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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeApr 4th 2012

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.

• CommentRowNumber2.
• CommentAuthorEmily Riehl
• CommentTimeApr 5th 2012

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?

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeApr 6th 2012

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 $X$.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeFeb 1st 2013
• (edited Feb 1st 2013)

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.

• CommentRowNumber5.
• CommentAuthorrognes
• CommentTimeFeb 4th 2013
The statement that the nerve of the category of non-degenerate simplices is a model for the barycentric subdivision is still wrong. At least if by barycentric subdivision you mean Kan (normal) subdivision, as the link to barycentric subdivision suggests. Think about X = \Delta^2/\partial\Delta^2, for instance. It helps if the simplicial set is what Waldhausen calls non-singular, i.e., each non-degenerate simplex is embedded. Then the category of non-degenerate simplices is a partially ordered set, and its nerve is the same as Kan's subdivision.
• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeFeb 4th 2013

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.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeFeb 4th 2013

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.)

• CommentRowNumber8.
• CommentAuthorTim_Porter
• CommentTimeApr 9th 2019

added a ‘de’ to non-generate! plus some hyphens in nondegenerate.

• CommentRowNumber9.
• CommentAuthorHurkyl
• CommentTimeOct 28th 2020

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.

• CommentRowNumber10.
• CommentAuthorJohn Dougherty
• CommentTimeMar 22nd 2021

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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