A Model Structure On The Category Of Small Acyclic Categories

TL;DR

This paper demonstrates that the Thomason model structure on small categories can be restricted to acyclic categories, establishing a cofibrantly generated model structure that is Quillen equivalent and preserves key properties.

Contribution

It introduces a new model structure on acyclic categories derived from the Thomason model structure, showing it is cofibrantly generated, proper, and combinatorial.

Findings

The Thomason model structure restricts to acyclic categories with Quillen equivalence.

The category of acyclic categories is proper and combinatorial under this structure.

The model structure's generating cofibrations are explicitly characterized.

Abstract

In this paper, we show that the Thomason model structure restricts to a Quillen equivalent cofibrantly generated model structure on the category of acyclic categories, whose generating cofibrations are the same as those generating the Thomason model structure. To understand the Thomason model structure, we need to have a closer look at the (barycentric) subdivision endofunctor on the category of simplicial sets. This functor has a well known right adjoint, called Kan's Ex functor. Taking the subdivision twice and then the fundamental category yields a left adjoint of an adjunction between the category of simplicial sets and the category of small categories, whose right adjoint is given by applying the Ex functor twice on the nerve of a category. This adjunction lifts the cofibrantly generated Quillen model structure on simplicial sets to a cofibrantly generated model structure on the category of small categories, the Thomason model structure. The generating sets are given by the image of the generating sets of the Quillen model structure on simplicial sets under the aforementioned adjunction. We furthermore show that the category of acyclic categories is proper and combinatorial with respect to said model structure. That is weak equivalences behave nicely with respect to pushouts along fibrations and cofibrations, and cofibrations satisfy certain smallness conditions which allow us to work with sets instead of proper classes.