Ideally exact categories

TL;DR

This paper introduces ideally exact categories as a non-pointed analogue of semi-abelian categories, providing new definitions and showing they support quotient objects via intrinsic ideals, expanding categorical algebra.

Contribution

It defines ideally exact categories, establishes their properties, and connects them to existing structures like semi-abelian categories and cotoposes, introducing the notion of essentially nullary monads.

Findings

Ideally exact categories admit quotient objects via intrinsic ideals.

Bourn protomodularity ensures cartesian monads are essentially nullary.

All semi-abelian categories and cotoposes are examples of ideally exact categories.

Abstract

The purpose of this paper is to initiate a development of a new non-pointed counterpart of semi-abelian categorical algebra. We are making, however, only the first step in it by giving equivalent definitions of what we call ideally exact categories, and showing that these categories admit a description of quotient objects by means of intrinsically defined ideals, in spite of being non-pointed. As a tool we involve a new notion of essentially nullary monad, and show that Bourn protomodularity condition makes cartesian monads essentially nullary. All semi-abelian categories, all non-trivial Bourn protomodular varieties of universal algebras, and all cotoposes are ideally exact.