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=a6165e9deae3a47b9401f946e8c3087d85a33cdb;hb=24506613cb3b7c8e1cbe45a55f29de40d92bdbd3;hpb=7b00a5a3dc3f7208f67ed5c87faf22b351e14b0c
diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index a6165e9d..b148fe6e 100644
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -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, 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 `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)
+
(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 `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
= (join G') -v- (M (unit G')) -v- γ
??
= (unit G') <=< γ
+
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`.