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Some tidying up and additions at simplex category, in particular a section on its 2-categorical structure, and more on universal properties.
I’ve edited the definition to focus more on the augmented simplex category $\Delta_a$ instead of the ’topologists’ $\Delta$’, but I haven’t changed their names, because it seemed to me that that was the best way to keep everyone involved in the discussion at that page happy. (I also changed the ordinal sum functor from $+$ to $\oplus$, after Tim’s suggestion.)
The addition of natural numbers extends to a tensor product-type functor on both $\Delta$ and $\Delta_a$. If we visualise an object, $[n]$ of $\Delta_a$, as above, as a totally ordered set $\{0 \lt 1 \lt \cdots \lt n-1\}$, then from two such $[m]$ and $[n]$, we can form a new one by making all the elements of $[n]$ strictly greater than those in $[m]$.
I would add between these two sentences something along the lines of:
which can be seen as the disjoint union (⊎, the coproduct of Set) of the underlying sets with the order augmented so that all of [m] is less than all of [n].
and edit the following text to flow with this.
The link to tensor could conceivably handle this but it doesn’t discuss the basic case of disjoint union which is used here. Anyway, it think it is probably best to state this very simple notion at this point.
I agree, although in my defence I didn’t write that bit. I’ve edited it following your suggestion.
Remember, though, that the best thing to do if you come across something you don’t like about an nLab page is just to go ahead and change it yourself! Just make sure you let us know here what you’ve done.
From About:
… if you feel existing material needs to be changed, you can do so. If you feel further material needs to be added, different perspectives need to be amplified, you can add new paragraph, headed by a suitable headline. Be bold: The $n$Lab will be the better the more people decide to contribute to it.
There was some weird formatting of the definitions right at the beginning of Simplex category - Definition. I have fixed that.
at simplex category in the section on the duality to intervals, I have added a pointer to Joyal’s 1997 preprint. I gather that’s where this duality was first made explicit, is that right?
Urs, my feeling is that it’s been known much longer than that. I mean, I was in the audience when he was first exposing his approach to higher categories via $\Theta$, and I think I myself knew it at the time. But I’d have to hunt down an actual reference, and meanwhile Joyal’s article is a reference. :-)
I changed the wording at the end of the section on duality to intervals (again, although my sense of history is hazy, I’m reasonably sure that this duality had been known for a long time, as a special case of the Stone duality between finite posets and finite distributive lattices; perhaps one should go back to Joyal’s original proof – sometime in the 70’s I think – that the topos of simplicial sets classifies strict intervals, for more on the history of this observation).
I added a wee bit of simple(x) combinatorics, explaining how to count the number of morphisms $\mathbf{n} \to \mathbf{m} \in \Delta_a$.
I added just a bit more, plus a slight rearrangement so that I could better see how Chu-Vandermonde applies. Hope that’s alright.
Edit: Actually, I had to fix it because $\mathbf{m}$ has $m+1$ elements, etc.
While I was at it, I wrote a short article Chu-Vandermonde identity.
(Huh – why does it tell me that the article doesn’t exist? How about here.)
Re: #9, I thought the conventions in the article were that $\mathbf{m} \in \Delta_a$ has $m$ elements and $[m] \in \Delta$ has $m+1$ elements, so if we’re counting morphisms in $\Delta_a$ then the formula and the examples should be bumped back. Otherwise #10 looks nice.
Edit: okay, after a few attempts I’ve bumped all the values to what I think are the correct ones for $\Delta_a$, you can have a look.
Oh, sorry Noam. I’ll take care of it.
Edit: Looks like you got’em all. Thanks, and sorry again.
oh, I noticed that you already gave a couple derivations of the number of morphisms in $\Delta_a$ here on the nForum a few years ago. The second one via the image factorization is I think essentially the same as the one currently in the article, but it seems like a cleaner way of putting it. Perhaps we could import that explanation (and/or the direct “bijective” argument) into the article?
I don’t particularly mind, although I have mixed feelings about the second method. (I’m not convinced it’s any cleaner.) The first method is probably at least worth a remark. Alternatively, we can always link to the nForum discussion (which I assume is “internet-stable”).
I went ahead and added a link.
In some countries. It’s not a great name.
Our for the moment mysterious editor made the previous edit (v71) to this page. I am reverting the change made in that edit to the definition of the augmented simplex category as it would make it over-complicated, but am retaining the content added in the edit as a remark following the definition.
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