Coherent and ideal actions in ideally exact categories with an application to varieties of universal algebras

The notion of action is pervasive in algebra, as it formalizes how algebraic structures act on and interact with other objects. A central role is played by internal actions, where both the acting and the acted object belong to the same category. For instance, the categorical description of a group action as a functor is equivalent to a group homomorphism from a group G to the group of automorphisms of a group X. In 2005, Borceux, Janelidze and Kelly introduced the notion of internal action in semi-abelian categories: an internal action of an object B on an object X is described as an algebra on X for a specified monad. In this talk we introduce the notions of internal coherent action and internal ideal action, defined in coherent to generalise internal actions to the context of ideally exact categories. We prove that every ideal action is coherent, and we call BAT any ideally exact context with a good theory of actions, i.e., in which all coherent actions are ideal and all morphisms of such actions are ideal. Here, the acronym BAT is inspired by the notion of BIT-variety, where BIT stands for Buona (good, in Italian) Ideal Theory, introduced by A. Ursini. Analogously, BAT stands for Buona Action Theory. Moreover, we show that if a variety U is obtained from a semi-abelian variety V by freely adding nullary operations together with suitable identities, then the ideally exact context determined by the associated free–forgetful monadic adjunction with cartesian unit is BAT if and only if a compatibility condition between coherent actions and the added identities is satisfied. Our setting applies in particular to certain categories of interest in algebraic logic, including MV-algebras and product algebras, and unital non-associative F-algebras. Finally, we study the BAT context of the dual of the category of sets.