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Jonas Frey has raised the question of the notation $[n]$ in the entry for simplex category. I would go along with his choice of notation as it is the one I use myself. (I was surprised to see another convention being used.)
The notation that starts from $[-1] = \emptyset$ is explained at the end of that section. The other notation, the one that starts with $[0]$, is used by Mac Lane in his description of the simplex category in CWM, and I think it makes more sense in the context of the augmented or algebraists’ simplex category. But if you think the passage in question should be rephrased to draw more attention to the topologists’ convention then I certainly won’t object, if you think the change is warranted.
There is no question in the common usage nowdays, one distinguished $[n]$ the linear order with $n$ arrows, $n+1$ objects and by $\mathbf{n}$ the set with $n$ elements, or linear order with $n$ objects. We should follow common usage. Talking “topologist” or “algebraic” for some convention is not fair, as many algebraists do it differently from what such users like to call algebraists and so on. I see nothing topological about a convention and somebody’s social perception should not be part fo a terminology.
[comment removed, I posted accidentally to the wrong thread !]
Regarding ’topologists” vs. ’algebraists”, that is Street’s terminology. I don’t think it’s meant in a taxonomic spirit.
We should follow common usage.
No, clarity is more important than conformity. I think the page is perfectly clear as it stands, and I don’t see how that would be improved by switching to two subtly different notations, especially since the different conventions are already explained there. But if anyone else wants to grub through the page changing $[n]$s to $\mathbf{n}$s, then work away.
But sometimes nonconformity is the sworn enemy of clarity. Everyone working with simplicial sets writes $X_0$ for the set of 0-simplices, which is the image of the object of $\Delta$ with zero (nonidentity) arrows. Therefore, if you want to read anything at all other than the nLab page in question, it is much more confusing to start out calling that object [1] rather than [0].
Well, yes, but the problem is that there are two different categories that want to be called $\Delta$. If you’re mostly talking about unaugmented simplicial sets then one convention suggests itself, but for people (like me) who don’t particularly care about simplicial sets but want to think about the universal property of the augmented category, as well as that of the 2-category it underlies, that convention doesn’t make sense. We need to define one category or the other first, and to me it doesn’t seem to matter which convention we choose as long as that choice, and the fact that there is a choice, are made clear. I think the page does that, but I won’t be offended if anyone disagrees, or if there’s a consensus that the other convention is better enough to justify a switch. (What does get up my nose, however, is preciousness about notation. I get quite enough of that from the computer scientists at work.)
I don’t know the meaning of “preciousness” in this context. If it means “caring about”, then I’m sorry – but notation sometimes does matter!
Can we use non-conflicting notation for the two conventions, as we can for instance with $g\circ f = f;g$ for the two choices about composition? For instance, we could denote by $[1]$ and also by $\langle 2 \rangle$ the object with one non-identity arrow, and similarly $[0] = \langle 1\rangle$, $[-1] = \langle 0 \rangle$, etc. That way anyone can use whatever numbering scheme they prefer, and the notation makes it clear what numbering scheme is in use.
My worry is more of consistency. I have not really got the time, at the moment, to worry about the convention used. I prefer the [n] corresponding to n-arrows as that makes most sense in the nerve, and have a sneaking Ehresmannian liking for arrows as fundamental rather than objects! However I am not sure that we do not have inconsistencies in simplicial notation overall. (I have just had a reminder about an article I agreed to write of which only the ground work has been done as yet … oh dear! so will not be checking for inconsistencies in a consistent way for a bit.)
@Mike I agree with your idea. [1] and $\langle 2\rangle$ is a good convention. What worries me is that i am fairly sure that we have [n] having n-arrows in other pages than this.
if anyone else wants to grub through the page changing [n]s to $\mathbf{n}$s, then work away.
It is less usual, as Mike pointed out to do it in terms of $\mathbf{n}$ and confusing for standard users and for agreement with other pages.
in 9 above, I should have added that [n] is the n-arrow case in most sources on homotopy theory.
As I said, if others think it’s a good idea to change the convention used on the page, then I don’t have the slightest problem with that. I’m sorry if my earlier comments came across as bad-tempered, but I must confess that Jonas Frey’s original comment annoyed me somewhat, because it struck me as unhelpful and more of a dig than a contribution, even though that was probably the furthest thing from his mind.
Mike wrote:
I don’t know the meaning of “preciousness” in this context. If it means “caring about”, then I’m sorry – but notation sometimes does matter!
By ’precious’ I mean caring too much about irrelevant details. I wasn’t accusing you (Mike) of that, rather it was the impression I got from the original comment. Perhaps that was unfair of me. Of course I agree that questions of notation can be important.
Zoran wrote:
if anyone else wants to grub through the page changing $[n]$s to $\mathbf{n}$s, then work away.
It is less usual, as Mike pointed out to do it in terms of $\mathbf{n}$ and confusing for standard users and for agreement with other pages.
My point all along has been that people who choose either convention have good reasons for doing so, and that explaining both of them clearly and carefully is more important than not upsetting homotopy theorists. If we can agree on a not-too-confusing notation to distinguish between the two conventions (and Mike’s suggestion, or something like it, is fine by me), then let’s go ahead and change the page, but let’s also make sure that we don’t obscure an important part of the subject for the sake of conformity.
A general rule of thumb is to present both notations on the page with an explanation of the sources and who uses what under various situations. Then choose one for the page and make it clear what your choice was and explain how your choice relates to what some others may be accustomed to.
Is the specific notation “$[n]$” used in print for the total order with $n$ objects (as opposed to some other notation involving the number $n$)?
Finn, I’m sorry if my comment seemed annoying to you. I realize that I shouldn’t have used the exclamation mark, which made it seem quite harsh I guess. I just thought it is these little things that tend to be quite confusing when you are looking something up at different places, and I hoped this remark would be helpful to later readers of the page. It was not meant as a criticism, I’m sorry if I conveyed this impression.
By the way, it should also be mentioned that the offset between the notation $[n]$ for objects of $\Delta$ and $\mathbf{n+1}$ for objects of $\Delta_a$ can be viewed as a feature, not a bug.
The reason is that $\Delta_a$ can be viewed as a theory (in a sense generalizing Lawvere theories) of monoids: As stated in the nlab page, a monoid is the same as a tensor product preserving functor $M:\Delta_a\to \mathbf{Set}$. Via this identification, $M(\mathbf{n})$ is the set of $n$-tuples of elements of the monoid. Note that when we view a monoid as a category, an $n$-tuple of elements becomes a chain of $n$ composable morphisms.
Now $\Delta$ (or rather $\Delta^{\mathrm{op}}$) can be viewed as a theory of categories: a category is the same thing as a functor $C:\Delta^{\mathrm{op}}\to\mathbf{Set}$ satisfying the Segal condition. Under this identification, $C([n])$ are the strings of $n$ comosable morphisms of the category.
Finally, the fact that monoids are special categories corresponds to the fact that there is an embedding $\Delta_a^{\mathrm{op}}\to\Delta$ which sends $\mathbf{n}$ to $[n]$. Via this embedding, $\Delta_a$ can be identified with the subcategory of $\Delta^{\mathrm{op}}$ of monotone maps which preserve least and greatest elements.
This point of view of $\Delta$ as a theory (i.e., an algebraist’s rather than a topoloigist’s thing) is elaborated in Paul-André Melliès’ Segal condition meets computational effects.
I have edited the entry according to my taste and according to what seems to have become consensus here. Among other things I changed the notation to $\mathbf{n} = [n-1]$ for $\mathbf{n} \in \Delta_a$.
(I did this already yesterday, but then right the minute when I posted the announcement here the night train that I was on started crossing the Alps and my cell-phone internet connection disappeared).
No worries, Jonas. I was probably just being a bit narky. That paper you cite by Melliès is a very nice one. It’d be great if you could add some of what you say to the page itself. (While we’re on the subject, I found your work on the tripos-to-topos construction very interesting – I’m sure it too would be a very welcome addition to the nLab.)
Mike wrote:
Is the specific notation “$[n]$” used in print for the total order with $n$ objects (as opposed to some other notation involving the number $n$)?
I had thought that Mac Lane used it, but now that I check again I see he uses just $n$.
Edit: Urs’s edits look good to me. I’ve changed a few more $[n]$s to $\mathbf{n}$s.
