add mappable laws

[lambda.git] / topics / week7_introducing_monads.mdwn

diff --git a/topics/week7_introducing_monads.mdwn b/topics/week7_introducing_monads.mdwn
index b04f517..adcf0cc 100644 (file)
--- a/topics/week7_introducing_monads.mdwn
+++ b/topics/week7_introducing_monads.mdwn
@@ -84,7 +84,7 @@ For instance, the following are Kleisli arrows:
 In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <code><u>Q</u></code> is <code><u>bool</u></code>).
 
 Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where
-if `α list` is our box type, we can write the second arrow as
+if `α list` is our box type, we can write the second type as:
 
 <code><u>int</u> -> <u>int list</u></code>
 
@@ -107,11 +107,11 @@ Here are the types of our crucial functions, together with our pronunciation, an
 
 <code>&lt;=&lt; or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
 
-<code>&gt;=&gt; or mpmoc (flip mcomp): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
+<code>&gt;=&gt; (flip mcomp, should we call it mpmoc?): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
 
 <code>&gt;&gt;= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
 
-<code>=&lt;&lt; or mdnib (flip mbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
+<code>=&lt;&lt; (flip mbind, should we call it mdnib?) (Q -> <u>R</u>) -> (<u>Q</u>) -> (<u>R</u>)</code>
 
 <code>join: <span class="box2">P</span> -> <u>P</u></code> 
 
@@ -127,7 +127,11 @@ certain useful guarantees.
 if there is a `map` function defined for that box type with the type given above. This
 has to obey the following Map Laws:
 
-    TODO LAWS
+    <code>map (id : α -> α) = (id : <u>α</u> -> <u>α</u>)</code>  
+    <code>map (g ○ f) = (map g) ○ (map f)</code>
+
+    Essentially these say that `map` is a homomorphism from `(α -> β, ○, id)` to <code>(<u>α</u> -> <u>β</u>, ○', id')</code>, where `○'` and `id'` are `○` and `id` restricted to arguments of type <code><u>_</u></code>.
+
 
 *   ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
        if there are in addition `map2`, `mid`, and `mapply`.  (Given either