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Stream: theory: mathematics

Topic: Categorical understanding of subdivision

Brendan Murphy (Mar 27 2024 at 18:28):

Let $\mathsf{Poly}$ denote the category of polytopes (defined as the convex hulls of finite collections of points) embedded in countable dimensional Euclidean space $\R^{\infty} = \mathrm{colim}_n \R^n$ with affine maps between them. Recall that a polytopal complex is a finite collection of polytopes closed under taking faces and where the pairwise intersection of any two polytopes in the collection is a face of each. I believe we can understand these as certain objects in $\operatorname{Psh}(\mathsf{Poly})$. Any polytope has an underlying subset of $\R^{\infty}$ (by definition) and any polytopal complex does as well, just by taking the union of the underlying sets of each polytope in it. If $P$ is a polytope then a subdivision of $P$ is a polytopal complex $K$ such that $|K| = |P|$. Is there a more categorical definition of this property in terms of $\mathsf{Poly}$ and its presheaf category? The awkward thing is that there's no obvious map from $K$ to $P$ or vice-versa which exhibits $K$ as a subdivision of $P$, or at least I don't think so.

Morgan Rogers (he/him) (Mar 28 2024 at 10:13):

I remember seeing a talk by Vincenzo Marra about sheaves on a site of polygons (or possibly polyhedra) which seems vaguely relevant but I couldn't find a record of it or a paper where he discussed it; maybe it was a seminar I attended in Milano at some point.

Spencer Breiner (Mar 28 2024 at 22:30):

Not particularly on topic, but the barycentric subdivision of a simplicial set can be viewed as a complex built from flags in the original complex.

Brendan Murphy (Mar 28 2024 at 22:37):

Right, there's some very nice combinatorial/categorical constructions of specific subdivisions. But it doesn't seem like the concept of "a subdivision" can be easily framed categorically

Amar Hadzihasanovic (Mar 29 2024 at 06:21):

I think that if "homeomorphism of realisations" is part of your definition of subdivision, then it is hopeless to make it categorical, since these homeomorphisms seem to me almost never canonical/natural.

Even in the special case of barycentric subdivision, you do not get a canonical isomorphism between a simplicial set and its subdivision. The best you get is the natural "last vertex" map from the subdivision to the original simplicial set, which is a weak equivalence but not a homeomorphism.

Amar Hadzihasanovic (Mar 29 2024 at 06:28):

I have some thoughts on subdivision of abstract polytopes — I think you can fruitfully model subdivisions as (dual to) certain maps between (abstract) faces, such that the fibre of a face is the set of faces in the interior of its subdivision.

But such maps are not "tracked" by any map of embedded convex polytopes; you really need to change the point of view.

Amar Hadzihasanovic (Mar 29 2024 at 06:30):

The "homeomorphic realisation" is then not a defining property, but a theorem about the order complexes of two abstract polytopes related by a subdivision map.

Amar Hadzihasanovic (Mar 29 2024 at 06:35):

(Note that I "have done" this — in work coming very soon — not for polytopes but for "directed polytopes" of the kind that show up in the theory of n-categorical pasting diagrams; so I do not know whether, and how much of this can be generalised to the "undirected" setting — I only think that some of it can be transported, but haven't actually tried!)

Brendan Murphy (Mar 30 2024 at 03:31):

I'd love to see the paper about subdivision of abstract polytopes and the duality you mention!

Brendan Murphy (Mar 30 2024 at 03:33):

That said I don't find the reasoning you lay out very convincing for it being "hopeless". Notably arbitrary homeomorphism are not allowed in the category I mentioned, only affine ones, and the definition I gave was actually in terms of equality of realizations and not isomorphism

Brendan Murphy (Mar 31 2024 at 00:24):

Morgan Rogers (he/him) said:

I remember seeing a talk by Vincenzo Marra about sheaves on a site of polygons (or possibly polyhedra) which seems vaguely relevant but I couldn't find a record of it or a paper where he discussed it; maybe it was a seminar I attended in Milano at some point.

I found some relevant slides ("A topos for piecewise-linear geometry, and its logic") and it's very interesting! There's a topology on the category of polytopes I defined above where covering families are generated by jointly surjective finite families of injective affine maps. This isn't subcanonical, and the sheafified representables contained PL maps instead of affine ones. If we enlarge to the category of polytopal complexes and PL maps we do get a subcanonical topology

Brendan Murphy (Mar 31 2024 at 00:25):

So there's a rough correspondence between the notion of subdivision, the notion of a PL map, and this topology

Brendan Murphy (Mar 31 2024 at 00:28):

(the slides were linked in the paper "Model categories for o-minimal geometry" by @Reid Barton and @Johan Commelin which is a really wonderful take on tame topology)

Morgan Rogers (he/him) (Mar 31 2024 at 13:35):

Hooray, I'm glad you managed to find them and that they were relevant!