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I added a bit to category of simplices, including the fact that the category of nondegenerate simplices is final and thus colimits can be computed using only that, and that the nerve of the category of simplices itself is colimit-preserving.
This relates to a question I was thinking about today. How do you prove that the “last vertex map” $N(\Delta \downarrow X) \to X$ is a weak equivalence for any simplicial set? Because the colimits over the category of simplices are homotopy colimits, by your remarks on cocontinuity, it would suffice to prove this in the case $X=\Delta^n$. But then both $N(\Delta \downarrow \Delta^n)$ and $\Delta^n$ are nerves of categories, and I suppose it’s reasonably clear that the functor $\Delta \downarrow \Delta^n \to [n]$ is final. Does this make any sense?
Hmm, yes, that makes sense. Actually, 4.2.3.14 in Higher Topos Theory asserts that $N(\Delta\downarrow X)\to X$ is a final map for any simplicial set $X$.
Somebody kindly pointed out by email to me that there was a false statement in the entry category of simplicies (that the non-degenerate $n$-simplices of $X$ are equivalently the monos $\Delta^n \to X$).
I have fixed that and in the course of this I have tried to slightly polish the entry a bit more. Added formal proposition-environments, stated the relation to barycentric subdivision and added a textbook reference.
More could be done here. But I am out of time now.
Maybe we should rather add more comments to the entry on barycentric subdivision then. For instance Lurie in HTT, variant 4.2.3.15 uses the term as I did.
Okay, I have edited a bit further.
(But I do need to look into something else now. Feel free to hit the edit-button if there is more you’d want to see clarified.)
The article here says that the inclusion of the full subcategory of nondegenerate simplices has a left adjoint when every nondegenerate simplex of $X$ has nondegerate faces. But HTT variant 4.2.3.15 claims this is true for every $X$.
Is HTT correct here? That the assumption can be omitted?
I’m inclined to think HTT is correct, but I’ve been confused on this point in the past… but I think my confusion might have involved this very nLab page, so my instincts are all messed up here.
The condition on $X$ is necessary. If the inclusion has a left adjoint $L$ then the unit $\eta_{\sigma} : \sigma \to L\sigma$ exhibits the unique factorization of $\sigma$ as a surjection $\eta_{\sigma}$ followed by a nondegenerate simplex $L\sigma$. If $\sigma$ is nondegenerate then the adjunction factors the injection $\delta^{i} : d_{i}\sigma \to \sigma$ through the surjection $\eta_{d_{i}\sigma}$, making $\eta_{d_{i}\sigma}$ the identity and so $d_{i}\sigma$ nondegenerate.
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