X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=advanced_topics%2Fmonads_in_category_theory.mdwn;h=b148fe6e44c4a6f3ef2d8c31f0a2c5f979d94d53;hp=044d97332b942d86762e6ba2becf7dda514a8d2b;hb=24506613cb3b7c8e1cbe45a55f29de40d92bdbd3;hpb=06d07bead9501b9450d8c7a1c30926a83385c38a diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 044d9733..b148fe6e 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -194,7 +194,7 @@ We also need to designate for `M` a "join" transformation, which is a natural tr These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion. -Let φ and γ be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now `(M γ)` will also be a natural transformation, formed by composing the functor `M` with the natural transformation γ. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, (M γ), and φ, and abbreviate it as follows: +Let φ and γ be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now (M γ) will also be a natural transformation, formed by composing the functor `M` with the natural transformation γ. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, (M γ), and φ, and abbreviate it as follows: 
 	γ <=< φ  =def.  ((join G') -v- (M γ) -v- φ)
@@ -202,23 +202,29 @@ Let φ and γ be members of `T`, that is they
 
 Since composition is associative I don't specify the order of composition on the rhs.
 
-In other words, `<=<` is a binary operator that takes us from two members φ and γ of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes its written φ >=> γ where that's the same as γ <=< φ.)
+In other words, `<=<` is a binary operator that takes us from two members φ and γ of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes it's written φ >=> γ where that's the same as γ <=< φ.)
 
-φ is a transformation from `F` to `MF'` which = `MG`; `(M γ)` is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite γ <=< φ will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`.
+φ is a transformation from `F` to `MF'`, where the latter = `MG`; (M γ) is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite γ <=< φ will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`.
 
 Now we can specify the "monad laws" governing a monad as follows:
 
 	(T, <=<, unit) constitute a monoid
 
-That's it. (Well, perhaps we're cheating a bit, because γ <=< φ isn't fully defined on `T`, but only when `F` is a functor to `MF'` and `G` is a functor from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied:
+That's it. Well, there may be a wrinkle here. I don't know whether the definition of a monoid requires the operation to be defined for every pair in its set. In the present case, γ <=< φ isn't fully defined on `T`, but only when φ is a transformation to some `MF'` and γ is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied:
+
+
+	    (i) γ <=< φ is also in T
+
+	   (ii) (ρ <=< γ) <=< φ  =  ρ <=< (γ <=< φ)
 
-	(i) γ <=< φ is also in T
-	(ii) (ρ <=< γ) <=< φ  =  ρ <=< (γ <=< φ)
 	(iii.1) unit <=< φ  =  φ                 (here φ has to be a natural transformation to M(1C))
+
 	(iii.2)                φ  =  φ <=< unit  (here φ has to be a natural transformation from 1C)
+

 
-If φ is a natural transformation from `F` to `M(1C)` and γ is `(φ G')`, that is, a natural transformation from `PG` to `MG`, then we can extend (iii.1) as follows:
+If φ is a natural transformation from `F` to `M(1C)` and γ is (φ G'), that is, a natural transformation from `FG` to `MG`, then we can extend (iii.1) as follows:
 
+
 	γ = (φ G')
 	  = ((unit <=< φ) G')
 	  = ((join -v- (M unit) -v- φ) G')
@@ -226,11 +232,13 @@ If φ is a natural transformation from `F` to `M(1C)` and γ is `(φ
 	  = (join G') -v- (M (unit G')) -v- γ
 	  ??
 	  = (unit G') <=< γ
+

 
-where as we said γ is a natural transformation from some `PG'` to `MG'`.
+where as we said γ is a natural transformation from some `FG'` to `MG'`.
 
-Similarly, if φ is a natural transformation from `1C` to `MF'`, and γ is `(φ G)`, that is, a natural transformation from `G` to `MF'G`, then we can extend (iii.2) as follows:
+Similarly, if φ is a natural transformation from `1C` to `MF'`, and γ is (φ G), that is, a natural transformation from `G` to `MF'G`, then we can extend (iii.2) as follows:
 
+
 	γ = (φ G)
 	  = ((φ <=< unit) G)
 	  = (((join F') -v- (M φ) -v- unit) G)
@@ -238,8 +246,9 @@ Similarly, if φ is a natural transformation from `1C` to `MF'`, and γ
 	  = ((join F'G) -v- (M (φ G)) -v- (unit G))
 	  ??
 	  = γ <=< (unit G)
+

 
-where as we said γ is a natural transformation from `G` to some `MF'G`.
+where as we said γ is a natural transformation from `G` to some `MF'G`.