cat theory tweaks

Signed-off-by: Jim Pryor <profjim@jimpryor.net>

diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 44ab786..768b854 100644 (file)
--- 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, <code>&gamma; &lt;=&lt; &phi;</code> isn't fully defined on `T`, but only when <code>&phi;</code> is a transformation to some `MF'` and <code>&gamma;</code> is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied:
 
 <pre>
 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, <code>&gamma; &lt;=&lt; &phi;</code> isn't fully defined on `T`, but only when <code>&phi;</code> is a transformation to some `MF'` and <code>&gamma;</code> is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied:
 
 <pre>

-           (i) &gamma; <=< &phi; is also in T
+           (i) &gamma; `<=<` &phi; is also in T

 
 

-          (ii) (&rho; <=< &gamma;) <=< &phi;  =  &rho; <=< (&gamma; <=< &phi;)
+          (ii) (&rho; `<=<` &gamma;) `<=<` &phi;  =  &rho; `<=<` (&gamma; `<=<` &phi;)

 
 

-       (iii.1) unit <=< &phi;  =  &phi;                 (here &phi; has to be a natural transformation to M(1C))
+       (iii.1) unit `<=<` &phi;  =  &phi;                 (here &phi; has to be a natural transformation to M(1C))

 
 

-       (iii.2)                &phi;  =  &phi; <=< unit  (here &phi; has to be a natural transformation from 1C)
+       (iii.2)                &phi;  =  &phi; `<=<` unit  (here &phi; has to be a natural transformation from 1C)

 </pre>
 
 If <code>&phi;</code> is a natural transformation from `F` to `M(1C)` and <code>&gamma;</code> is <code>(&phi; G')</code>, that is, a natural transformation from `FG` to `MG`, then we can extend (iii.1) as follows:
 
 <pre>
        &gamma; = (&phi; G')
 </pre>
 
 If <code>&phi;</code> is a natural transformation from `F` to `M(1C)` and <code>&gamma;</code> is <code>(&phi; G')</code>, that is, a natural transformation from `FG` to `MG`, then we can extend (iii.1) as follows:
 
 <pre>
        &gamma; = (&phi; G')

-         = ((unit <=< &phi;) G')
+         = ((unit `<=<` &phi;) G')

          = ((join -v- (M unit) -v- &phi;) G')
          = (join G') -v- ((M unit) G') -v- (&phi; G')
          = (join G') -v- (M (unit G')) -v- &gamma;
          ??
          = ((join -v- (M unit) -v- &phi;) G')
          = (join G') -v- ((M unit) G') -v- (&phi; G')
          = (join G') -v- (M (unit G')) -v- &gamma;
          ??

-         = (unit G') <=< &gamma;
+         = (unit G') `<=<` &gamma;

 </pre>
 
 where as we said <code>&gamma;</code> is a natural transformation from some `FG'` to `MG'`.
 </pre>
 
 where as we said <code>&gamma;</code> is a natural transformation from some `FG'` to `MG'`.


@@ -240,12 +240,12 @@ Similarly, if <code>&phi;</code> is a natural transformation from `1C` to `MF'`,
 
 <pre>
        &gamma; = (&phi; G)
 
 <pre>
        &gamma; = (&phi; G)

-         = ((&phi; <=< unit) G)
+         = ((&phi; `<=<` unit) G)

          = (((join F') -v- (M &phi;) -v- unit) G)
          = ((join F'G) -v- ((M &phi;) G) -v- (unit G))
          = ((join F'G) -v- (M (&phi; G)) -v- (unit G))
          ??
          = (((join F') -v- (M &phi;) -v- unit) G)
          = ((join F'G) -v- ((M &phi;) G) -v- (unit G))
          = ((join F'G) -v- (M (&phi; G)) -v- (unit G))
          ??

-         = &gamma; <=< (unit G)
+         = &gamma; `<=<` (unit G)

 </pre>
 
 where as we said <code>&gamma;</code> is a natural transformation from `G` to some `MF'G`.
 </pre>
 
 where as we said <code>&gamma;</code> is a natural transformation from `G` to some `MF'G`.