On (Enriched) Left Bousfield Localization of Model Categories

TL;DR

This paper establishes the existence and properties of enriched left Bousfield localizations in model categories, enabling the construction of new models such as homotopy limits, Postnikov towers, and presheaves with descent conditions.

Contribution

It proves the existence of enriched left Bousfield localizations and characterizes their fibrations, leading to new model categories for various homotopical constructions.

Findings

Existence of enriched left Bousfield localizations verified.

Characterization of fibrations in localized model categories.

Construction of models for homotopy limits, Postnikov towers, and presheaves.

Abstract

I verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and I prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. I also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition.