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 3e34f85..29b6feb 100644 (file)
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -121,13 +121,13 @@ Consider four categories <b>B</b>, <b>C</b>, <b>D</b>, and <b>E</b>. Let `F` be
 <pre>
        - <b>B</b> -+ +--- <b>C</b> --+ +---- <b>D</b> -----+ +-- <b>E</b> --
                 | |        | |            | |
-        F: -----&rarr; G: -----&rarr;     K: -----&rarr;
-                | |        | |  | &eta;     | |  | &psi;
+        F: ------> G: ------>     K: ------>
+                | |        | |  | &eta;       | |  | &psi;
                 | |        | |  v         | |  v
-                | |    H: -----&rarr;     L: -----&rarr;
-                | |        | |  | &phi;     | |
+                | |    H: ------>     L: ------>
+                | |        | |  | &phi;       | |
                 | |        | |  v         | |
-                | |    J: -----&rarr;         | |
+                | |    J: ------>         | |
        -----+ +--------+ +------------+ +-------
 </pre>
 
@@ -154,7 +154,7 @@ by naturalness of <code>&eta;</code>, is:
        &phi;[C2] &#8728; &eta;[C2] &#8728; G(f) = J(f) &#8728; &phi;[C1] &#8728; &eta;[C1]
 </pre>
 
-Hence, we can define <code>(&phi; -v- &eta;)[x]</code> as: <code>&phi;[x] &#8728; &eta;[x]</code> and rely on it to satisfy the constraints for a natural transformation from `G` to `J`:
+Hence, we can define <code>(&phi; -v- &eta;)[\_]</code> as: <code>&phi;[\_] &#8728; &eta;[\_]</code> and rely on it to satisfy the constraints for a natural transformation from `G` to `J`:
 
 <pre>
        (&phi; -v- &eta;)[C2] &#8728; G(f) = J(f) &#8728; (&phi; -v- &eta;)[C1]
@@ -173,7 +173,7 @@ I'll assert without proving that vertical composition is associative and has an
 
 <pre>
        (&phi; -h- &eta;)[C1]  =  L(&eta;[C1]) &#8728; &psi;[G(C1)]
-                                          =  &psi;[H(C1)] &#8728; K(&eta;[C1])
+                                 =  &psi;[H(C1)] &#8728; K(&eta;[C1])
 </pre>
 
 Horizontal composition is also associative, and has the same identity as vertical composition.