There are different notions of computation, the most popular being monads, applicative functors, and arrows. In this article we show that these three notions can be seen as monoids in a monoidal category. We demonstrate that at this level of abstract
ion one can obtain useful results which can be instantiated to the different notions of computation. In particular, we show how free constructions and Cayley representations for monoids translate into useful constructions for monads, applicative functors, and arrows. Moreover, the uniform presentation of all three notions helps in the analysis of the relation between them.
Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of second-order fu
nctionals which are polymorphic over a class of functors. Types polymorphic over a class of functors are easily representable in languages such as Haskell, but are difficult to analyse and reason about. The concrete representation provided by the theorem is easier to analyse, but it might not be as convenient to implement. Therefore, depending on the task at hand, the change of representation may prove valuable in one direction or the other. We showcase the usefulness of the representation theorem with a range of examples. Concretely, we show how the representation theorem can be used to show that traversable functors are finitary containers, how parameterised coalgebras relate to very well-behaved lenses, and how algebraic effects might be implemented in a functional language.
