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Edoardo Lanari
Edoardo Lanari
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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?

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 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 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?

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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Edoardo Lanari
Edoardo Lanari
  • 1.4k
  • 8
  • 15

Equivalent definitions of Cartesian Fibrations between Quasi-Categories

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 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 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?