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Nice exact categories are coexact

James Richard Andrew Gray

Abstract

Several important types of categories have been shown to be both exact and coexact (in the sense of Barr). The first type consists of abelian categories, which due to their self-dual definition, can be seen to be both exact and coexact by Tierney's characterization of them as additive exact categories. The next type consists of elementary toposes which are well-known to be exact, but have also been shown to be coexact and coprotomodular by Bourn. In this paper we study a condition weaker than extensivity and equivalent to additivity for pointed categories. We show that for a finitely cocomplete category this condition together with exactness implies coexactness and coprotomodularity. As a special case we obtain that a finitely cocomplete pretopos is coexact.

Nice exact categories are coexact

Abstract

Several important types of categories have been shown to be both exact and coexact (in the sense of Barr). The first type consists of abelian categories, which due to their self-dual definition, can be seen to be both exact and coexact by Tierney's characterization of them as additive exact categories. The next type consists of elementary toposes which are well-known to be exact, but have also been shown to be coexact and coprotomodular by Bourn. In this paper we study a condition weaker than extensivity and equivalent to additivity for pointed categories. We show that for a finitely cocomplete category this condition together with exactness implies coexactness and coprotomodularity. As a special case we obtain that a finitely cocomplete pretopos is coexact.

Paper Structure

This paper contains 4 sections, 24 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lextensive categories
  3. Additive categories
  4. Condition \ref{['conditions']} in the exact context

Key Result

Lemma 2.1

For each morphism $f:A \to B$ in a lextensive category, the diagram is a pullback.

Figures (1)

  • Figure 1: Construction of biproducts

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 31 more