(i') ((join G') (M γ) φ) etc are also in T
+
-
-
+
(ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
==>
- (ρ <=< γ) is a transformation from G to MR', so
- (ρ <=< γ) <=< φ becomes: ((join R') (M (ρ <=< γ)) φ)
+ (ρ <=< γ) is a transformation from G to MR', so
+ (ρ <=< γ) <=< φ becomes: ((join R') (M (ρ <=< γ)) φ)
which is: ((join R') (M ((join R') (M ρ) γ)) φ)
- similarly, ρ <=< (γ <=< φ) is:
+ similarly, ρ <=< (γ <=< φ) is:
((join R') (M ρ) ((join G') (M γ) φ))
- substituting these into (ii), and helping ourselves to associativity on the rhs, we get:
- ((join R') (M ((join R') (M ρ) γ)) φ) = ((join R') (M ρ) (join G') (M γ) φ)
+ substituting these into (ii), and helping ourselves to associativity on the rhs, we get:
+ ((join R') (M ((join R') (M ρ) γ)) φ) = ((join R') (M ρ) (join G') (M γ) φ)
- which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields:
- ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (M ρ) (join G') (M γ) φ)
+ which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields:
+ ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (M ρ) (join G') (M γ) φ)
- which by lemma 1, with ρ a transformation from G' to MR', yields:
- ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (join MR') (MM ρ) (M γ) φ)
-
- which will be true for all ρ,γ,φ just in case:
- ((join R') (M join R')) = ((join R') (join MR')), for any R'.
+ which by lemma 1, with ρ a transformation from G' to MR', yields:
+ ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (join MR') (MM ρ) (M γ) φ)
- which will in turn be true just in case:
-
- (ii') (join (M join)) = (join (join M))
+ [-- Are the next two steps too cavalier? --]
+ which will be true for all ρ, γ, φ only when:
+ ((join R') (M join R')) = ((join R') (join MR')), for any R'
+ which will in turn be true when:
+ (ii') (join (M join)) = (join (join M))
+
+
(iii.1) (unit G') <=< γ = γ
when γ is a natural transformation from some FG' to MG'
- (iii.1) (unit F') <=< φ = φ
==>
- (unit F') is a transformation from F' to MF', so:
- (unit F') <=< φ becomes: (join F') (M unit F') φ
- which is: (join F') (M unit F') φ
- substituting in (iii.1), we get:
- ((join F') (M unit F') φ) = φ
-
- which will be true for all φ just in case:
+ (unit G') is a transformation from G' to MG', so:
+ (unit G') <=< γ becomes: ((join G') (M (unit G')) γ)
+ which is: ((join G') ((M unit) G') γ)
- ((join F') (M unit F')) = the identity transformation, for any F'
+ substituting in (iii.1), we get:
+ ((join G') ((M unit) G') γ) = γ
- which will in turn be true just in case:
-
- (iii.1') (join (M unit) = the identity transformation
+ which is:
+ (((join (M unit)) G') γ) = γ
+ [-- Are the next two steps too cavalier? --]
+ which will be true for all γ just in case:
+ for any G', ((join (M unit)) G') = the identity transformation
+ which will in turn be true just in case:
+ (iii.1') (join (M unit)) = the identity transformation
+
- (iii.2) γ = γ <=< (unit G)
+
+ (iii.2) γ = γ <=< (unit G)
when γ is a natural transformation from G to some MR'G
- (iii.2) φ = φ <=< (unit F)
==>
- φ is a transformation from F to MF', so:
- unit <=< φ becomes: (join F') (M φ) unit
- substituting in (iii.2), we get:
- φ = ((join F') (M φ) (unit F))
- --------------
- which by lemma (2), yields:
- ------------
- φ = ((join F') ((unit MF') φ)
+ γ <=< (unit G) becomes: ((join R'G) (M γ) (unit G))
+
+ substituting in (iii.2), we get:
+ γ = ((join R'G) (M γ) (unit G))
+
+ which by lemma 2, yields:
+ γ = (((join R'G) ((unit MR'G) γ)
- which will be true for all φ just in case:
+ which is:
+ γ = (((join (unit M)) R'G) γ)
- ((join F') (unit MF')) = the identity transformation, for any F'
+ [-- Are the next two steps too cavalier? --]
- which will in turn be true just in case:
+ which will be true for all γ just in case:
+ for any R'G, ((join (unit M)) R'G) = the identity transformation
