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1. • CommentRowNumber1.
   • CommentAuthorJohn Baez
   • CommentTimeAug 22nd 2017
   • (edited Aug 22nd 2017)
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   Author: John Baez
   Format: MarkdownItexOn the article [[augmented simplicial set]], an augmented simplicial set was defined as a presheaf on the full subcategory $\Delta_+$ of $Cat$ consisting of free categories over finite linear directed graphs. This is fairly convoluted, so I've added a simpler description of a skeleton of this category. My main point, though, is that this category is denoted $\Delta_+$. On the article [[semi-simplicial set]], a semi-simplicial set is defined as a presheaf on the category of finite linearly ordered sets and injective order-preserving maps. This is _also_ denoted $\Delta_+$. This is a bit unfortunate. On [[augmented simplicial set]] the alternative notation $\Delta_a$ is suggested for the category on which augmented simplicial sets are presheaves. So, one possible solution is to change the notation to this. However, I suspect that augmented simplicial sets are used more commonly than semi-simplicial sets, at least on the nLab, so this might cause more damage. Can someone fix things someday? The reason I bring this up is that I'd like to write a bit about augmented semi-simplicial sets, but right now I can't, due to notational conflicts. <img src = "http://math.ucr.edu/home/baez/emoticons/tongue2.gif" alt = ""/>

   On the article augmented simplicial set, an augmented simplicial set was defined as a presheaf on the full subcategory $\Delta_+$ of $Cat$ consisting of free categories over finite linear directed graphs. This is fairly convoluted, so I’ve added a simpler description of a skeleton of this category. My main point, though, is that this category is denoted $\Delta_+$.

   On the article semi-simplicial set, a semi-simplicial set is defined as a presheaf on the category of finite linearly ordered sets and injective order-preserving maps. This is also denoted $\Delta_+$.

   This is a bit unfortunate. On augmented simplicial set the alternative notation $\Delta_a$ is suggested for the category on which augmented simplicial sets are presheaves. So, one possible solution is to change the notation to this. However, I suspect that augmented simplicial sets are used more commonly than semi-simplicial sets, at least on the nLab, so this might cause more damage. Can someone fix things someday?

   The reason I bring this up is that I’d like to write a bit about augmented semi-simplicial sets, but right now I can’t, due to notational conflicts.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeAug 22nd 2017
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   Author: Mike Shulman
   Format: MarkdownItexI would also incline towards renaming the shape category for semi-simplicial sets. One possible name for this category is $\vec{\Delta}$, since it is the category of "up-going maps" in the [[Reedy category]] structure on $\Delta$. A more verbose name would be $\Delta_{inj}$.

   I would also incline towards renaming the shape category for semi-simplicial sets. One possible name for this category is $\vec{\Delta}$, since it is the category of “up-going maps” in the Reedy category structure on $\Delta$. A more verbose name would be $\Delta_{inj}$.

3. • CommentRowNumber3.
   • CommentAuthorJohn Baez
   • CommentTimeAug 22nd 2017
   • (edited Aug 22nd 2017)
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   Author: John Baez
   Format: MarkdownItexNice idea. But over on [[Reedy category]], the category of "up-going maps" on $\Delta$ is called $\Delta_+$! Do most people use $\vec{\Delta}$? If so, I'll dive in and change that page! (By the way, how do you notate the subcategory of "down-going maps"?)

   Nice idea. But over on Reedy category, the category of “up-going maps” on $\Delta$ is called $\Delta_+$! Do most people use $\vec{\Delta}$? If so, I’ll dive in and change that page!

   (By the way, how do you notate the subcategory of “down-going maps”?)

