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nLab > Latest Changes: category of simplices

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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(Δ↓X)→XN(\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=Δ nX=\Delta^n. But then both N(Δ↓Δ n)N(\Delta \downarrow \Delta^n) and Δ n\Delta^n are nerves of categories, and I suppose it’s reasonably clear that the functor Δ↓Δ n→[n]\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(Δ↓X)→XN(\Delta\downarrow X)\to X is a final map for any simplicial set XX.

    • 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 nn-simplices of XX are equivalently the monos Δ n→X\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.

    diff, v15, current

    • CommentRowNumber9.
    • CommentAuthorHurkyl
    • CommentTimeOct 29th 2020

    The article here says that the inclusion of the full subcategory of nondegenerate simplices has a left adjoint when every nondegenerate simplex of XX has nondegerate faces. But HTT variant 4.2.3.15 claims this is true for every XX.

    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 XX is necessary. If the inclusion has a left adjoint LL then the unit η σ:σ→Lσ\eta_{\sigma} : \sigma \to L\sigma exhibits the unique factorization of σ\sigma as a surjection η σ\eta_{\sigma} followed by a nondegenerate simplex LσL\sigma. If σ\sigma is nondegenerate then the adjunction factors the injection δ i:d iσ→σ\delta^{i} : d_{i}\sigma \to \sigma through the surjection η d iσ\eta_{d_{i}\sigma}, making η d iσ\eta_{d_{i}\sigma} the identity and so d iσd_{i}\sigma nondegenerate.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMar 10th 2022

    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?

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2022

    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.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2022

    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 Δ 1×Δ 1\Delta^1\times \Delta^1 and collapse the diagonal to a point. Then for that degenerate 1-simplex σ\sigma, the category σ↓(Δ↓X) nondeg\sigma \downarrow (\Delta\downarrow X)_{nondeg} is, I think, not connected.

    • CommentRowNumber14.
    • CommentAuthorHurkyl
    • CommentTimeMar 15th 2022

    Sounds right. Embedding the square in Δ 3\Delta^3 so that I can name the vertices 0123 and so that XX is the quotient by the 03 simplex….

    The only factorizations Δ 1↪Δ n→X\Delta^1 \hookrightarrow \Delta^n \to X of σ\sigma with Δ n→X\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 Δ n\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 Δ 1→Δ n\Delta^1 \to \Delta^n part,

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2022
    • CommentRowNumber16.
    • CommentAuthorManuel Araújo
    • CommentTimeNov 23rd 2024

    “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.

  17. 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.

    only commenting

    diff, v19, current

  18. The identity map on every simplex factors through a degenerate simplex (by the simplicial identity d 0s 0=idd_0s_0=\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.

  19. Good points! I should have said factoring through a degeneracy of lower dimension, i.e. if one has a simplex of dimension nn, I think what is intended is to exclude factoring through the degeneracy taking an (n−1)(n-1)-simplex to an nn-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.

    only commenting

    diff, v21, current

  20. Good points! I should have said factoring through a degeneracy of lower dimension, i.e. if one has a simplex of dimension nn, I think what is intended is to exclude factoring through the degeneracy taking an (n−1)(n-1)-simplex to an nn-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.

    only commenting

    diff, v21, current

  21. I think the answer here may clarify what is going on.

    only commenting

    diff, v22, current

  22. 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!

    only commenting

    diff, v22, current

  23. Consider a morphism in the category of simplices of XX, i.e. a commuting triangle consisting of f:Δ n→Δ n′f:\Delta^n\to \Delta^{n'}, c:Δ n→Xc:\Delta^n \to X and c′:Δ n′→Xc':\Delta^{n'} \to X such that c′f=cc'f=c.

    If c,c′c,c' are nondegenerate, then it’s not possible for ff to be the inclusion of a degenerate face of c′c', as the face of c′c' corresponding to restriction along ff is exactly cc, which is assumed to be nondegenerate.

  24. Take for example XX to be the model for the circle with a single 00-simplex and single non-degenerate 1-simplex, and let cc and c′c' be maps Δ 2→Δ 1→X\Delta^{2} \rightarrow \Delta^{1} \rightarrow X which pick the non-degenerate 1-simplex out, with the two projections Δ 2→Δ 1\Delta^{2} \rightarrow \Delta^{1} being different. The taking ff to be Δ 2→Δ 1→Δ 2\Delta^{2} \rightarrow \Delta^{1} \rightarrow \Delta^{2}, where the projection is the one used in cc, and the face map being one which is projected onto in c′c', one gets a commuting diagram.

    The idea as I understood it was that such an ff 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 Δ n→X\Delta^{n} \rightarrow X picking out a non-degenerate simplex.

    only commenting

    diff, v23, current

  25. Replaced “for the (non-full) subcategory on the non-degenerate simplices.” with “for the (full) subcategory on the non-degenerate simplices.”

    diff, v24, current

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