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In the paper by Verity and Riehl "Fibrations and Yoneda lemma in an ${\infty}$-cosmos" (https://arxiv.org/abs/1506.05500), they prove a Yoneda lemma that holds in any ${\infty}$-cosmos (see Corollary 6.2.13). Therefore, in particular, it holds in $\textbf{qCat}$.

If we unravel their definition of cartesian fibration, we find out that it is almost representably defined, in the sense that an isofibration $p:E \to B$ is a cartesian fibration in an ${\infty}$-cosmos iff $map(A,E) \to map(A,B)$ is (almost) a cartesian fibration of quasicategories for every $A$. More precisely, instead of the usual requirement of a suitable lift for a certain edge which has to be cartesian with respect to the inner fibration $map(A,p)$,they ask for a weaker property: this lift $\chi$ has to be such that $map(A,p)$ has the right lifting property only against the inclusion $\Lambda^2[2] \to \Delta[2]$ when the restriction to $\Delta^{\{1,2\}}$ is $\chi$ itself (so it is Lurie's definition of cartesian edge where we restrict ourselves to n=2). They also have a conservativity requirement, but I am not exactly sure if and how this fits into my question.

Does this produce an equivalent notion of cartesian fibration (I suspect it does not)? Or at least, is it true that $p:X \to Y$ inner fibration of quasicategories is a cartesian fibration iff $map(A,p)$ is a cartesian fibration in this weaker sense for every quasicategory (or even any simplicial set) A?

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Riehl and Verity prove that their definition agrees with Lurie's in Corollary 4.1.24 (cf. Remark 2.4.1.4 in HTT).

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  • $\begingroup$ Thanks! I read the paper a year ago and I forgot they proved it $\endgroup$ – Edoardo Lanari Dec 17 '16 at 23:38
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Here's an outline of a proof that Riehl-Verity (RV) cartesian fibrations are the same as Joyal-Lurie (JL) cartesian fibrations. There are some details that I'm not sure actually hold though.

First show that $p: E \to B$ is a JL cartesian fibration iff the cotensor $p^{\Delta^n} : E^{\Delta^n} \to B^{\Delta^n}$ is an RV cartesian fibration for every $n$. In fact, I think that a lift is JL-cartesian iff its degeneracies in $E^{\Delta^n}$ are RV-cartesian for every $n$.

Then use Riehl and Verity's Theorem 4.1.10, which says that an isofibration $p: E \to B$ is an RV cartesian fibration iff the induced functor $E\downarrow E \to B \downarrow p$ admits a right adjoint right inverse. Now cotensoring with $\Delta^n$ induces should commute with taking comma categories, and I think (or hope) that it should be 2-functorial on the homotopy 2-category. So if $E\downarrow E \to B \downarrow p$ admits a right adjoint right inverse, then $E^{\Delta^n} \downarrow E^{\Delta^n} \to B^{\Delta^n} \downarrow p^{\Delta^n}$ does too, and so $p^{\Delta^n}: E^{\Delta^n} \to B^{\Delta^n}$ is RV-cartesian, and $p$ is JL cartesian.

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