4. • CommentRowNumber4.
   • CommentAuthorKarol Szumiło
   • CommentTimeAug 22nd 2017
   • (edited Aug 22nd 2017)
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   Author: Karol Szumiło
   Format: MarkdownItexI don't think that there is a truly standard notation, everybody just uses something ad hoc. For what it's worth, I've been using the following notation inspired by the book by Fritsch and Piccinini. Given a simplicial operator $\phi$ they write $\phi = \phi^\sharp \phi^\flat$ for its standard decomposition with $\phi^\sharp$ a face operator and $\phi^\flat$ a degeneracy operator. I think they stop there, but I have adopted this for face/degeneracy operators in any Reedy category $R$ and also started writing $R_\sharp$ and $R_\flat$ for its direct and inverse parts. (It makes sense in that morphisms of $R_\sharp$ move up the Reedy degree and those of $R_\flat$ move down.) Consequently, I denote the semisimplicial indexing category by $\Delta_\sharp$.

   I don’t think that there is a truly standard notation, everybody just uses something ad hoc. For what it’s worth, I’ve been using the following notation inspired by the book by Fritsch and Piccinini. Given a simplicial operator $\phi$ they write $\phi = \phi^\sharp \phi^\flat$ for its standard decomposition with $\phi^\sharp$ a face operator and $\phi^\flat$ a degeneracy operator. I think they stop there, but I have adopted this for face/degeneracy operators in any Reedy category $R$ and also started writing $R_\sharp$ and $R_\flat$ for its direct and inverse parts. (It makes sense in that morphisms of $R_\sharp$ move up the Reedy degree and those of $R_\flat$ move down.) Consequently, I denote the semisimplicial indexing category by $\Delta_\sharp$.

5. • CommentRowNumber5.
   • CommentAuthorJohn Baez
   • CommentTimeAug 22nd 2017
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   Author: John Baez
   Format: MarkdownItexI like that notation, Karol. That would make me happy. But I'd also like to hear Mike's opinion.

   I like that notation, Karol. That would make me happy. But I’d also like to hear Mike’s opinion.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeAug 23rd 2017
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   Author: Mike Shulman
   Format: MarkdownItexI prefer $\overset{\to}{\Delta}$ and $\overset{\leftarrow}{\Delta}$, with $\phi = \overset{\to}{\phi} \overset{\leftarrow}{\phi}$. I think this is rather more common, for instance [Riehl-Verity](https://arxiv.org/abs/1304.6871) use it, as does Hirschhorn in his book on model categories, and I [used it too](https://arxiv.org/abs/1507.01065). I've never seen a musical notation for Reedy categories before.

   I prefer $\overset{\to}{\Delta}$ and $\overset{\leftarrow}{\Delta}$, with $\phi = \overset{\to}{\phi} \overset{\leftarrow}{\phi}$. I think this is rather more common, for instance Riehl-Verity use it, as does Hirschhorn in his book on model categories, and I used it too. I’ve never seen a musical notation for Reedy categories before.

7. • CommentRowNumber7.
   • CommentAuthorHarry Gindi
   • CommentTimeAug 24th 2017
   • (edited Aug 24th 2017)
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   Author: Harry Gindi
   Format: MarkdownItexJacob Lurie uses, if I'm not mistaken, $\Delta^R$ for this subcategory and $\Delta^L$ for the subcategory built from the surjections, which agrees in arrow-direction with Mike's suggestion. Jacob uses that notation generally for Reedy categories, and the arrow notation is pretty much identical in spirit. I believe Joyal also uses the superscript $R$ and $L$ notation for factorization systems. On the other hand, Cisinski uses the $\Delta_+$ and $\Delta_-$ notation, so I don't think there's really any consistency of use between the different writers. I think $\Delta_a$ is good for the augmented simplex category, and then whatever notation gets chosen for the injections should just be made consistent. I'm not in love with the overarrow notation, and I think I prefer the $R$ and $L$ notation best, but I guess whoever is going to go in and do all of the cleanup edits first (not me) has the prerogative to use whatever he wants so long as it is consistent.

   Jacob Lurie uses, if I’m not mistaken, $\Delta^R$ for this subcategory and $\Delta^L$ for the subcategory built from the surjections, which agrees in arrow-direction with Mike’s suggestion. Jacob uses that notation generally for Reedy categories, and the arrow notation is pretty much identical in spirit. I believe Joyal also uses the superscript $R$ and $L$ notation for factorization systems.

   On the other hand, Cisinski uses the $\Delta_+$ and $\Delta_-$ notation, so I don’t think there’s really any consistency of use between the different writers.

