Let A be a ring and MA the category of right A-modules. It is well known in module theory that any A-bimodule B is an A-ring if and only if the functor − ⊗A B : MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor − ⊗A C : MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of −⊗A B and comodules (or coalgebras) of − ⊗A C...

This paper introduces coalgebraic monads as a unified model of term algebras covering fundamental examples such as initial algebras, final coalgebras, rational terms and term graphs. We develop a general method for obtaining finitary coalgebraic monads which allows us to generalise the notion of rational term and term graph to categories other than Set. As an application we sketch part of the c...

We consider the equivalence of Lawvere theories and finitary monads on Set from the perspective of Endf (Set)-enriched category theory, where Endf (Set) is the category of finitary endofunctors of Set. We identify finitary monads with one-object Endf (Set)-categories, and ordinary categories admitting finite powers (i.e., n-fold products of each object with itself) with Endf (Set)-categories ad...

Motivated by a desire to gain a better understanding of the “dimensionby-dimension” decompositions of certain prominent monads in higher category theory, we investigate descent theory for endofunctors and monads. After setting up a basic framework of indexed monoidal categories, we describe a suitable subcategory of Cat over which we can view the assignment C 7→ Mnd(C) as an indexed category; o...

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...

The formal theory of monads, originally introduced by Street in [22] and developed further by Lack and Street [17] provides a mathematically efficient treatment of several aspects of the theory of monads [1]. For example, it exhibits a universal property of the category of algebras for a monad and provides a clear explanation for Beck’s axioms for a distributive law [2]. Over the past few years...

For a generalisation of the classical theory of Hopf algebra over fields, A. Bruguières and A. Virelizier study opmonoidal monads on monoidal categories (which they called bimonads). In a recent joint paper with S. Lack the same authors define the notion of a pre-Hopf monad by requiring only a special form of the fusion operator to be invertible. In previous papers it was observed by the presen...

Arrows are an extension of the well-established notion of a monad in functional programming languages. This article presents several examples and constructions, and develops denotational semantics of arrows as monoids in categories of bifunctors C×C→ C. Observing similarities to monads – which are monoids in categories of endofunctors C→ C – it then considers Eilenberg-Moore and Kleisli constru...