γ = (φ G')
= ((unit <=< φ) G')
+ since unit is a natural transformation to M(1C), this is:
= (((join 1C) v (M unit) v φ) G')
= (((join 1C) G') v ((M unit) G') v (φ G'))
= ((join (1C G')) v (M (unit G')) v γ)
= ((join G') v (M (unit G')) v γ)
 since (unit G') is a natural transformation to MG',
 this satisfies the definition for <=<:
+ since (unit G') is a natural transformation to MG', this is:
= (unit G') <=< γ
@@ 245,11 +247,11 @@ Similarly, if ρ
is a natural transformation from `1C` to `MR'`,
γ = (ρ G)
= ((ρ <=< unit) G)
+ = since ρ is a natural transformation to MR', this is:
= (((join R') v (M ρ) v unit) G)
= (((join R') G) v ((M ρ) G) v (unit G))
= ((join (R'G)) v (M (ρ G)) v (unit G))
 since γ = (ρ G) is a natural transformation to MR'G,
 this satisfies the definition <=<:
+ since γ = (ρ G) is a natural transformation to MR'G, this is:
= γ <=< (unit G)
@@ 265,16 +267,16 @@ Summarizing then, the monad laws can be expressed as:
(ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
(iii.1) (unit G') <=< γ = γ
 when γ is a natural transformation from some FG' to MG'
+ whenever γ is a natural transformation from some FG' to MG'
 (iii.2) γ = γ <=< (unit G)
 when γ is a natural transformation from G to some MR'G
+ (iii.2) γ = γ <=< (unit G)
+ whenever γ is a natural transformation from G to some MR'G
The standard categorytheory presentation of the monad laws

+Getting to the standard categorytheory presentation of the monad laws
+
In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`.
Let's remind ourselves of some principles:
+Let's remind ourselves of principles stated above:
* composition of morphisms, functors, and natural compositions is associative
* functors "distribute over composition", that is for any morphisms `f` and `g` in `F`'s source category: F(g ∘ f) = F(g) ∘ F(f)
* if η
is a natural transformation from `F` to `G`, then for every f:C1→C2
in `F` and `G`'s source category C: η[C2] ∘ F(f) = G(f) ∘ η[C1]
.
+* if η
is a natural transformation from `G` to `H`, then for every f:C1→C2
in `G` and `H`'s source category C: η[C2] ∘ G(f) = H(f) ∘ η[C1]
.
+
+* (η F)[X] = η[F(X)]
+
+* (K η)[X] = K(η[X])
+
+* ((φ v η) F) = ((φ F) v (η F))
Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
Recall that join is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` in C, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism f:C1→C2
in C:
+Recall that `join` is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` in C, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism f:C1→C2
in C:
(1) join[C2] ∘ MM(f) = M(f) ∘ join[C1]
Next, consider the composite transformation ((join MG') v (MM γ))
.
+Next, let γ
be a transformation from `G` to `MG'`, and
+ consider the composite transformation ((join MG') v (MM γ))
.
* γ
is a transformation from `G` to `MG'`, and assigns elements `C1` in C a morphism γ\*: G(C1) → MG'(C1)
. (MM γ)
is a transformation that instead assigns `C1` the morphism MM(γ\*)
.
+* γ
assigns elements `C1` in C a morphism γ\*:G(C1) → MG'(C1)
. (MM γ)
is a transformation that instead assigns `C1` the morphism MM(γ\*)
.
* `(join MG')` is a transformation from `MMMG'` to `MMG'` that assigns `C1` the morphism `join[MG'(C1)]`.
+* `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.
Composing them:
@@ 311,17 +320,17 @@ Composing them:
(2) ((join MG') v (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*).
Next, consider the composite transformation ((M γ) v (join G))
.
+Next, consider the composite transformation ((M γ) v (join G))
:
 (3) This assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
+ (3) ((M γ) v (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
So for every element `C1` of C:
((join MG') v (MM γ))[C1], by (2) is:
 join[MG'(C1)] ∘ MM(γ*), which by (1), with f=γ*: G(C1)→MG'(C1) is:
+ join[MG'(C1)] ∘ MM(γ*), which by (1), with f=γ*:G(C1)→MG'(C1) is:
M(γ*) ∘ join[G(C1)], which by 3 is:
((M γ) v (join G))[C1]
@@ 329,33 +338,34 @@ So for every element `C1` of C:
So our **(lemma 1)** is:
 ((join MG') v (MM γ)) = ((M γ) v (join G)), where γ is a transformation from G to MG'.