   I think $\Delta_a$ is good for the augmented simplex category, and then whatever notation gets chosen for the injections should just be made consistent. I’m not in love with the overarrow notation, and I think I prefer the $R$ and $L$ notation best, but I guess whoever is going to go in and do all of the cleanup edits first (not me) has the prerogative to use whatever he wants so long as it is consistent.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeAug 24th 2017
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   Author: Mike Shulman
   Format: MarkdownItexFor some reason I don't usually think of "R" as referring to something that *goes* to the right rather than being *placed* on the right, so I would have a harder time remembering what $\Delta^R$ means. Actually, since the real intuition is that they go up and down, maybe $\Delta^{\uparrow}$ and $\Delta^{\downarrow}$ would be even better.

   For some reason I don’t usually think of “R” as referring to something that goes to the right rather than being placed on the right, so I would have a harder time remembering what $\Delta^R$ means. Actually, since the real intuition is that they go up and down, maybe $\Delta^{\uparrow}$ and $\Delta^{\downarrow}$ would be even better.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeAug 24th 2017
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   Author: Mike Shulman
   Format: MarkdownItexBy the way, Hovey also uses $+/-$, while the Dwyer-Hirschhorn-Kan-Smith book on model categories also uses $\to/\leftarrow$; that's not a completely independent data point because of Hirschhorn, but it at least means his coauthors didn't have too strong feelings otherwise. A quick glance at Reedy's original note introducing the Reedy model structure doesn't show any notation for these subcategories at all. Are there other references we should consult? A more verbose but possibly more self-evident notation for the augmented simplex category might be $\Delta_{\ge -1}$.

   By the way, Hovey also uses $+/-$, while the Dwyer-Hirschhorn-Kan-Smith book on model categories also uses $\to/\leftarrow$; that’s not a completely independent data point because of Hirschhorn, but it at least means his coauthors didn’t have too strong feelings otherwise. A quick glance at Reedy’s original note introducing the Reedy model structure doesn’t show any notation for these subcategories at all. Are there other references we should consult?

   A more verbose but possibly more self-evident notation for the augmented simplex category might be $\Delta_{\ge -1}$.

10. • CommentRowNumber10.
    • CommentAuthorCharles Rezk
    • CommentTimeAug 24th 2017
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    Author: Charles Rezk
    Format: MarkdownItexAs I remember it, the $\to/\leftarrow$ notation was chosen by Kan (who called them the "direct" and "indirect" subcategories), and was adopted by Hirschhorn. Publication order is not a good guide here: DHKS was already in development during the 90s, and I regularly saw drafts of it (or heard about it from Dan) when I was a graduate student (c. 1993-1996). It had a great deal of influence on both Hirshhorn and Hovey's books. I kind of like the $\sharp/\flat$ notation.

    As I remember it, the $\to/\leftarrow$ notation was chosen by Kan (who called them the “direct” and “indirect” subcategories), and was adopted by Hirschhorn. Publication order is not a good guide here: DHKS was already in development during the 90s, and I regularly saw drafts of it (or heard about it from Dan) when I was a graduate student (c. 1993-1996). It had a great deal of influence on both Hirshhorn and Hovey’s books.

    I kind of like the $\sharp/\flat$ notation.

11. • CommentRowNumber11.
    • CommentAuthorHarry Gindi
    • CommentTimeAug 24th 2017
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    Author: Harry Gindi
    Format: MarkdownItex@Charles I thought Lurie uses $\sharp/\flat$ for all of that marked sSet stuff, which would introduce another level of confusion along the line. @Mike I like the uparrow and downarrow superscript idea a lot, probably the best of all of them. For the augmented simplex category though, I dislike it because the indexing is usually increased by one in the augmented category, for better or for worse.

    @Charles I thought Lurie uses $\sharp/\flat$ for all of that marked sSet stuff, which would introduce another level of confusion along the line.

    @Mike I like the uparrow and downarrow superscript idea a lot, probably the best of all of them. For the augmented simplex category though, I dislike it because the indexing is usually increased by one in the augmented category, for better or for worse.