Maths - Category Theory - List and Set Monads

For a more theoretical approach see this page.

List Monad

We can introduce a functor which takes any type and turns it into a list of that type. list
We can draw this as an endo-functor. list

List - Natural Transformations

The unit transform creates a single valued list list unit
The natural way to create a mult transform is a transform which flatterns a list of lists into a flat list. list mult

Lists Form a Free Monoid

list as free monoid Lists form a free monoid, that is, this diagram commutes.

This concept can be extended to other free structures see [R.M. Burstall and P.J. Landin, "Programs and their proofs, an algebraic approach", Machine Intelligence 4(1969)]

Lists as monads

map :: (x -> y) -> (M x -> M y)

This map applies function (x -> y) to every element of M

properties:

where:

map is an example of a functor:

map :: (x -> y) -> (M x -> M y)
map :: f -> M f

That is, we can use M to represent the result of mapping elements and the result of mapping a function of elements. So M is a functor, a higher order function.

As usual we will call the two natural transformations 'unit' and 'join' and so adding these to the map gives:

  [] notation   M notation
map :: (x -> y) -> [x] -> [y]   map :: (x -> y) -> (M x -> M y)
unit :: x -> [x]   unit :: x -> M x
join :: [[ x ]]-> [x]   join :: M ( M x) -> M x

List Monads in Haskell

The class (defines type) of any monad is:

class Monad m where
  return :: a -> m a
  (>>=) :: m a -> (a -> m b) -> m b

The definition depends on what type, in this case the list monad can be defined like this:

instance Monad List where
  return x = [x]
  xs >>= f = concatMap f xs

Comprehensions

[ t | q ]

where:

defined by rules:

where:

Equivalence between comprehensions and monads

Monad Comprehension  
map :: (x -> y) -> (M x -> M y) [ t | x <- u] = map(λ x -> t) u  
unit :: x -> M x [ t | /\ ] = unit t  
join :: M ( M x) -> M x [ t | (p,q) ] = join [[ t | q] | p]  

Sets as monads

We can generalise to set notation

Monad Comprehension  
mapset :: f x = { f x | x∈ x }
mapset :: f x = [ f x | x <- x ]
 
unitset :: x = { x } unitset :: x = [ x ]  
joinset
x
= U
x
joinset
x
= [ x | x <-
x
, x <- x ]
 

where:

U = union
x element of set
x
= set of x
x
= set of set of x

comprehension notation:

[ (x,y) | x <- x , y <- y ]set

 


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see also:

http://www.mathreference.com/

Correspondence about this page

Book Shop - Further reading.

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Terminology and Notation

Specific to this page here:

 

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