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I tried to brush-up Warsaw circle and in the process created a bunch of simple stubs:
I liked the reworking of the Warsaw circle. For polyhedron, the term is also used for a space which has a triangulation, i.e. the geometric realisation of a simplicial complex. (cf. Spanier p.113) I think the definition of triangulation is non-standard. I do not know of a special term for a space together with a homeomorphism to the realisation of a simplicial set. The classical notion of triangulation has simplicial complex. (From my point of view, the wikipedia entry is c**p.)
I liked the reworking of the Warsaw circle.
Okay, good.
For polyhedron, the term is also used […]
Please put that into the entry somewhere.
I would suggest, in fact that cubulation be given a separate entry as it is important in its own right, and at present is sitting on the page labelled triangulation. I am not sure (i) how best to do this technically, and (ii) how to handle the comparison at the bottom ot the current entry.
There is a hanging completion wish on the simplicial complex page which is relevant to this! (Possibly from Todd?)
If it’s the query about PL structures, then yes, that was me.
Since I wrote up a lot on that cubulation page, I’ll see if I can get it separated out in a smooth way.
Thanks.
I would love to have more on PL stuff. Any ideas. I have some stuff in later chapters of the Menagerie that provides ideas on PL-microbundles but not on the basic stuff which is needed first. :-(
I have now completed the assembly of cubulation as a separate entry, and did a little reworking of it.
Regarding the standard meaning of triangulation as involving simplicial complexes, not simplicial sets: agreed. I’m not sure of the best way to “fix” this, but I’m inclined not to worry too much since I believe a space is (classically) triangulable if and only if it is homeomorphic to the geometric realization of a simplicial set. (Sketch of proof by Peter May, seen on the categories list recently: the subdivision of a simplicial set is a regular simplicial set, the geometric realization of a regular simplicial set is a regular CW complex, a regular CW complex is triangulable. The other direction is I think easier: the subdivision of a simplicial complex gives an ordered simplicial complex which can be turned into a simplicial set, and the realizations of the subdivision and the simplicial set match.)
Which brings up another discussion: at subdivision I sketched out some cross-relations between simplicial complexes, posets, and simplicial sets, and made a claim that various functors all wind up in the same place in $Top$. But I didn’t properly check that it all works as asserted. Presumably the answer can be sorted out with a close reading of Fritsch and Piccinini, Cellular Structures in Topology, but I don’t have that book here and Google Books doesn’t give me enough of a view to tell.
I think the result is in the old paper by Curtis and he gave a reference, (memory here) possibly to the Moore seminar or Cartan seminars from a few years earlier. It is probably in the book by R. Piccinini and R. Fritsch as you said.
Thanks, Tim. My main worry now is whether the claims at the page subdivision are correct, or if not how to fix them.
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.
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.
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