This paper provides an introduction to the theory of monads. The main result of the paper is a folkloric proof of Beck’s monadicity theorem which gives an explicit construction of the equivalence involved. Several examples of monads are presented which illustrate the variety of guises in which monads can appear.

For programmers, Monads are a well-known way to represent an abstract data structure, but without knowing its true nature. Admittedly, the definition of Monads in Category Theory is too subtle and obscure so makes people to avoid from trying to understand them. In this paper, we attempt to explore popular Monads in a purely functional programming language, Haskell, with a Category-theoretic min...

Relative monads (Altenkirch, Chapman, and Uustalu 2010) are a recent generalisation of ordinary monads to cover similar structures where the underlying functor need not be an endofunctor. Our interest in this generalisation was triggered by some structures from programming theory that, in many ways, are strikingly similar to monads (even respecting the same laws), have similar programming appli...

We present a coalgebraic approach to trace equivalence semantics based on lifting behaviour endofunctors for deterministic action to Kleisli categories of monads for non-deterministic choice. In Set , this gives a category with ordinary transition systems as objects and with morphisms characterised in terms of a linear notion of bisimulation. The final object in this category is the canonical a...

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras” or “Kleisli monoids” relative to a “monad” on a bicategory. However, the meanings of these wor...

Eklund et al. (2002) present a graphical technique aimed at simplifying the verification of various category-theoretic constructions, notably the composition of monads. In this note we take a different approach involving string rewriting. We show that a given tuple (T,μ, η) is a monad if and only if T is a terminal object in a certain category of strings and rewrite rules, and that this fact ca...

We consider sets of monad rules derived by focussing on the Kleisli category of a monad, and from these we derive some constructions for compound monads. Under certain conditions these constructions correspond to a distributive law connecting the monads. We also show how these relate to some constructions for compound monads described previously.