Notes on model structures on preorders

TL;DR

This paper characterizes when preorders can have model category structures with specified cofibrant and fibrant objects, providing classification results and constructions that connect topologies, matroids, and Boolean algebras.

Contribution

It offers necessary and sufficient conditions for model structures on preorders and introduces a construction linking topologies and matroids to Boolean algebra model structures.

Findings

Classified all model structures on small Boolean algebras.

Derived conditions for model structures with given cofibrant and fibrant objects.

Analyzed Bousfield localizations and colocalizations among these structures.

Abstract

Given subsets $\mathcal{C},\mathcal{F}$ of a preorder $\mathcal{A}$, we give necessary and sufficient conditions for $\mathcal{A}$ to admit the structure of a model category whose cofibrant objects are $\mathcal{C}$ and whose fibrant objects are $\mathcal{F}$. We give various classification results for model structures on preorders by describing model structures in terms of their fibrant and cofibrant objects, or in terms of their (co)fibrant replacment (co)monads. This leads to a construction which takes topologies and matroids as input, and produces model structures on Boolean algebras. We carry out some detailed case studies, calculating all model structures on small Boolean algebras, and all the Bousfield localization and colocalization relations between them.