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Left Bousfield localization without left properness

David White, Michael Batanin

Abstract

Given a combinatorial (semi-)model category $M$ and a set of morphisms $C$, we establish the existence of a semi-model category $L_C M$ satisfying the universal property of the left Bousfield localization in the category of semi-model categories. Our main tool is a semi-model categorical version of a result of Jeff Smith, that appears to be of independent interest. Our main result allows for the localization of model categories that fail to be left proper. We give numerous examples and applications, related to the Baez-Dolan stabilization hypothesis, localizations of algebras over operads, chromatic homotopy theory, parameterized spectra, $C^*$-algebras, enriched categories, dg-categories, functor calculus, and Voevodsky's work on radditive functors.

Left Bousfield localization without left properness

Abstract

Given a combinatorial (semi-)model category and a set of morphisms , we establish the existence of a semi-model category satisfying the universal property of the left Bousfield localization in the category of semi-model categories. Our main tool is a semi-model categorical version of a result of Jeff Smith, that appears to be of independent interest. Our main result allows for the localization of model categories that fail to be left proper. We give numerous examples and applications, related to the Baez-Dolan stabilization hypothesis, localizations of algebras over operads, chromatic homotopy theory, parameterized spectra, -algebras, enriched categories, dg-categories, functor calculus, and Voevodsky's work on radditive functors.

Paper Structure

This paper contains 12 sections, 13 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Smith's theorem for locally presentable semi-model categories
  4. Left Bousfield localization for locally presentable semi-model categories
  5. Applications
  6. Radditive functors
  7. Inverting Unary Operations
  8. Localizing categories of algebras over colored operads
  9. Stabilization and Spectra
  10. Enriched categories
  11. Functor Calculus
  12. Other possible directions

Key Result

Theorem A

Suppose that $\mathcal{M}$ is a combinatorial semi-model category whose generating cofibrations have cofibrant domains, and $\mathcal{C}$ is a set of morphisms of $\mathcal{M}$. Then there is a semi-model structure $L_{\mathcal{C}}(\mathcal{M})$ on $\mathcal{M}$, whose weak equivalences are the $\ma

Theorems & Definitions (44)

  • Theorem A
  • Theorem B
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 34 more