+ which will in turn be true just in case:
(iii.2') (join (unit M)) = the identity transformation
Collecting the results, our monad laws turn out in this format to be:
-
- when φ a transformation from F to MF', γ a transformation from F' to MG', ρ a transformation from G' to MR' all in T:
+
+ For all ρ, γ, φ in T,
+ where φ is a transformation from F to MF',
+ γ is a transformation from G to MG',
+ ρ is a transformation from R to MR',
+ and F'=G and G'=R:
- (i') ((join G') (M γ) φ) etc also in T
+ (i') ((join G') (M γ) φ) etc also in T
- (ii') (join (M join)) = (join (join M))
+ (ii') (join (M join)) = (join (join M))
(iii.1') (join (M unit)) = the identity transformation
- (iii.2')(join (unit M)) = the identity transformation
+ (iii.2') (join (unit M)) = the identity transformation
Getting to the functional programming presentation of the monad laws
--------------------------------------------------------------------
-In functional programming, unit is usually called "return" and the monad laws are usually stated in terms of return and an operation called "bind" which is interdefinable with <=< or with join.
-
-Additionally, whereas in category-theory one works "monomorphically", in functional programming one usually works with "polymorphic" functions.
+In functional programming, `unit` is sometimes called `return` and the monad laws are usually stated in terms of `unit`/`return` and an operation called `bind` which is interdefinable with `<=<` or with `join`.
The base category C will have types as elements, and monadic functions as its morphisms. The source and target of a morphism will be the types of its argument and its result. (As always, there can be multiple distinct morphisms from the same source to the same target.)
-A monad M will consist of a mapping from types C1 to types M(C1), and a mapping from functions f:C1→C2 to functions M(f):M(C1)→M(C2). This is also known as "fmap f" or "liftM f" for M, and is called "function f lifted into the monad M." For example, where M is the list monad, M maps every type X into the type "list of Xs", and maps every function f:x→y into the function that maps [x1,x2...] to [y1,y2,...].
+A monad `M` will consist of a mapping from types `'t` to types `M('t)`, and a mapping from functions f:C1→C2 to functions M(f):M(C1)→M(C2). This is also known as liftM f for `M`, and is pronounced "function f lifted into the monad M." For example, where `M` is the list monad, `M` maps every type `'t` into the type `'t list`, and maps every function f:x→y into the function that maps `[x1,x2...]` to `[y1,y2,...]`.
+
+In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad."
+A "monadic value" is any member of a type `M('t)`, for any type `'t`. For example, any `int list` is a monadic value for the list monad. We can think of these monadic values as the result of applying some function `phi`, whose type is `F('t)->M(F'('t))`. `'t` here is any collection of free type variables, and `F('t)` and `F'('t)` are types parameterized on `'t`. An example, with `M` being the list monad, `'t` being `('t1,'t2)`, `F('t1,'t2)` being `char * 't1 * 't2`, and `F'('t1,'t2)` being `int * 't1 * 't2`:
+
+ let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
+
+
+
+
+Now where `gamma` is another function of type F'('t) → M(G'('t)), we define:
+
+
+ gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a)
+ = ((join G') -v- (M gamma) -v- phi) a
+ = (gamma <=< phi) a
+
-A natural transformation t assigns to each type C1 in C a morphism t[C1]: C1→M(C1) such that, for every f:C1→C2:
- t[C2] ∘ f = M(f) ∘ t[C1]
+Hence:
+
+
+ gamma <=< phi = fun a -> (gamma =<< phi a)
+
+
+`gamma =<< phi a` is called the operation of "binding" the function gamma to the monadic value `phi a`, and is usually written as `phi a >>= gamma`.
