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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.
Back on this inclusion of the subcategory of nondegenerate simplices. We’ve observed that if every nondegenerate simplex has nondegenerate faces, then the inclusion is a right adjoint, and John says the converse also holds. This implies in particular that the inclusion is a final functor, so that colimits over the category of simplices can be computed on the nondegenerate simplices. But being final is a weaker statement than being a right adjoint, so can the inclusion of nondegenerate simplices be final even if not every nondegenerate simplex has nondegenerate faces? Are there simplicial sets for which this inclusion fails to be final?
By the way, I had a look at the current version of HTT 4.2.3.15, and I think it includes this condition: the restriction to nondegenerate simplices only happens in step (5), after we’ve already passed to a category of simplices several times in a way that ensures that at this point the faces of nondegenerate simplices are all nondegenerate. Perhaps that was an update in the past 1.5 years.
I guess it isn’t too hard to come up with examples where the inclusion of nondegenerate simplices isn’t final. E.g. take a square $\Delta^1\times \Delta^1$ and collapse the diagonal to a point. Then for that degenerate 1-simplex $\sigma$, the category $\sigma \downarrow (\Delta\downarrow X)_{nondeg}$ is, I think, not connected.
Sounds right. Embedding the square in $\Delta^3$ so that I can name the vertices 0123 and so that $X$ is the quotient by the 03 simplex….
The only factorizations $\Delta^1 \hookrightarrow \Delta^n \to X$ of $\sigma$ with $\Delta^n \to X$ nondegenerate are through the 013 and 023 simplices. There are no morphisms out of these factorizations, since they have to be monic on the $\Delta^n$ term, and these are the top simplices. There are no morphisms into these factorizations, because all of the other ones have a nonmonic $\Delta^1 \to \Delta^n$ part,
I have added pointer to
and polished up the other reference items
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