Coherent and ideal actions in ideally exact categories
Manuel Mancini, Giuseppe Metere, Federica Piazza
TL;DR
This work introduces the notions of coherent and ideal actions within ideally exact categories to unify how units and ideals act internally across algebraic structures. It shows that ideal actions are always coherent and identifies BAT contexts where the converse holds, linking actions to semidirect products through monadic adjunctions and diagrammatic characterizations. The theory is specialized to concrete settings—unit-closed varieties of non-associative algebras, MV-algebras, hoops and product algebras, and a non-varietal Set^op example—where coherence and ideality align and can be described by explicit bottom-element conditions or structural properties. These results provide a coherent framework for internal actions across diverse algebraic categories and clarify when semidirect-product-like constructions behave well in non-pointed, ideally exact contexts.
Abstract
In the context of ideally exact categories, we introduce the notions of internal coherent action and internal ideal action that generalise different aspects of unital actions of rings and algebras. We prove that every ideal action is coherent, and that the converse statement holds in some relevant ideally exact contexts. Furthermore, a connection with G. Janelidze's notion of semidirect product in ideally exact categories is analysed.