+ ((join MG') v (MM γ)) = ((M γ) v (join G)),
+ where as we said γ is a natural transformation from G to MG'.
Next recall that unit is a natural transformation from `1C` to `M`. So for elements `C1` in C, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism f:a→b
in C:
+Next recall that `unit` is a natural transformation from `1C` to `M`. So for elements `C1` in C, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism f:C1→C2
in C:
 (4) unit[b] ∘ f = M(f) ∘ unit[a]
+ (4) unit[C2] ∘ f = M(f) ∘ unit[C1]
Next consider the composite transformation ((M γ) v (unit G))
:
+Next, consider the composite transformation ((M γ) v (unit G))
:
 (5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
+ (5) ((M γ) v (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
Next consider the composite transformation ((unit MG') v γ)
.
+Next, consider the composite transformation ((unit MG') v γ)
:
 (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
+ (6) ((unit MG') v γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
So for every element C1 of C:
((M γ) v (unit G))[C1], by (5) =
 M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*: G(C1)→MG'(C1) is:
+ M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*:G(C1)→MG'(C1) is:
unit[MG'(C1)] ∘ γ*, which by (6) =
((unit MG') v γ)[C1]
@@ 363,130 +373,241 @@ So for every element C1 of C:
So our **(lemma 2)** is:
 (((M γ) v (unit G)) = ((unit MG') v γ)), where γ is a transformation from G to MG'.
+ (((M γ) v (unit G)) = ((unit MG') v γ)),
+ where as we said γ is a natural transformation from G to MG'.
Finally, we substitute ((join G') v (M γ) v φ)
for γ <=< φ
in the monad laws. For simplicity, I'll omit the "v".
 for all φ,γ,ρ in T, where φ is a transformation from F to MF', γ is a transformation from G to MG', R is a transformation from R to MR', and F'=G and G'=R:
+ 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) γ <=< φ etc are also in T
+ (i) γ <=< φ etc are also in T
==>
 (i') ((join G') (M γ) φ) etc are also in T

+ (i') ((join G') (M γ) φ) etc are also in T
+
 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
++ (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
 which is: (join R') (M ((join R') (M ρ) γ)) φ
 substituting in (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 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)
+ (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)
 which will be true for all ρ,γ,φ just in case:
+ similarly, ρ <=< (γ <=< φ) is:
+ ((join R') (M ρ) ((join G') (M γ) φ))
 ((join R') (M join R')) = ((join R') (join MR')), for any R'.
+ 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 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:
+ [ Are the next two steps too cavalier? ]
 (ii') (join (M join)) = (join (join M))
+ 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 F') <=< φ = φ
++ (iii.1) (unit G') <=< γ = γ
+ when γ is a natural transformation from some FG' to MG'
==>
 (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
+ (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)
 which will be true for all φ just in case:
+ substituting in (iii.1), we get:
+ ((join G') ((M unit) G') γ) = γ
 ((join F') (M unit F')) = the identity transformation, for any F'
+ which is:
+ (((join (M unit)) G') γ) = γ
 which will in turn be true just in case:
+ [ Are the next two steps too cavalier? ]
 (iii.1') (join (M unit) = the identity transformation
+ 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) φ = φ <=< (unit F)
++ (iii.2) γ = γ <=< (unit G)
+ when γ is a natural transformation from G to some MR'G
==>
 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
+In categorytheory presentations, you may see `unit` referred to as η
, and `join` referred to as μ
. Also, instead of the monad `(M, unit, join)`, you may sometimes see discussion of the "Kleisli triple" `(M, unit, =<<)`. Alternatively, `=<<` may be called ⋆
. These are interdefinable (see below).
7. 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 categorytheory one works "monomorphically", in functional programming one usually works with "polymorphic" functions.
+Getting to the functional programming presentation of the monad laws
+
+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 lift_{M} 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)]
+
+
+[ I intentionally chose this polymorphic function because simpler ways of mapping the polymorphic monad operations from functional programming onto the category theory notions can't accommodate it. We have all the F, MF' (unit G') and so on in order to be able to be handle even phis like this. ]
+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),
+ and 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
+ whenever γ is a natural transformation from some FG' to MG'
+ ==>
+ (unit G') <=< gamma = gamma
+ whenever 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)
+ whenever γ is a natural transformation from G to some MR'G
+ ==>
+ gamma = gamma <=< (unit G)
+ whenever 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`
+