Subdivision of simplicial sets, but not the barycentric one

Question

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.

Answer 1

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.

Answer 2

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).