Left and right Bousfield localization on lattices

TL;DR

This paper explores how Bousfield localization impacts model category structures on lattices, providing a combinatorial framework to understand these changes and their implications.

Contribution

It introduces a minimal generating system of morphisms that describe the effect of Bousfield localization on transfer systems in lattice-based model categories.

Findings

Describes the effect of Bousfield localization on transfer systems.

Provides a minimal generating set of morphisms responsible for structural changes.

Offers new insights into model categories on posets.

Abstract

The key information of a model category structure on a poset is encoded in a transfer system, which is a combinatorial gadget, originally introduced to investigate homotopy coherence structures in equivariant homotopy theory. We describe how a transfer system associated with in a model structure on a lattice is affected by left and right Bousfield localization and provide a minimal generating system of morphisms which are responsible for the change in model structure. This leads to new concrete insights into the behavior of model categories on posets in general.