Let $\Sigma$ be a~simplicial set with finitely many nondegenerate simplices, homeomorphic to a $d$-sphere.
I would like to construct some ''derived'' simplicial set $\Sigma'$ that is already a simplicial complex, together with a simplicial map $m: \Sigma'\to \Sigma$ that is homotopic to a homeomorphism. Is such construction possible / algorithmic?
My attemps:
I found in Fritsch, Piccinini, Cellular structures in topology that the barycentric subdivision of $\Sigma$ is a regular simplicial set and its geometric realisation is a regular CW complex. This sounds nice, as a regular CW complex can be subdivided into a simplicial complex.
However, I'm not sure how to do it to stay in the realm of simplicial sets: formally, the second barycentric subdivision $Sd^2(\Sigma)$ is not a simplicial complex in general. In this text, page 90ff, I found a claim that $Sd^k(\Sigma)$ is a simplicial complex for some $k$ iff $\Sigma$ satisfies that faces of nondegenerate simplices are always nondegenerate.
I think that I could somehow preprocess degenerate faces and change them to be non-degenerate, and preserve the topology by erecting cones over them, but I'm not sure to which extent this works...
Thanks for possible hints.
Ok, I've found the answer in Proposition 3.5. here.