| dc.description.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 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 and Raptis’ model structure on posets these form a square-shaped diagram of Quillen equivalences in which the subsquare of right adjoints commutes.
To establish the model structure referred to above, we first argue that simplicial sets have an analogue of mapping spaces. Namely, let X^K denote the simplicial set whose n-simplices are the simplicial maps Delta^n x K -> X. We prove that X^K is non-singular whenever X is non-singular.
As a related result that is interesting in its own right, we prove that the left Quillen functor of the Quillen equivalence that we establish has an alternate description. The Barratt nerve - denoted B - is the endofunctor that takes a simplicial set to the nerve of the poset of its non-degenerate simplices. The ordered simplicial complex BSd X, namely the Barratt nerve of the Kan subdivision Sd X, is a triangulation of the original simplicial set X in the sense that there is a natural map BSd X -> X whose geometric realization is homotopic to some homeomorphism. This is a refinement to the result that any simplicial set can be triangulated. A simplicial set is said to be regular if each of its non-degenerate simplices is embedded along its n-th face. That BSd X -> X is a triangulation of X is a consequence of the fact that the Kan subdivision makes simplicial sets regular and that B X is a triangulation of X whenever X is regular. In this dissertation, we argue that B - interpreted as a functor from regular to non-singular simplicial sets - is not just any triangulation, but in fact the best. We mean this in the sense that B is the left Kan extension of barycentric subdivision along the Yoneda embedding. Consequently, BSd is indeed the left Quillen functor of a Quillen equivalence - as stated. | |