Talk:Monads/Maybe monad: Difference between revisions
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::::More concretely, defining Just and Nothing, Left and Right, or Lists, for example, gives us a type but does not give us a monad.
::::The monad instance for a type consists of well-formed monadic functions (bind, and return/pure) which enable monadic enchaining of computations *embedded in that type*. List is not a monad, nor are the Maybe or Either types in themselves. The List, Maybe and Either monads consists of the bind and return function instances written for each of those types. [[User:Hout|Hout]] ([[User talk:Hout|talk]]) 23:02, 2 October 2019 (UTC)
===Some notes===
A monad consists of a pair of functions (specialised for a particular data-type or other computational context), which simplify the pipelining of computations which are wrapped in that context.
The two functions are sometimes called return and bind, or pure and bind, or '''η''' and '''μ'''.
* pure/return simply wraps a raw value in a context (for example, wraps a number in a list).
* bind takes two arguments: (1) a wrapped or context-embedded value, and (2) a function which applies to a raw value but returns a wrapped value.
When applied, bind takes care of the mechanics of:
# Extracting the raw value from the enclosing context,
# Applying the supplied function to it, and returning the wrapped output value.
To use the metaphor of postal correspondence, pure / return places the letter in an envelope.
Bind assists the writer by discarding the envelope, and offering its contents to the reader for perusal and digestion.
(The writer (or supplied write-and-wrap function) drafts a response, and places it in a new envelope, continuing the chain of correspondence)
The data-type or context (the envelope or functor) is not itself the monad. The monad consists of the functor-specific pair of functions which abstract away the boiler-plate required for wrapping and unwrapping, (extraction from a context, and embedding in that context), to facilitate a chain of computations that are embedded in some enclosing context.
A little more formally [https://en.wikipedia.org/wiki/Monad_(category_theory)#Formal_definition] (see definition 1.5 in Moggi, which refers in turn to MacLane) A monad over a category C is a triple (T, '''η''', '''μ''') where T:C -> C is the functor (for programming purposes, the context of the computation), '''η''' is the function which simply 'lifts' a raw value into that context (places a letter in an envelope), and '''μ''' is the function from the incoming envelope containing one message, to the outgoing envelope, which contains the new derived (responding) message. [[User:Hout|Hout]] ([[User talk:Hout|talk]]) 15:12, 3 October 2019 (UTC)
===Foundational references===
S. MacLane. Categories for the Working Mathematician. Springer Verlag, 1971.
[https://core.ac.uk/download/pdf/21173011.pdf | Moggi 1991, Notions of computation and monads]
[https://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf | Wadler 1992, Monads for functional programming]
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