cat theory tweaks

[lambda.git] / advanced_topics / monads_in_category_theory.mdwn

diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 90f77cb..2467079 100644 (file)
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -422,7 +422,6 @@ Finally, we substitute <code>((join G') -v- (M &gamma;) -v- &phi;)</code> for <c
        ==>
                         (unit G') is a transformation from G' to MG', so:
                         (unit G') <=< &gamma; becomes: ((join G') (M unit G') &gamma;)
        ==>
                         (unit G') is a transformation from G' to MG', so:
                         (unit G') <=< &gamma; becomes: ((join G') (M unit G') &gamma;)

-                                                   which is: ((join G') (M unit G') &gamma;)

 
                         substituting in (iii.1), we get:
                         ((join G') (M unit G') &gamma;) = &gamma;
 
                         substituting in (iii.1), we get:
                         ((join G') (M unit G') &gamma;) = &gamma;


@@ -439,7 +438,7 @@ Finally, we substitute <code>((join G') -v- (M &gamma;) -v- &phi;)</code> for <c
         (iii.2) &gamma;  =  &gamma; <=< (unit G)
                 when &gamma; is a natural transformation from G to some MR'G
        ==>
         (iii.2) &gamma;  =  &gamma; <=< (unit G)
                 when &gamma; is a natural transformation from G to some MR'G
        ==>

-                        unit <=< &gamma; becomes: (join R'G) (M &gamma;) unit
+                        unit <=< &gamma; becomes: ((join R'G) (M &gamma;) unit)

                        
                         substituting in (iii.2), we get:
                         &gamma; = ((join R'G) (M &gamma;) (unit G))
                        
                         substituting in (iii.2), we get:
                         &gamma; = ((join R'G) (M &gamma;) (unit G))


@@ -457,8 +456,12 @@ Finally, we substitute <code>((join G') -v- (M &gamma;) -v- &phi;)</code> for <c
 
 Collecting the results, our monad laws turn out in this format to be:
 
 
 Collecting the results, our monad laws turn out in this format to be:
 

-</pre>
-       when &phi; a transformation from F to MF', &gamma; a transformation from F' to MG', &rho; a transformation from G' to MR' are all in T:
+<pre>
+       For all &rho;, &gamma;, &phi; in T,
+       where &phi; is a transformation from F to MF',
+       &gamma; is a transformation from G to MG',
+       &rho; is a transformation from R to MR',
+       and F'=G and G'=R:

 
            (i') ((join G') (M &gamma;) &phi;) etc also in T
 
 
            (i') ((join G') (M &gamma;) &phi;) etc also in T