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Proto-exact categories and injective Banach modules

Jack Kelly

Abstract

We develop the basic theory of covers and envelopes in proto-exact categories. As an application, we prove the existence of enough injectives for categories of Banach modules over arbitrary Banach rings.

Proto-exact categories and injective Banach modules

Abstract

We develop the basic theory of covers and envelopes in proto-exact categories. As an application, we prove the existence of enough injectives for categories of Banach modules over arbitrary Banach rings.
Paper Structure (29 sections, 62 theorems, 62 equations)

This paper contains 29 sections, 62 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Proto-exact and parabelian categories
  3. The definition
  4. Some useful facts
  5. Exact functors
  6. Exact subcategories
  7. The obscure axiom
  8. Split morphisms
  9. Epimorphic and admissible generators
  10. Proto-exact structures on monad algebras
  11. Cotorsion and enveloping theory
  12. Projectives and injectives.
  13. General cotorsion theory
  14. Coherence and deconstructibility
  15. Deconstructibility
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $R$ be a Banach ring. Then the quasi-abelian category $\mathrm{Ban}_{R}$ of Banach $R$-modules has enough injectives.

Theorems & Definitions (155)

  • Theorem 1.1: Theorem \ref{['thm:injban']}
  • Proposition 2.1
  • Definition 2.2: MR3970975 Definition 2.4.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 145 more