Ideally Exact Categories

George Janelidze

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.

Keywords: Ideally exact category, semi-abelian category, Barr exact category, Bourn protomodular category, essentially nullary monad, cartesian monad, ideal

2020 MSC: 18E13, 18C15, 18E08, 08B99

Theory and Applications of Categories, Vol. 41, 2024, No. 11, pp 414-425.

Published 2024-04-05.

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