I’m with @Rhymoid on this one, I believe all Monads have two (!!) transformers. My construction is a bit different, and far less complete. I’d like to be able to take this sketch into a proof, but I think I’m either missing the skills/intuition and/or it may be quite involved. Due to Kleisli, every monad … Read more
Here’s an even simpler answer: A Monad is something that “computes” when monadic context is collapsed by join :: m (m a) -> m a (recalling that >>= can be defined as x >>= y = join (fmap y x)). This is how Monads carry context through a sequential chain of computations: because at each … Read more
Clearly, >>= is not a way to represent function composition. Function composition is simply done with .. However, I don’t think any of the articles you’ve read meant this, either. What they meant was “upgrading” function composition to work directly with “monadic functions”, i.e. functions of the form a -> m b. The technical term … Read more
That particular phrasing is by James Iry, from his highly entertaining Brief, Incomplete and Mostly Wrong History of Programming Languages, in which he fictionally attributes it to Philip Wadler. The original quote is from Saunders Mac Lane in Categories for the Working Mathematician, one of the foundational texts of Category Theory. Here it is in … Read more