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Flat model structures for accessible exact categories

Jack Kelly

Abstract

We develop techniques for constructing model structures on chain complexes valued in accessible exact categories, and apply this to show that for a closed symmetric monoidal, locally presentable exact category $\mathpzc{E}$ with exact filtered colimits and enough flat objects, the flat cotorsion pair on $\mathpzc{E}$ induces an exact model structure on $\mathrm{Ch}(\mathpzc{E})$. Further we show that when enriched over $\mathbb{Q}$ such categories furnish convenient settings for homotopical algebra - in particular that they are Homotopical Algebra Contexts, and admit powerful Koszul duality theorems. As an example, we consider categories of sheaves valued in monoidal locally presentable exact categories.

Flat model structures for accessible exact categories

Abstract

We develop techniques for constructing model structures on chain complexes valued in accessible exact categories, and apply this to show that for a closed symmetric monoidal, locally presentable exact category with exact filtered colimits and enough flat objects, the flat cotorsion pair on induces an exact model structure on . Further we show that when enriched over such categories furnish convenient settings for homotopical algebra - in particular that they are Homotopical Algebra Contexts, and admit powerful Koszul duality theorems. As an example, we consider categories of sheaves valued in monoidal locally presentable exact categories.
Paper Structure (64 sections, 128 theorems, 162 equations)

This paper contains 64 sections, 128 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Layout of the Paper
  3. Acknoweldgements
  4. Generalities and Conventions
  5. Notation and Conventions
  6. Conventions for Monoidal Model Categories
  7. Exact Category Generalities
  8. The Left Heart
  9. Monoidal Exact Categories and the $\otimes$-Pure Exact Structure
  10. Smallness Conditions and Exactness of Colimits
  11. Accessible Exact Categories
  12. The General Theory
  13. $\lambda$-Pure Morphisms
  14. The $\lambda$-Pure Exact Structure
  15. Deconstructibility and Structure Theorems
  16. ...and 49 more sections

Key Result

Theorem 1.1

Let $\mathpzc{E}$ be a purely $\lambda$-accessible closed symmetric monoidal exact category with a generator such that Then there is a model structure on both $\mathrm{Ch}(\mathpzc{E})$ and $\mathrm{Ch}_{\ge0}(\mathpzc{E})$ where If $\mathpzc{E}$ is $\lambda$-presentable and $\mathpzc{E}$ is enriched over $\mathbb{Q}$, then with these model structures $\mathrm{Ch}(\mathpzc{E})$ and $\mathrm{Ch}_

Theorems & Definitions (279)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['thm:Kflat']}
  • Lemma 1.3: Lemma \ref{['lem:accdec']}
  • Theorem 1.4: Corollary \ref{['cor:modelstruturealg']}
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 269 more