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1. • CommentRowNumber1.
   • CommentAuthorKarol Szumiło
   • CommentTimeDec 5th 2013
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   Author: Karol Szumiło
   Format: MarkdownItexIs there standard name for a simplicial set $X$ such that (a) faces of non-degenerate simplices of $X$ are non-degenerate, and for an $X$ such that (b) for each non-degenerate $x \in X_m$ the map $x \colon \Delta[m] \to X$ is injective? Similarly, are there standard names for categories whose nerves satisfy (a), respectively (b)?

   Is there standard name for a simplicial set $X$ such that

   (a) faces of non-degenerate simplices of $X$ are non-degenerate,

   and for an $X$ such that

   (b) for each non-degenerate $x \in X_m$ the map $x \colon \Delta[m] \to X$ is injective?

   Similarly, are there standard names for categories whose nerves satisfy (a), respectively (b)?

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeDec 5th 2013
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   Author: Zhen Lin
   Format: MarkdownItexI think condition (a) is satisfied if and only if the simplicial set is obtained by left Kan extension along $\mathbf{\Delta}_m \hookrightarrow \mathbf{\Delta}$, where $\mathbf{\Delta}_m$ contains only the monomorphisms. So perhaps you could say "$X$ has free degeneracies"? And condition (b) is closely related to the question of whether or not the simplicial set is a [[concrete presheaf]].

   I think condition (a) is satisfied if and only if the simplicial set is obtained by left Kan extension along $\mathbf{\Delta}_m \hookrightarrow \mathbf{\Delta}$, where $\mathbf{\Delta}_m$ contains only the monomorphisms. So perhaps you could say “$X$ has free degeneracies”? And condition (b) is closely related to the question of whether or not the simplicial set is a concrete presheaf.

3. • CommentRowNumber3.
   • CommentAuthorKarol Szumiło
   • CommentTimeDec 5th 2013
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   Author: Karol Szumiło
   Format: MarkdownItexI am aware of the characterization of (a) by "free degeneracies". That's perhaps not a bad name, but I doubt that it is standard. I'm not familiar with concrete sheaves and I don't understand your remark. The definition of a concrete sheaf requires a certain map to be injective, I fail to see how this map is related to the one of condition (b).

   I am aware of the characterization of (a) by “free degeneracies”. That’s perhaps not a bad name, but I doubt that it is standard.

   I’m not familiar with concrete sheaves and I don’t understand your remark. The definition of a concrete sheaf requires a certain map to be injective, I fail to see how this map is related to the one of condition (b).

4. • CommentRowNumber4.
   • CommentAuthorZhen Lin
   • CommentTimeDec 5th 2013
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   Author: Zhen Lin
   Format: MarkdownItexIf $X$ is a concrete simplicial set, then it has property (b), and the converse is true if no two simplices of $X$ have the same boundary.

   If $X$ is a concrete simplicial set, then it has property (b), and the converse is true if no two simplices of $X$ have the same boundary.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeDec 5th 2013
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   Author: Marc Hoyois
   Format: MarkdownItexA simplicial set satisfying (b) is sometimes called a *regular*. This appears in the [erratum to Hovey's book](http://hopf.math.purdue.edu/Hovey/model-err.pdf), for example.

   A simplicial set satisfying (b) is sometimes called a regular. This appears in the erratum to Hovey’s book, for example.

6. • CommentRowNumber6.
   • CommentAuthorKarol Szumiło
   • CommentTimeDec 5th 2013
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   Author: Karol Szumiło
   Format: MarkdownItex> If $X$ is a concrete simplicial set, then it has property (b), and the converse is true if no two simplices of $X$ have the same boundary. Well, simplicial sets that satisfy (b) and such that no two simplices have the same boundary are simply (ordered) simplicial complexes. That's a standard terminology, but this condition is much stronger than just (b) so this is not what I am after. > A simplicial set satisfying (b) is sometimes called a *regular*. This appears in the [erratum to Hovey's book](http://hopf.math.purdue.edu/Hovey/model-err.pdf), for example. That's actually the first time I see "regular" as a name for this property. On the other hand, I have seen it used more than once for a certain weaker property (e.g. on p. 46 of [this paper](http://www.tac.mta.ca/tac/volumes/12/2/12-02.pdf)). Are there other sources that use "regular" for condition (b)?

   If $X$ is a concrete simplicial set, then it has property (b), and the converse is true if no two simplices of $X$ have the same boundary.

   Well, simplicial sets that satisfy (b) and such that no two simplices have the same boundary are simply (ordered) simplicial complexes. That’s a standard terminology, but this condition is much stronger than just (b) so this is not what I am after.

   A simplicial set satisfying (b) is sometimes called a regular. This appears in the erratum to Hovey’s book, for example.

   That’s actually the first time I see “regular” as a name for this property. On the other hand, I have seen it used more than once for a certain weaker property (e.g. on p. 46 of this paper). Are there other sources that use “regular” for condition (b)?

7. • CommentRowNumber7.
   • CommentAuthorMarc Hoyois
   • CommentTimeDec 6th 2013
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   Author: Marc Hoyois
   Format: MarkdownItex> That's actually the first time I see "regular" as a name for this property. On the other hand, I have seen it used more than once for a certain weaker property (e.g. on p. 46 of [this paper](http://www.tac.mta.ca/tac/volumes/12/2/12-02.pdf)). Are there other sources that use "regular" for condition (b)? I don't know any. I mistakenly assumed Hovey was repeating the standard definition.

   That’s actually the first time I see “regular” as a name for this property. On the other hand, I have seen it used more than once for a certain weaker property (e.g. on p. 46 of this paper). Are there other sources that use “regular” for condition (b)?

   I don’t know any. I mistakenly assumed Hovey was repeating the standard definition.

8. • CommentRowNumber8.
   • CommentAuthorTim_Porter
   • CommentTimeDec 6th 2013
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   Author: Tim_Porter
   Format: MarkdownItexI seem to remember a discussion of something along those lines in the old article of Curtis. I have not my copy to hand, but have a look there. Ah I found it. He discusses a notion called `polyhedral' simplicial set and states (section 12) >If K is polyhedral and $x\in K$, is non-degenerate, then the vertices of x are all distinct, and each face $d_i x$ must also be nondegenerate. The double barycentric subdivision of any simplicial set is polyhedral, I think.

   I seem to remember a discussion of something along those lines in the old article of Curtis. I have not my copy to hand, but have a look there. Ah I found it. He discusses a notion called ‘polyhedral’ simplicial set and states (section 12)

   If K is polyhedral and $x\in K$, is non-degenerate, then the vertices of x are all distinct, and each face $d_i x$ must also be nondegenerate.

   The double barycentric subdivision of any simplicial set is polyhedral, I think.

9. • CommentRowNumber9.
   • CommentAuthorKarol Szumiło
   • CommentTimeDec 6th 2013
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   Author: Karol Szumiło
   Format: MarkdownItex"Polyhedral simplicial sets" in the sense of Curtis are the same as ordered simplicial complexes. This condition is strictly stronger than (b) since in a polyhedral simplicial set every simplex is determined by its vertices. > The double barycentric subdivision of any simplicial set is polyhedral, I think. The double subdivision of $\Delta[2] / \Delta[1]$ is not polyhedral. What's true is that the subdivision of any simplicial set is regular in the sense of Jardine's paper mentioned above. It is also true that the subdivision of the nerve of the "thin category of simplices" is the nerve of a poset and hence polyhedral. By the "thin category of simplices" I mean the category of all elements but only with face operators as morphisms.

   “Polyhedral simplicial sets” in the sense of Curtis are the same as ordered simplicial complexes. This condition is strictly stronger than (b) since in a polyhedral simplicial set every simplex is determined by its vertices.

   The double barycentric subdivision of any simplicial set is polyhedral, I think.

   The double subdivision of $\Delta[2] / \Delta[1]$ is not polyhedral. What’s true is that the subdivision of any simplicial set is regular in the sense of Jardine’s paper mentioned above. It is also true that the subdivision of the nerve of the “thin category of simplices” is the nerve of a poset and hence polyhedral. By the “thin category of simplices” I mean the category of all elements but only with face operators as morphisms.