From 636bebd66c03835ae35ec9b38b82485c4ae6460b Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Tue, 2 Nov 2010 09:30:44 -0400 Subject: [PATCH] cat theory tweaks Signed-off-by: Jim Pryor --- advanced_topics/monads_in_category_theory.mdwn | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 44ab7866..768b8542 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -213,25 +213,25 @@ Now we can specify the "monad laws" governing a monad as follows: 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
+	    (i) γ `<=<` φ is also in T
 
-	   (ii) (ρ <=< γ) <=< φ  =  ρ <=< (γ <=< φ)
+	   (ii) (ρ `<=<` γ) `<=<` φ  =  ρ `<=<` (γ `<=<` φ)
 
-	(iii.1) unit <=< φ  =  φ                 (here φ has to be a natural transformation to M(1C))
+	(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)
+	(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')
+	  = ((unit `<=<` φ) G')
 	  = ((join -v- (M unit) -v- φ) G')
 	  = (join G') -v- ((M unit) G') -v- (φ G')
 	  = (join G') -v- (M (unit G')) -v- γ
 	  ??
-	  = (unit G') <=< γ
+	  = (unit G') `<=<` γ
where as we said γ is a natural transformation from some `FG'` to `MG'`. @@ -240,12 +240,12 @@ Similarly, if φ is a natural transformation from `1C` to `MF'`, 
 	γ = (φ G)
-	  = ((φ <=< unit) G)
+	  = ((φ `<=<` unit) G)
 	  = (((join F') -v- (M φ) -v- unit) G)
 	  = ((join F'G) -v- ((M φ) G) -v- (unit G))
 	  = ((join F'G) -v- (M (φ G)) -v- (unit G))
 	  ??
-	  = γ <=< (unit G)
+	  = γ `<=<` (unit G)
where as we said γ is a natural transformation from `G` to some `MF'G`. -- 2.11.0