Homotopy Theory of Non-singular Simplicial Sets

Abstract

A simplicial set is said to be non-singular if its non-degenerate simplices are embedded. Let $sSet$ denote the category of simplicial sets. We prove that the full subcategory $nsSet$ whose objects are the non-singular simplicial sets admits a model structure such that $nsSet$ becomes is Quillen equivalent to $sSet$ equipped with the standard model structure due to Quillen. The model structure on $nsSet$ is right-induced from $sSet$ and it makes $nsSet$ a proper cofibrantly generated model category. Together with Thomason's model structure on small categories (1980) and Raptis' model structure on posets (2010) these form a square-shaped diagram of Quillen equivalent model categories in which the subsquare of right adjoints commutes.