# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

## Comments on Theorem 5.4.4.1

Go back to the page of Theorem 5.4.4.1.

Comment #756 by Tim Holzschuh on

Typo in the proof: "From the implication $(1)\implies (3)$ shows that $\sigma′$ is also locally $q$-cartesian when viewed as an edge of the simplicial set $\mathcal{C}_{/Z}$.": maybe "The implication ... shows that... "?

Also shortly before the end of the proof there is a latex line that doesn't get parsed properly (for me at least) right after "together with the following translation of conditions $(c)$ and $(d)$:".

Comment #760 by Kerodon on

Yep. Thanks!

Comment #1264 by Carles Sáez on

There are some typos in the proof: 1) Along the proof the same map is sometimes called $\mu_0$ and sometimes $f_0$. 2) Maps $u,v,w,w'$ are sometimes called $f,g,h,h'$. 3) In condition c), $\operatorname{N}_{\bullet }( \{ 0 < i-1 < i \} )$ should be $\operatorname{N}_{\bullet }( \{ j< i-1 < i \} )$. 4) After the first square diagram, "therefore by extended" should be "therefore be extended". 5) In condition $(*_s)$, the simplex $\tau_2$ should be the simplex $\tau_s$.

Comment #1697 by Bogdan on

Since $\mathcal{C}$ is an $\infty$-category

It should be "since $\mathcal{C}$ is an $(\infty, 2)$-category".

Comment #1699 by Kerodon on

Yep. Thanks!

Comment #2020 by Dennis Chen on

Hey! So I might be missing something, but the induction on the conditions $(1_n)$ seems to miss the case $n=3$. I'm thinking of the line "In particular, our inductive hypothesis guarantees that the simplex $\sigma''$ satisfies condition $(1_m)$ for $3 \leq m < n$." It seems like this hypothesis assumes the $n= 3$ case... is it obviously true or something? Thanks!

Comment #2035 by Kerodon on

When $n=3$, the assertion that $(\ast_m)$ holds for $3 \leq m < n$ is vacuous.

There are also:

• 2 comment(s) on Section 5.4: $(\infty ,2)$-Categories
• 2 comment(s) on Subsection 5.4.4: The Local Thinness Criterion

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