Classes of semigroup actions from the point of view of semimonads

When considering semigroup actions, there are several subclasses of actions that pop up when trying to imitate the theory of monoid actions. For example, we can consider the category UAct of unitary actions and the category FAct of firm actions, but we can also consider the category CoUAct of counitary actions and the category CoFAct of cofirm actions (usually denoted CAct).

We will give an introduction to monads and semimonads and generalize the notions of firmness and unitarity to the setting of algebras over a semimonad.

One motivation behind our generalization is that it allows us to make explicit how unitarity is dual to counitarity and how firmness is dual to cofirmness. This allows us to, for example, prove things about unitary algebras, and from that get for free results about counitary algebras.