+
+With these definitions, our monadic laws become:
+
+
+
+ Where phi is a polymorphic function of type F('t) -> M(F'('t))
+ gamma is a polymorphic function of type G('t) -> M(G'('t))
+ rho is a polymorphic function of type R('t) -> M(R'('t))
+ and F' = G and G' = R,
+ and a ranges over values of type F('t),
+ b ranges over values of type G('t),
+ and c ranges over values of type G'('t):
+
+ (i) γ <=< φ is defined,
+ and is a natural transformation from F to MG'
+ ==>
+ (i'') fun a -> gamma =<< phi a is defined,
+ and is a function from type F('t) -> M(G'('t))
+
+
+
+ (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
+ ==>
+ (fun a -> (rho <=< gamma) =<< phi a) = (fun a -> rho =<< (gamma <=< phi) a)
+ (fun a -> (fun b -> rho =<< gamma b) =<< phi a) = (fun a -> rho =<< (gamma =<< phi a))
+
+ (ii'') (fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)
+
+
+
+ (iii.1) (unit G') <=< γ = γ
+ when γ is a natural transformation from some FG' to MG'
+ ==>
+ (unit G') <=< gamma = gamma
+ when gamma is a function of type F(G'('t)) -> M(G'('t))
+
+ fun b -> (unit G') =<< gamma b = gamma
+
+ (unit G') =<< gamma b = gamma b
+
+ Let return be a polymorphic function mapping arguments of any
+ type 't to M('t). In particular, it maps arguments c of type
+ G'('t) to the monadic value (unit G') c, of type M(G'('t)).
+
+ (iii.1'') return =<< gamma b = gamma b
+
+
+
+ (iii.2) γ = γ <=< (unit G)
+ when γ is a natural transformation from G to some MR'G
+ ==>
+ gamma = gamma <=< (unit G)
+ when gamma is a function of type G('t) -> M(R'(G('t)))
+
+ gamma = fun b -> gamma =<< (unit G) b
+
+ As above, return will map arguments b of type G('t) to the
+ monadic value (unit G) b, of type M(G('t)).
+
+ gamma = fun b -> gamma =<< return b
+
+ (iii.2'') gamma b = gamma =<< return b
+
-The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2).
+Summarizing (ii''), (iii.1''), (iii.2''), these are the monadic laws as usually stated in the functional programming literature:
+* `fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)`
+ Usually written reversed, and with a monadic variable `u` standing in
+ for `phi a`:
-In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question.
+ `u >>= (fun b -> gamma b >>= rho) = (u >>= gamma) >>= rho`
-For an example of the latter, let φ be a function that takes arguments of some (schematic, polymorphic) type C1 and yields results of some (schematic, polymorphic) type M(C2). An example with M being the list monad, and C2 being the tuple type schema int * C1:
+* `return =<< gamma b = gamma b`
- let φ = fun c → [(1,c), (2,c)]
+ Usually written reversed, and with `u` standing in for `gamma b`:
-φ is polymorphic: when you apply it to the int 0 you get a result of type "list of int * int": [(1,0), (2,0)]. When you apply it to the char 'e' you get a result of type "list of int * char": [(1,'e'), (2,'e')].
+ `u >>= return = u`
-However, to keep things simple, we'll work instead with functions whose type is settled. So instead of the polymorphic φ, we'll work with (φ : C1 → M(int * C1)). This only accepts arguments of type C1. For generality, I'll talk of functions with the type (φ : C1 → M(C1')), where we assume that C1' is a function of C1.
+* `gamma b = gamma =<< return b`
-A "monadic value" is any member of a type M(C1), for any type C1. For example, a list is a monadic value for the list monad. We can think of these monadic values as the result of applying some function (φ : C1 → M(C1')) to an argument of type C1.
+ Usually written reversed:
+ `return b >>= gamma = gamma b`
+