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I have an issue concerning a property of "left cancellation" for groupoidal cartesian fibrations of ∞-cosmoi (but everything works fine in a 2-category as well).

A 1-cell $p: E \to B$ is called groupoidal cartesian fibration if every 2-cell $\alpha:b\rightarrow p\circ e$ in $hom(X,B)$ admits an essentially unique lift $\chi:e' \rightarrow e$ in $hom (X,E)$, where essentially unique means that any two lifts are isomorphic via a 2-cell that projects to an identity cell under $p$.

What I want to prove is that if $p$ and $p\circ q$ are groupoidal cartesian fibrations then $q$ is such .

I managed to find a 2-cell $\tilde{\chi}$ such that $q\tilde {\chi}=\alpha \gamma$, where $\gamma$ is an iso projecting to an identity under $p$, but I don't see how to proceed from here.

Thanks in advance for any idea/hint !

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  • $\begingroup$ It's not true. If it were true then every morphism of groupoids would be a fibration. $\endgroup$
    – Zhen Lin
    Commented Sep 11, 2015 at 10:34
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    $\begingroup$ They are assumed to be fibrations, I skipped this detail since it seemed unessential to the proof. The reference is "Fibrations and Yoneda's lemma in an ∞-cosmos" by Riehl and Verity, Lemma 4.2.6. $\endgroup$
    – Edoardo Lanari
    Commented Sep 11, 2015 at 10:39

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