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Todd had created subdivision.
I interlinked that with the entry Kan fibrant replacement, where the subdivision $nerve \circ Face$ appears.
The topic has resurfaced in the ’topological stubs’ thread. I made the following comments:
Looking at that page I think it may be worth being a bit more ’expansive’ and putting in a simple example e.g. the subdivision of $\Delta^2$, giving the poset etc. and then showing what the nerve looks like. I would use the term barycentric subdivision as well and briefly about subdivisions in simplicial complexes. (This list is for me to do unless you feel like doing bits of it.) I will try and go through the proof later.
There are other subdivisions as well and these might be mentioned.
and then:
I forgot to mention ordinal subdivision. This is a neat and useful version that is related geometric edgewise subdivision. This was studied by my student Phil Ehlers and a version was written up by me based on his work but extended.
P.J.Ehlers and T. Porter Ordinal subdivision and special pasting in quasicategories, Advances in Math. 217 (2007), No 2. pp 489-518, doi:10.1016/j.aim.2007.05.023 (Preprint available as 05.03 on Bangor preprint server)
NB. Not all subdivisions are functorial and in TQFTs one has to use the more ’archaic’ geometric definition.
Has anyone any comments on this. At present we are saying that subdivision is functorial and that is not the usual meaning of it (see Spanier for instance), but (and this is why I am not just going in an changing things) somehow subdivision should not be dependent on geometric realisation either and that is how it comes across if one goes to the classical form.
This all started because of triangulations and there is the nice result that the nerves of open covers of triangulable spaces are cofinally the same as the simplicial complexes of triangulations (or some statement along those lines). Does anyone see how to give a definition of subdivision so that (i) the classical form is a consequence and (ii) the functorial forms are instances of it? (I don’t know of one but ….)
I think something is wrong with subdivision. I can’t believe that a correct description of the subdivision functor on simplicial sets only needs to know whether one nondegenerate simplex is a face of another one, rather than in what way(s) it is so. Also, taking the nondegenerate simplices of a simplicial set is not a functor, since a simplicial map can take nondegenerate simplices to degenerate ones.
I think you are right. I have not got a source to cross check with. Something along those lines is near the end of the Curtis article which I have somewhere!!!!! I would guess that the ’non-degenerate’ should be removed. Your other comment needs thinking about. The poset of faces of a simplex does not have the problem you mention. If we translate that over to a general simplicial set, (? Kan extension?) then it should work. My thought is that subdividing the models should induce the functor on everything. I do not particularly like the entry’s treatment of barycentric subdivision as I tend towards a more geometric picture, but could live with this one (if it was corrected).
The definition I’ve seen elsewhere (e.g. Goerss-Jardine) goes as you say, by first defining the subdivision of simplices as the nerve of its poset of simplices and face inclusions, then Kan extending to give a cocontinuous endofunctor of SSet.
The reason I came to look at subdivision is that I had another question to ask: what is the relationship between $sd X$, for a simplicial set X, and the opposite of the nerve of its category of simplices? They look very similar to me (though not quite identical).
You say not quite identical which suggests you have some ’easyish’ example in which they diverge.
(Somewhere I had a copy of Dana May Latch and Rudolf Fritsch on homotopy inverse for nerve. It may be worth checking online: One moment please.:-) Here it is Homotopy inverses for nerve Rudolf Fritsch and Dana May Latch Source: Bull. Amer. Math. Soc. (N.S.) Volume 1, Number 1 (1979), 258-262. )
That was well received at the time and they wrote with Bob Thomason shortly afterwards on a related subject.
Well, if X is the terminal simplicial set $\Delta^0$, then its subdivision is itself, whereas its category of simplices is the chaotic category on $\mathbb{N}$.
I like that :-) Nice example.
But of course in that example, the two are equivalent categories. And moreover they become isomorphic (in that example) if, in the category of simplices, we quotient making all degeneracy morphisms into identities. (I think this seemingly violent operation should always produce an equivalent category, since every degeneracy morphism in a category of simplices is an isomorphism between distinct objects. I just said it that way instead of “take the nondegenerate simplices” because that makes it look functorial.)
Are there examples in which that procedure doesn’t make the two isomorphic? In particular, there are presumably simplicial sets whose subdivision is not the nerve of a category, or I would have heard about it, right?
Oops I should know better than that. The category of simplices of $\Delta^0$ is $\Delta$, the simplex category. So maybe they are not that similar after all…
Off the top of my head, etc. I cannot think of one as all the ones I think up are really simplicial complexes and they clearly won’t work! The circle does not work if my scribbles were correct.
