In denotational semantics and functional programming, the terms monad morphism, monad layering, monad constructor, and monad transformer have by now accumulated 20 years of twisted history. The exchange between Eric Kidd and sigfpe about the probability monad prompted me to investigate this history
Generalising Monoids The word 'monad' is derived from the word 'monoid'. The explanation usually given is that there is an analogy between monoids and monads. On the surface, this seems a bit unlikely. The join operation in a monad is supposed to correspond to the binary operator in the monoid, but join is a completely different kind of thing, certainly not a binary operator in any usual sense. I'm going to make this analogy precise so that it's clear that both monoids and monads are examples of the same construction. In fact, I'm going to write some Haskell code to define monoids and monads in almost exactly the same way. I was surprised to find I could do this because instances of Haskell's Monoid and Monad aren't even the same kind of thing (where I'm using 'kind' in its technical sense). But it can be done.
Suppose someone stole all the monads but one, which monad would you want it to be? If you're a Haskell programmer you wouldn't be too bothered, you could just roll your own monads using nothing more than functions. But suppose someone stole do-notation leaving you with a version that only supported one type of monad. Which one would you choose? Rolling your own Haskell syntax is hard so you really want to choose wisely. Is there a universal monad that encompasses the functionality of all other monads? About a year ago I must have skimmed this post because the line "the continuation monad is in some sense the mother of all monads" became stuck in my head. So maybe Cont is the monad we should choose. This post is my investigation of why exactly it's the best choice. Along the way I'll also try to give some insight into how you can make practical use the continuation monad.
S. Goncharov, L. Schröder, и T. Mossakowski. Mathematical Foundations of Computer Science, том 4162 из Lecture Notes in Computer Science, стр. 447-458. Springer; Berlin; http://www.springer.de, (2006)
L. Schröder, и T. Mossakowski. Fundamental Approaches to Software Engineering (FASE 2003), том 2621 из Lecture Notes in Computer Science, стр. 261--277. Springer; Berlin; http://www.springer.de, (2003)
D. Walter, L. Schröder, и T. Mossakowski. Algebra and Coalgebra in Computer Science, том 3629 из Lecture Notes in Computer Science, стр. 424-438. Springer; Berlin; http://www.springer.de, (2005)