On the representability of actions of unital algebras

Abstract

Working in the setting of ideally exact categories, we investigate the representability of actions of unital non-associative algebras over a field. We show that, in general, such categories fail to be action representable: for instance, the category of all unital algebras is not even action accessible. We then consider this problem in the context of operadic, action accessible, unit-closed varieties. Using the construction of the external weak actor, we prove that for any algebra $X$ in such a variety $\mathsf{V}$, the canonical map into its external weak actor is an isomorphism if and only if $X$ is unital. Consequently, the ideally exact category $\mathsf{V}_1$ of unital algebras in $\mathsf{V}$ is action representable, and the actor of $X$ is $X$ itself. Finally, we prove action representability for unital Poisson algebras via an explicit construction of the universal strict general actor.