Menu
Every time someone promises to "explain monads", my interest is piqued, only to be replaced by frustration when the alleged "explanation" is a long list of examples terminated by some off-hand remark that the "mathematical theory" behind the "esoteric ideas" is "too complicated to explain at this point".
Now I'm asking for the opposite. I have a solid grasp on category theory and I'm not afraid of diagram chasing, Yoneda's lemma or derived functors (and indeed on monads and adjunctions in the categorical sense).
Could someone give me a clear and concise definition of what a monad is in functional programming? The fewer examples the better: sometimes one clear concept says more than a hundred timid examples. Haskell would do nicely as a language for demonstration though I'm not picky.
579
In functional programming, a monad is a software design pattern with a structure that combines program fragments (functions) and wraps their return values in a type with additional computation.
So in simple words, a monad is a rule to pass from any type X to another type T(X) , and a rule to pass from two functions f:X->T(Y) and g:Y->T(Z) (that you would like to compose but can't) to a new function h:X->T(Z) .
A monad is an algebraic structure in category theory, and in Haskell it is used to describe computations as sequences of steps, and to handle side effects such as state and IO. Monads are abstract, and they have many useful concrete instances. Monads provide a way to structure a program.
A monad is a certain type of endofunctor. For example, if and are a pair of adjoint functors, with left adjoint to , then the composition is a monad. If and are inverse functors, the corresponding monad is the identity functor. In general, adjunctions are not equivalences—they relate categories of different natures.
This question has some good answers: Monads as adjunctions
More to the point, Derek Elkins' "Calculating Monads with Category Theory" article in TMR #13 should have the sort of constructions you're looking for: http://www.haskell.org/wikiupload/8/85/TMR-Issue13.pdf
Finally, and perhaps this is really the closest to what you're looking for, you can go straight to the source and look at Moggi's seminal papers on the topic from 1988-91: http://www.disi.unige.it/person/MoggiE/publications.html
See in particular "Notions of computation and monads".
My own I'm sure too condensed/imprecise take:
Begin with a category Hask whose objects are Haskell types, and whose morphisms are functions. Functions are also objects in Hask, as are products. So Hask is Cartesian closed. Now introduce an arrow mapping every object in Hask to MHask which is a subset of the objects in Hask. Unit! Next introduce an arrow mapping every arrow on Hask to an arrow on MHask. This gives us map, and makes MHask a covariant endofunctor. Now introduce an arrow mapping every object in MHask which is generated from an object in MHask (via unit) to the object in MHask which generates it. Join! And from the that, MHask is a monad (and a monoidal endofunctor to be more precise).
I'm sure there is a reason why the above is deficient, which is why I'd really direct you, if you're looking for formalism, to the Moggi papers in particular.
75
If you love us? You can donate to us via Paypal or buy me a coffee so we can maintain and grow! Thank you!
Donate Us With
