X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;f=advanced_topics%2Fmonads_in_category_theory.mdwn;h=4941e80b99875eb3f9179c704da83e63d4d3ae66;hb=4f87cdd50334c5d9dddbdad136fac4008d8ce6ff;hp=fe12655bac9e24310213d1616279771202565bd0;hpb=2f14eb20bd2428f346f8d5c8caa35b67eb043096;p=lambda.git diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index fe12655b..4941e80b 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -208,9 +208,13 @@ In other words, `<=<` is a binary operator that takes us from two members 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. -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 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
@@ -288,7 +292,7 @@ Next, consider the composite transformation ((join MG') -v- (MM γ))<
 Composing them:
 
 
-	(2) ((join MG') -v- (MM γ)) assigns to `C1` the morphism join[MG'(C1)] ∘ MM(γ*).
+	(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)).
@@ -347,7 +351,7 @@ So our **(lemma 2)** is:
-Finally, we substitute ((join G') -v- (M γ) -v- φ) for γ <=< φ in the monad laws. For simplicity, I'll omit the "-v-". +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: