Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonads, and, dually, initial algebras form both monads and comonads. In developing these theories we s...

We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads pr...

In this paper, we show that the category of directed complete posets with bottom elements (cpos) endowed with an action of a monoid $M$ on them forms a monoidal category. It is also proved that this category is symmetric closed.

Around twenty years ago two important developments happened in the areas of modularity and reuse in programming languages. On the one hand, Moggi showed how computational effects found in impure languages could be simulated using the notion of monads from category theory. Inspired by Moggi’s work, Wadler showed how monads could be used to structure (purely functional) programs. On the other han...

We exhibit sufficient conditions for a monoidal monad T on a monoidal category C to induce a monoidal structure on the Eilenberg–Moore category CT that represents bimorphisms. The category of actions in CT is then shown to be monadic over the base category C.

For a small quantaloid Q we consider four fundamental 2-monads T on Q-Cat, given by the presheaf 2-monad P and the copresheaf 2-monad P†, as well as their two composite 2-monads, and establish that they all laxly distribute over P. These four 2-monads therefore admit lax extensions to the category Q-Dist of Q-categories and their distributors. We characterize the corresponding (T,Q)-categories ...

This paper introduces guarded and strongly guarded monads as a unified model of a variety of different term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary guarded monads which allows us to define and prove properties of the rational and term graph monads. Furthermore, our treat...

We exhibit sufficient conditions for a monoidal monad T on a monoidal category C to induce a monoidal structure on the Eilenberg–Moore category CT that represents bimorphisms. The category of actions in CT is then shown to be monadic over the base category C.