Kerodon

In the second bullet displayed in the proof of Proposition 2.4.5.8, the equality

$\\\{ j, j +1,...,k −1,k\\\} = I_0$

should be an inclusion

$\\\{j, j +1,...,k −1,k\\\} \supseteq I_0$

Indeed, we aren't specifying a simplex and then "pushforwarding" our tuple along this simplex: we're directly specifying the tuple inside the simplicial set. Thus, the relevant part of the proof of Lemma 2.4.4.16 is the one which establishes condition (2'), and it does so by explicitly showing that the generators [of the $m$-th simplicial layer of the path category of $N(Q)$] —say, with domain $x$ and codomain $y$— correspond to tuples $\overrightarrow{J} = (J_0, ..., J_m)$ of ordered subsets of $Q$ starting at $x$ and ending at $y$ and such that $J_0 \supseteq ... \supseteq ... J_m=\\\{x,y\\\}$, with no further restriction on $J_0$.

Yep. Thanks!