Right Delocalization of Model Categories

TL;DR

This paper introduces the concept of right Bousfield delocalization in model categories, providing examples, an existence theorem, and showing how delocalization interacts with diagram categories.

Contribution

It defines right Bousfield delocalization, demonstrates how to construct new model structures via intersection of weak equivalences, and explores its preservation in diagram categories.

Findings

Defined right Bousfield delocalization and provided examples.

Proved an existence theorem for right Bousfield delocalization.

Showed delocalization is often preserved in diagram model categories.

Abstract

Model categories have long been a useful tool in homotopy theory, allowing many generalizations of results in topological spaces to other categories. Giving a localization of a model category provides an additional model category structure on the same base category, which alters what objects are being considered equivalent by increasing the class of weak equivalences. In some situations, a model category where the class of weak equivalences is restricted from the original one could be more desirable. In this situation we need the notion of a delocalization. In this paper, right Bousfield delocalization is defined, we provide examples of right Bousfield delocalization as well as an existence theorem. In particular, we show that given two model category structures $\scr{M}_1$ and $\scr{M}_2$ we can define an additional model category structure $\scr{M}_1 \cap \scr{M}_2$ by defining the class of weak equivalences to be the intersection of the $\scr{M}_1$ and $\scr{M}_2$ weak equivalences. In addition we consider the model category on diagram categories over a base category (which is endowed with a model category structure) and show that delocalization is often preserved by the diagram model category structure.