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)

@@ -258,23 +260,23 @@ where as we said γ is a natural transformation from `G` to so Summarizing then, the monad laws can be expressed as:
-	For all γ, φ in T for which ρ <=< γ and  γ <=< φ are defined:
+	For all ρ, γ, φ in T for which ρ <=< γ and γ <=< φ are defined:

-	    (i) γ <=< φ is also in T
+	    (i) γ <=< φ etc are also in T

(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 category-theory presentation of the monad laws ------------------------------------------------------------ +Getting to the standard category-theory 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:
+		     (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)

-	     ((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)
+			 similarly, ρ <=< (γ <=< φ) is:
+							((join R') (M ρ) ((join G') (M γ) φ))

-			which will be true for all ρ,γ,φ just in case:
+		 	 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)

-	      ((join R') (M join R')) = ((join R') (join MR')), for any R'.
+			 [-- Are the next two steps too cavalier? --]

-			which will in turn be true just in case:
-
-	(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

+	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
+
+ +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: -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] +
+	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
+