classified by $R$. This is a presentable fibration, so it is also a cocartesian fibration that is classified by a covariant functor $R^{ladj}$.

It’s clear $P$ and $R^{ladj}$ agree strictly objects and up to equivalence on arrows… but I realize now I have a gap in that I don’t actually know there is a natural equivalence between them. Is there any easy or general method for arriving at that conclusion from that sort of method? Or does closing this gap rely more on diving into the specifics of how $P$ is defined?

• CommentRowNumber8.
• CommentAuthorHurkyl
• CommentTimeMar 30th 2022
• (edited Mar 30th 2022)

Here’s a concrete conjecture.

Let $F : C^{op} \to \infty\!Cat$ and $G : C \to \infty\!Cat$. Suppose:

• For each object $c$ in $C$, we have $F(c) = G(c)$ (probably better to just require an equivalence)
• For each arrow $f : c \to d$ in $C$, an adjunction $G(f) \dashv F(f)$

I conjecture that this data can be extended to a natural equivalence $G \simeq F^{ladj}$, where $F^\ladj$ is the functor constructed from $F$ by reversing the action on arrows via taking left adjoints. To be more precise, I claim this data can be used to produce a bundle $p : X \to C$ such that $p$ is both a cocartesian fibration classified by $G$ and a cartesian fibration classified by $F$. I probably want to include some consistency constraint with the specified adjunctions as well.

The reason that there is hope that it is enough to observe the relationship between how $F$ and $G$ act on arrows is that uniqueness of adjoints might be able to fill in the rest of the missing data. This seems like a mess to actually work through, though.

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