TL;DR
This paper introduces a unified notion of exactness in category theory that generalizes various structures requiring finite limits and colimits, providing a broad framework for understanding categorical exactness.
Contribution
It proposes a new general concept of exactness applicable to multiple categorical structures, unifying them under a common theoretical framework.
Findings
Defines a new notion of exactness in finitely complete categories
Shows how various categorical structures are special cases
Provides a framework for cocompleteness in the 2-category of finitely complete categories
Abstract
Many kinds of categorical structure require the existence of finite limits, of colimits of some specified type, and of "exactness" conditions between the finite limits and the specified colimits. Some examples are the notions of regular, or Barr-exact, or lextensive, or coherent, or adhesive category. We introduce a general notion of exactness, of which each of the structures listed above, and others besides, are particular instances. The notion can be understood as a form of cocompleteness "in the lex world" -- more precisely, in the 2-category of finitely complete categories and finite-limit preserving functors.
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