The Bousfield localizations and colocalizations of the discrete model structure

TL;DR

This paper investigates the Bousfield localizations and colocalizations of discrete model categories, analyzing their homotopy categories, algebraic K-groups, and conditions for subcategories and monads to serve as fibrant object structures.

Contribution

It provides necessary and sufficient conditions for subcategories and monads to be realized as fibrant objects in these localized model structures.

Findings

Characterization of subcategories as fibrant object categories

Conditions for monads to be fibrant replacement monads

Analysis of algebraic K-groups in localizations and colocalizations

Abstract

We compute the Bousfield localizations and Bousfield colocalizations of discrete model categories, including the homotopy categories and the algebraic $K$-groups of these localizations and colocalizations. We prove necessary and sufficient conditions for a subcategory of a category to appear as the subcategory of fibrant objects for some such model structure. We also prove necessary and sufficient conditions for a monad to be the fibrant replacement monad of some such model structure.