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Suppose $K$ and $L$ are simplicial sets. When should one consider that $K$ is a subdivision of $L$? I ask with a view to defining some notion of ‘finer’ generalising that of ’finer triangulation‘ of a polyhedron.

If both the simplicial sets are ‘polyhedral’, then they will essentially be given by simplicial complexes together with an ordering of the set of vertices, and in that case the more geometric definition of subdivision can be applied, although it would be nicer if that definition did not rely on first taking the geometric realisation and could be handled just with the abstract simplicial complex formulation. Can this be extended in some cunning way to handle all simplicial sets?

Any simplicial set has a barycentric subdivision, but that subdivides everything in sight. That is not what I am looking for and is well known. Similarly for the ordinal subdivision that I explored years ago with Phil Ehlers. What I want is an idea of subdivision that might take, say, a single 1-simplex in $L$, replace it by a subdivided one and then generate up to higher dimensions, or add a new vertex as if it was in some specified 2-simplex and then to form a star subdivided version of that 2-simplex without altering other simplices unnecessarily.

Has anyone seen such a construction? I am looking for references, or an idea on what tools might give such an idea.

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2 Answers 2

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Of course, classically, a subdivision of a simplicial complex $K$ is defined to be a simplicial complex $L$ such that each simplex of $L$ is contained in a simplex of $K$ and each simplex of $K$ is the union of finitely many simplices of $L$, and we can ask for a functorial version. I don't know of a published analog for simplicial sets. It might be something like a functor $\operatorname{Sd}$ on the category of simplicial sets (maybe required to be induced from a functor from the simplicial category $\Delta$ to itself) together with a natural homeomorphism $\lvert\operatorname{Sd} X\rvert \to \lvert X\rvert$. Certainly there are known examples. Segal's paper “Configuration spaces and iterated loop spaces” introduces edgewise subdivision, and the first section of the paper “The cyclotomic trace and algebraic K-theory of spaces” by Bökstedt, Hsiang, and Madsen defines and exploits a variant of Segal's construction. Tim, I leave it to you to see whether or not that suits your needs.

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  • $\begingroup$ Thanks, Peter. For once I am not looking for something functorial! I also want, if possible, to avoid using geometric realisation. In the case of simplicial complexes (as used in Geometric Topology, for example) one may have that two such agree except near some subcomplex, where the simplexes in one are related to those in the other in the way you describe. I am looking for a characterisation of such contexts which would extend to the simplicial set case. I knew of Segal's paper and have used it. $\endgroup$
    – Tim Porter
    Feb 11, 2019 at 6:35
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For simplicity, let me talk about simplicial complexes. There, I think the subdivision you are looking for replaces a face $\sigma$ of your complex with $v_\sigma * \partial \sigma * \mathop{\mathrm{link} }\sigma$, where $v_\sigma$ is a new vertex, $\partial$ is the boundary of $\sigma$ and the link is $\{\tau \setminus \sigma : \sigma \subseteq \tau \}$. The asterisk of course denotes the join of simplicial complexes.

This operation is sometimes called stellar subdivision. A textbook source is Kozlov's Combinatorial Algebraic Topology, or Ewald's Combinatorial Convexity and Algebraic Geometry also covers the operation. Kozlov works with simplicial complexes, Ewald with cell complexes; it's likely that there's a source somewhere that works with simplicial sets (but I don't know it).

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  • $\begingroup$ Although I agree that that is a good subdivision it does not quite answer my question. The point is rather to have two simplicial sets, $K$ and $L$ and we want to say that $K$ is a subdivision of $L$. Perhaps one might say that we need extra data, e.g. a monomorphism from $L_0$ to $K_0$ satisfying some conditions. You might be able to dream up some such conditions by iterating your construction, and I agree that the join operation should be in their somewhere but exactly how is not 100% clear. $\endgroup$
    – Tim Porter
    Aug 6, 2021 at 6:01
  • $\begingroup$ Again your construction may help (I did know of that construction as it is clssical for simplicial complexes.) even for general simplicial sets, but would need to be restricted to the non-degenrate simplices, Thanks as it reminded me of things that needed to be recalled (by me)! $\endgroup$
    – Tim Porter
    Aug 6, 2021 at 6:02
  • $\begingroup$ Are you looking to detect whether K is obtained from L by repeated subdivisions computationally? Or in what sense do you want to detect subdivisions? Another term that might be along the lines of what you are looking for is "bistellar flip", although this is not in general straightforwardly a subdivision. But I'll mention that Frank Lutz and others have done computational work on bistellar flips for manifolds. $\endgroup$ Aug 6, 2021 at 19:04
  • $\begingroup$ My original question was simply along the lines of 'What should be the general definition / meaning of `$K$ is a subdivision of $L$' and from there to get to a notion of 'finer' subdivision, and (not in the question) to be able to work with the system of all finer 'triangulations' of a given simplicial set. (I needed this, at the time, for working with triangulated cobordisms in Topological Quantum Field Theory, but the question was independent of the constraints of that context.) $\endgroup$
    – Tim Porter
    Aug 7, 2021 at 5:20
  • $\begingroup$ I'd generally interpret being a subdivision as meaning "arising by repeated stellar subdivision." For example, the barycentric subdivision arises by iteratively subdividing all faces in a linear extension of reverse inclusion. OTOH, Stanley in Combinatorics and commutative algebra has an alternative that you might like more: he defines a subdivision of simplicial complex \Delta to be a complex \Delta' s.t. each face of \Delta' is contained in a face of \Delta, where his inclusion comes from a geometric realization. Should be easy to abstract out usefully, I think. $\endgroup$ Aug 9, 2021 at 19:12

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