I’ve changed a few more $[n]$s to $\mathbf{n}$s.
Thanks, I was sure I must have missed some, but currently I cannot spend as much time online as I’d need to…
Added references to CWM and Gabriel-Zisman at simplex category. I didn’t read the current thread yet - should there be some remark near the CWM reference stating that there $\Delta$ stands for the augmented simplex category?
Note also that there is a certain inconsistency between simplex category and augmented simplicial set: in the former, the augmented simplex category is denoted by $\Delta_a$, while in the latter it is $\Delta_+$ (in CWM, $\Delta^+$ stands for the non-augmented simplex category…).
CWM is usually optimised for general use and does not always get the best notation or conventions for the more specialist user of a particular topic, although generally it is very good.
The notation $\Delta_+$ for the augmented simplex category is certain to confuse me, because my immediate thought is that the + indicates, as apparently it does for Mac Lane, the set of finite positive ordinals (i.e., excluding the empty ordinal $0$). I would prefer $\Delta_a$ (where the ’a’ can stand for ’augmented’ or ’algebraist’s’).
Consistent cross-entry conventions are hard to establish and unlikely to be adhered to in all future by all contributors, and are therefore dangerous to rely on.
Every entry should declare its notation, as far as reasonable, and add remarks about possibly deviating notation, as far as feasible.
I apologize for resurrecting an old thread about notation, but I wonder how people feel about something other than boldface to denote the ordinal $\mathbf{n} = \{0 \lt \dots \lt n-1\}$? I like that there are two different notations to distinguish $\mathbf{n} = \{0 \lt \dots \lt n-1\}$ from $[n] = \{0 \lt \dots \lt n\}$, because this distinction came up naturally in a couple combinatorics-related articles I worked on earlier (order polynomial and zeta polynomial), and most recently at shuffle. But the boldface notation is a bit awkward (besides being time-consuming to typeset!) because it is difficult to distinguish the coproduct in $\Delta_a$ from the coproduct in Pos. Some alternative notations that would fix this problem are
Any votes or alternative suggestions?
(Of course, the reason this issue came up is because I was trying to be consistent about using the conventions from simplex category in other articles, and now I notice the irony of the last comment #22 in this thread. Still, I guess Urs was warning about relying on cross-entry conventions, and that wasn’t the case here because these other entries are redeclaring the notation, just trying to do it in a consistent way.)
(correction: what I meant to say is that with the boldface notation it is difficult to distinguish the tensor product in $\Delta_a$ from the coproduct in Pos.)
Whatever the convention is chosen for the notation of objects, the entry also needs to adjust the notation for arrows into a consistent convention. As currently defined, the codomains of maps $\sigma^n_i$ and $\delta^n_i$ are both $\mathbf{n}$, yet the string of adjunctions (for instance) shown below involves only superscript $n$, which fails to typecheck as the morphisms alternate between $\mathbf{n-1} \rightarrow \mathbf{n}$ and $\mathbf{n+1} \rightarrow \mathbf{n}$.
(Edit: fixed the misquoted notation in this comment)
You could write $\mathbf{m}\sqcup \mathbf{n}$ for the coproduct in $Pos$. In Emacs auctex, boldface is C-c C-f C-b
which I don’t find too bothersome.
You could write $\mathbf{m}\sqcup \mathbf{n}$ for the coproduct in $Pos$.
That would help for marking it as a coproduct, but the problem is that I also want to refer to the ordinal sum, and the notation $\mathbf{m + n}$ feels ambiguous to me. (It could be typeset as either “\mathbf{m+n}” or “\mathbf{m}+\mathbf{n}”, though the former is intended.)
Personally, I’d be very surprised if someone wrote $\mathbf{m}+\mathbf{n}$ and intended the coproduct of posets. But I see your point.
I suppose, In your situation, I’d probably go with something like $\underline{m}$.
I like to use $\oplus$ for ordinal sum since it is ‘oplus’, but there may be good reasons why not to use it!
I ended up going with the completely explicit notations $[1,n]$ vs $[0,n]$ for my examples, but am leaving simplex category as it is since I suppose there is not much appetite for changing notation, and no pressing need for maintaining cross-entry conventions as per #22.
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