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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeApr 4th 2012
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   Author: Mike Shulman
   Format: MarkdownItexI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorEmily Riehl
   • CommentTimeApr 5th 2012
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   Author: Emily Riehl
   Format: MarkdownItexThis 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?

   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?

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeApr 6th 2012
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   Author: Mike Shulman
   Format: MarkdownItexHmm, 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 $X$.

   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 $X$.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 1st 2013
   • (edited Feb 1st 2013)
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   Author: Urs
   Format: MarkdownItexSomebody 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorrognes
   • CommentTimeFeb 4th 2013
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   Author: rognes
   Format: TextThe 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.

   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.
6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2013
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   Author: Urs
   Format: MarkdownItexMaybe 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 4th 2013
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   Author: Urs
   Format: MarkdownItexOkay, 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.)

   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.)

8. • CommentRowNumber8.
   • CommentAuthorTim_Porter
   • CommentTimeApr 9th 2019
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   Author: Tim_Porter
   Format: MarkdownItexadded a `de' to non-generate! plus some hyphens in nondegenerate. <a href="https://ncatlab.org/nlab/revision/diff/category+of+simplices/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/category+of+simplices/15">v15</a>, <a href="https://ncatlab.org/nlab/show/category+of+simplices">current</a>

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

   diff, v15, current

9. • CommentRowNumber9.
   • CommentAuthorHurkyl
   • CommentTimeOct 29th 2020
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   Author: Hurkyl
   Format: MarkdownItexThe 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 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.

10. • CommentRowNumber10.
    • CommentAuthorJohn Dougherty
    • CommentTimeMar 22nd 2021
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    Author: John Dougherty
    Format: MarkdownItexThe 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.

    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.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMar 10th 2022
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    Author: Mike Shulman
    Format: MarkdownItexBack 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?

    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?

12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2022
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    Author: Mike Shulman
    Format: MarkdownItexBy 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.

    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.

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeMar 15th 2022
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    Author: Mike Shulman
    Format: MarkdownItexI 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.

    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.

14. • CommentRowNumber14.
    • CommentAuthorHurkyl
    • CommentTimeMar 15th 2022
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    Author: Hurkyl
    Format: MarkdownItexSounds 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,

    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,

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2022
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    Author: Urs
    Format: MarkdownItexI have added pointer to * [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Daniel M. Kan]], Section 5.9 of: *[[Model Categories and More General Abstract Homotopy Theory]]* (1997) &lbrack;[pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dhk.pdf), [[DwyerHirschhornKan-ModelCategories.pdf:file]]&rbrack; and polished up the other reference items <a href="https://ncatlab.org/nlab/revision/diff/category+of+simplices/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/category+of+simplices/18">v18</a>, <a href="https://ncatlab.org/nlab/show/category+of+simplices">current</a>

    I have added pointer to

    • William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Section 5.9 of: Model Categories and More General Abstract Homotopy Theory (1997) [pdf, DwyerHirschhornKan-ModelCategories.pdf:file]

    and polished up the other reference items

    diff, v18, current