  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.
