
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 -duality between the -category of small exact categories and the -category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the -category of regular toposes and show that the diagram of -(anti-)equivalences between three -categories commutes, the corresponding additive triangle is well-known.
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