Definable Categories

TL;DR

This paper introduces the concept of definable categories, explores their properties, and establishes a 2-duality with small exact categories, providing new insights into their structural relationships.

Contribution

It defines definable categories, characterizes definable subcategories as finite-injectivity classes, and establishes a 2-duality with small exact categories, including a new proof of the additive case.

Findings

Definable categories are equivalent to certain subcategories of locally finitely presentable categories.

A 2-duality exists between small exact categories and definable categories.

The paper introduces a third vertex involving regular toposes and shows the commutativity of a diagram of 2-equivalences.

Abstract

We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are precisely the finite-injectivity classes. We prove a $2$-duality between the $2$-category of small exact categories and the $2$-category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the $2$-category of regular toposes and show that the diagram of $2$-(anti-)equivalences between three $2$-categories commutes, the corresponding additive triangle is well-known.