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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(Δ↓X)→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=Δn. But then both N(Δ↓Δn) and Δn are nerves of categories, and I suppose it’s reasonably clear that the functor Δ↓Δn→[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(Δ↓X)→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 Δn→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 ησ:σ→Lσ exhibits the unique factorization of σ as a surjection ησ followed by a nondegenerate simplex Lσ. If σ is nondegenerate then the adjunction factors the injection δi:diσ→σ through the surjection ηdiσ, making ηdiσ the identity and so diσ 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 Δ1×Δ1 and collapse the diagonal to a point. Then for that degenerate 1-simplex σ, the category σ↓(Δ↓X)nondeg is, I think, not connected.
Sounds right. Embedding the square in Δ3 so that I can name the vertices 0123 and so that X is the quotient by the 03 simplex….
The only factorizations Δ1↪Δn→X of σ with Δn→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 Δn term, and these are the top simplices. There are no morphisms into these factorizations, because all of the other ones have a nonmonic Δ1→Δn part,
I have added pointer to
and polished up the other reference items
“Write (Δ↓X) nondeg↪(Δ↓X)
for the (non-full) subcategory on the non-degenerate simplices.” But it is actually is full, unless I am somehow completely confused here.
It is rather confusingly expressed, but the point I believe is that there are morphisms between non-degenerate simplices which factor through degenerate ones, and these are intended to be ignored. This does not just happen by magic, though, it of course makes sense to consider the full sub-category. Maybe what was intended was that one could set this up differently by replacing the full simplex category by the one without degeneracies and considering the slice category using the latter; then it does make sense to observe that the resulting category is not a full sub-category.
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The identity map on every simplex factors through a degenerate simplex (by the simplicial identity d0s0=id), so we can’t exclude those.
If one replaces the simplex category by one without degeneracies, one is simply keeping the monomorphisms. But, as the nlab page already states, every morphism in the category of simplices between nondegenerate simplices is already a monomorphism anyway, so what we get is again the full subcategory on nondegenerate simplices.
Good points! I should have said factoring through a degeneracy of lower dimension, i.e. if one has a simplex of dimension n, I think what is intended is to exclude factoring through the degeneracy taking an (n−1)-simplex to an n-simplex.
Whether keeping higher dimensional degeneracies is important for something, I don’t know; barycentric subdivision is subtle, and it is not impossible. But if not, since, as you point out, one gets the full sub-category from the more conceptual description in terms of the simplex category without degeneracies, it would be nice if one can just use that full sub-category.
It would be good to clarify all this on the nLab page anyhow.
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Good points! I should have said factoring through a degeneracy of lower dimension, i.e. if one has a simplex of dimension n, I think what is intended is to exclude factoring through the degeneracy taking an (n−1)-simplex to an n-simplex.
Whether keeping higher dimensional degeneracies is important for something, I don’t know; barycentric subdivision is subtle, and it is not impossible. But if not, since, as you point out, one gets the full sub-category from the more conceptual description in terms of the simplex category without degeneracies, it would be nice if one can just use that full sub-category.
It would be good to clarify all this on the nLab page anyhow.
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In other words, as I suggested, one has to exclude maps which e.g. take a simplex to a degenerate face. In particular, one will not have a semi-simplicial set, so actually the very point is to not replace the simplex category with the one without degeneracies; apologies for the red herring there, but it was good to discuss/clarify it!
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Consider a morphism in the category of simplices of X, i.e. a commuting triangle consisting of f:Δn→Δn′, c:Δn→X and c′:Δn′→X such that c′f=c.
If c,c′ are nondegenerate, then it’s not possible for f to be the inclusion of a degenerate face of c′, as the face of c′ corresponding to restriction along f is exactly c, which is assumed to be nondegenerate.
Take for example X to be the model for the circle with a single 0-simplex and single non-degenerate 1-simplex, and let c and c′ be maps Δ2→Δ1→X which pick the non-degenerate 1-simplex out, with the two projections Δ2→Δ1 being different. The taking f to be Δ2→Δ1→Δ2, where the projection is the one used in c, and the face map being one which is projected onto in c′, one gets a commuting diagram.
The idea as I understood it was that such an f is not allowed, one is only allowed injections.
In particular, the final sentence of Definition 2.2 is confusing I think. Non-degenerate is defined in terms of not being in the image of a degeneracy, whereas in the final sentence of Definition 2.2 it seems to mean when the source is the canonical morphism Δn→X picking out a non-degenerate simplex.
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