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.