There is also another way to look at the n-skeleton, described for instance in §8.1 of Cisinski's thesis (Les préfaisceaux...). For a Reedy category $A$, with degree function taking values on natural numbers, I denote $Sk^n A$ the sieve generated by the subcategory $A_{\le n}$ of $A$.

\o @DenisNardin

For any simplicial set $X$, the category of elements $\Delta/X$ inherits a structure of a Reedy category. The sieve $Sk^n(\Delta/X)$ of $\Delta/X$ corresponds to a sub-presheaf $Sk^n X$ of $X$. Notice that by definition $Sk^n(\Delta/X)$ is just $\Delta/Sk^n X$. The simplicial set $Sk^n X$ admits a easy explicit description, corresponding to the one already described above by the colimit.

(One has to be careful that the two descriptions do not agree in general for wilder Eilenberg-Zilber categories, like symmetric trees or E-Z categories with non-trivial automorphisms)

so I've found some errors in my preprint. Mostly minor (arrows pointing the wrong way) but some are more complicated to fix and will require bigger changes. Is it wiser to try to fix it up with minimal changes, or to do a "rm -rf" restart?

@JoeBerner up to you. a paper is meant to be a readable account of the situation. if the problem with your paper harms its narrative flow, then maybe a serious rewrite is in order. if the problem is purely technical and doesn't change the sketch you're trying to conjure with the narrative, then maybe don't