It was I who fucked up (excuse the language, but it seems to be appropriate) at subdivision, and I’ve even known this for some time, but hadn’t decided on how I wanted to fix it. The weird thing to me is that the composite which goes from simplicial complexes to posets and then back to simplicial complexes does seem to be subdivision (right?), and I thought at the time I was doing something similar starting at simplicial sets. Please feel free to dive in and fix the effin’ thing.
Don’t be too hard on yourself, Todd! I’m not very familiar with simplicial complexes, but it does look like the composite on that side is subdivision. And your functor “nerve” factors through “Flag” via the inclusion of simplicial complexes into simplicial sets, right?
There must be something slightly subtle here and it may pay to tease it out. The idea that seems to be way back in Kan’s earliest papers and possibly then in Barratt, is to do it on the models and extend. The models are simplicial complexes!! The subdivision functor on sSet is the Kan extension of that so perhaps what you wrote was not THAT far from what is the case. My feeling in handling simplicial sets is that the non-degenerate simplices are the generators (of course) so they tell everything else where to go! But generators are tricky since although somehow atomic they can easily be mapped to non-atomic stuff. Hence this problem.
I suffer from not having access to the old papers (and I mean 1950s) and my copy of Curtis (an original! :-)) is astray somewhere in the 8 piles of box files and assorted papers over in the corner of our dining room behind me.. lot of good it does there but each time I have taken it but I loose it for 18 months, so that is not much better. If someone can find Curtis he has a nice discussion of barycentric subdivision for simplicial sets. I would suggest using a fairly simple starting point and then working up to a description on the line that is there at the moment.
I made an attempt at correcting subdivision; the description for simplicial sets is kind of sketchy but I don’t have time to add details.
Coming back to my question in #5, HTT 4.2.3.15 claims that the nerve of the category of nondegenerate simplices is the same as the barycentric subdivision. This seems intuitively likely. It’s easy to identify the two functors acting on standard simplices, so the obvious thing to do is to show that the “nerve of the category of nondegenerate simplices” is a colimit-preserving functor. But I can’t quite see how to show this; I think I see why the “nerve of the category of simplices” is colimit-preserving (I wrote it out at category of simplices), but why does this extend to the subcategory of nondegenerate simplices? Ideas, anyone?
GRIPE: Looking back at subdivision, there is a case for saying that subdivision is much more general that this. Classically (and sometimes very usefully) one wants to consider constructions that are subdivisions but are not the barycentric one. There is a sop to this in the last paragraph but since subdivision further up the page is generalised barycentric subdivision only, and seems always to be used with ’the’ (the subdivision functor) it is odd to refer to other subdivisions when apparently only the barycentric one is considered legitimate!
Tim, I encourage you to do some rewriting in that case. I think you could add in the Idea section that there are several different kinds of subdivision, and then insert the word ’barycentric’ where the first definition is given (and elsewhere, where appropriate), and then perhaps define other notions of subdivision below it. This would certainly improve the page!
I have made a minor start!
Great! Hope to see more.
Mike, in response to #16, I don’t see HTT 4.2.3.15 mentioning the barycentric subdivision. Anyway, the claim that barycentric subdivision is the same as the nerve of the category of non-degenerate simplices sounds fishy to me. For example take $X$ to be a simplicial set generated by one 2-simplex with one of its 1-faces collapsed. If I’m not mistaken the nerve of the category of non-degenerate simplices of $X$ has 4 non-degenerate 2-simplices while $\mathrm{Sd} X$ has 6 (just like $\mathrm{Sd} \Delta[2]$).
EDIT: A more fundamental problem is that “the nerve of the category of non-degenerate simplices” doesn’t seem to be a functor in any obvious way.
@Karol: a good point. That would be 4.2.3.15 of the most recent arXiv version of HTT that I was looking at, which is apparently now 4 years out of date.
I see, I guess that’s why this remark didn’t make it to the final version.
I don’t think there is such a thing as a “final version” of HTT. At one point, at least, I thought that the arXiv v4 was newer than the physically published version, even though the official publication date is later. But although Lurie has apparently declined to continue updating the arXiv version, he shows no sign of ceasing to update the version on his own web site.
Added a paragraph Properties - Relation to the category of simplices and a pointer to Thomason’s article where this is discussed.
@Guest: Have a look at an example say the unit interval I with vertices 0 and 1 and simplices {0},{1},{0,1}. Now model Sd(I) and compare it with I in the way you have tried to do. What is your obvious inclusion map. You can try mapping vertices in Sd(I) to vertices in I, but where do you send the midpoint of the subdivided interval, as neither of the choices will give an isomorphism on realisation.
Edit: On the other hand, I do think that something is wrong here or at least unclear, as I do not see how you can have a morphism of simplicial complexes as described by the specification of $\alpha$. It is possible to define a map from $|X|$ to $|Sd(X)|$ by specifying it on the vertices and extending ‘linearly’ but not as described. Simplicial complexes are quite subtle!!
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