arXiv2026
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 $\mathcal{V}$, the canonical map into its external weak actor is an isomorphism if and only if $X$ is unital. Consequently, the ideally exact category $\mathcal{V}_1$ of unital algebras in $\mathcal{V}$ is action representable, and the actor of a unital algebra $X$ is $X$ itself. The action accessibility of $\mathcal{V}$ is not necessary for this conclusion: in characteristic zero, unital Jordan algebras form an action representable category, although the variety of all Jordan algebras is not action accessible. Finally, we prove action representability for the categories of unital Poisson algebras and unital commutative Poisson algebras.