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 33f9bf3..588d1a3 100644 (file)
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -258,23 +258,23 @@ where as we said <code>&gamma;</code> is a natural transformation from `G` to so
 Summarizing then, the monad laws can be expressed as:
 
 <pre>
 Summarizing then, the monad laws can be expressed as:
 
 <pre>

-       For all &gamma;, &phi; in T for which &rho; <=< &gamma; and  &gamma; <=< &phi; are defined:
+       For all &rho;, &gamma;, &phi; in T for which &rho; <=< &gamma; and &gamma; <=< &phi; are defined:

 
 

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

 
           (ii) (&rho; <=< &gamma;) <=< &phi;  =  &rho; <=< (&gamma; <=< &phi;)
 
        (iii.1) (unit G') <=< &gamma;  =  &gamma;
                when &gamma; is a natural transformation from some FG' to MG'
 
 
           (ii) (&rho; <=< &gamma;) <=< &phi;  =  &rho; <=< (&gamma; <=< &phi;)
 
        (iii.1) (unit G') <=< &gamma;  =  &gamma;
                when &gamma; is a natural transformation from some FG' to MG'
 

-       (iii.2) &gamma;  =  &gamma; <=< (unit G)
+       (iii.2)                     &gamma;  =  &gamma; <=< (unit G)

                when &gamma; is a natural transformation from G to some MR'G
 </pre>
 
 
 
                when &gamma; is a natural transformation from G to some MR'G
 </pre>
 
 
 

-The standard category-theory presentation of the monad laws
------------------------------------------------------------
+Getting to the standard category-theory presentation of the monad laws
+----------------------------------------------------------------------

 In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`.
 
 <!--
 In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`.
 
 <!--


@@ -288,22 +288,29 @@ Let's remind ourselves of some principles:
 
 *      functors "distribute over composition", that is for any morphisms `f` and `g` in `F`'s source category: <code>F(g &#8728; f) = F(g) &#8728; F(f)</code>
 
 
 *      functors "distribute over composition", that is for any morphisms `f` and `g` in `F`'s source category: <code>F(g &#8728; f) = F(g) &#8728; F(f)</code>
 

-*      if <code>&eta;</code> is a natural transformation from `F` to `G`, then for every <code>f:C1&rarr;C2</code> in `F` and `G`'s source category <b>C</b>: <code>&eta;[C2] &#8728; F(f) = G(f) &#8728; &eta;[C1]</code>.
+*      if <code>&eta;</code> is a natural transformation from `G` to `H`, then for every <code>f:C1&rarr;C2</code> in `G` and `H`'s source category <b>C</b>: <code>&eta;[C2] &#8728; G(f) = H(f) &#8728; &eta;[C1]</code>.
+
+*      <code>(&eta; F)[E] = &eta;[F(E)]</code> 
+
+*      <code>(K &eta;)[E} = K(&eta;[E])</code>
+
+*      <code>((&phi; -v- &eta;) F) = ((&phi; F) -v- (&eta; F))</code>

 
 Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
 
 
 
 Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
 
 

-Recall that join is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` in <b>C</b>, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism <code>f:C1&rarr;C2</code> in <b>C</b>:
+Recall that `join` is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` in <b>C</b>, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism <code>f:C1&rarr;C2</code> in <b>C</b>:

 
 <pre>
        (1) join[C2] &#8728; MM(f)  =  M(f) &#8728; join[C1]
 </pre>
 
 
 <pre>
        (1) join[C2] &#8728; MM(f)  =  M(f) &#8728; join[C1]
 </pre>
 

-Next, consider the composite transformation <code>((join MG') -v- (MM &gamma;))</code>.
+Next, let <code>&gamma;</code> be a transformation from `G` to `MG'`, and
+ consider the composite transformation <code>((join MG') -v- (MM &gamma;))</code>.

 
 

-*      <code>&gamma;</code> is a transformation from `G` to `MG'`, and assigns elements `C1` in <b>C</b> a morphism <code>&gamma;\*: G(C1) &rarr; MG'(C1)</code>. <code>(MM &gamma;)</code> is a transformation that instead assigns `C1` the morphism <code>MM(&gamma;\*)</code>.
+*      <code>&gamma;</code> assigns elements `C1` in <b>C</b> a morphism <code>&gamma;\*:G(C1) &rarr; MG'(C1)</code>. <code>(MM &gamma;)</code> is a transformation that instead assigns `C1` the morphism <code>MM(&gamma;\*)</code>.

 
 

-*      `(join MG')` is a transformation from `MMMG'` to `MMG'` that assigns `C1` the morphism `join[MG'(C1)]`.
+*      `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.

 
 Composing them:
 
 
 Composing them:
 


@@ -311,17 +318,17 @@ Composing them:
        (2) ((join MG') -v- (MM &gamma;)) assigns to C1 the morphism join[MG'(C1)] &#8728; MM(&gamma;*).
 </pre>
 
        (2) ((join MG') -v- (MM &gamma;)) assigns to C1 the morphism join[MG'(C1)] &#8728; MM(&gamma;*).
 </pre>
 

-Next, consider the composite transformation <code>((M &gamma;) -v- (join G))</code>.
+Next, consider the composite transformation <code>((M &gamma;) -v- (join G))</code>:

 
 <pre>
 
 <pre>

-       (3) This assigns to C1 the morphism M(&gamma;*) &#8728; join[G(C1)].
+       (3) ((M &gamma;) -v- (join G)) assigns to C1 the morphism M(&gamma;*) &#8728; join[G(C1)].

 </pre>
 
 So for every element `C1` of <b>C</b>:
 
 <pre>
        ((join MG') -v- (MM &gamma;))[C1], by (2) is:
 </pre>
 
 So for every element `C1` of <b>C</b>:
 
 <pre>
        ((join MG') -v- (MM &gamma;))[C1], by (2) is:

-       join[MG'(C1)] &#8728; MM(&gamma;*), which by (1), with f=&gamma;*: G(C1)&rarr;MG'(C1) is:
+       join[MG'(C1)] &#8728; MM(&gamma;*), which by (1), with f=&gamma;*:G(C1)&rarr;MG'(C1) is:

        M(&gamma;*) &#8728; join[G(C1)], which by 3 is:
        ((M &gamma;) -v- (join G))[C1]
 </pre>
        M(&gamma;*) &#8728; join[G(C1)], which by 3 is:
        ((M &gamma;) -v- (join G))[C1]
 </pre>


@@ -329,33 +336,34 @@ So for every element `C1` of <b>C</b>:
 So our **(lemma 1)** is:
 
 <pre>
 So our **(lemma 1)** is:
 
 <pre>

-       ((join MG') -v- (MM &gamma;))  =  ((M &gamma;) -v- (join G)), where &gamma; is a transformation from G to MG'.
+       ((join MG') -v- (MM &gamma;))  =  ((M &gamma;) -v- (join G)),
+       where as we said &gamma; is a natural transformation from G to MG'.

 </pre>
 
 
 </pre>
 
 

-Next recall that unit is a natural transformation from `1C` to `M`. So for elements `C1` in <b>C</b>, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism <code>f:a&rarr;b</code> in <b>C</b>:
+Next recall that `unit` is a natural transformation from `1C` to `M`. So for elements `C1` in <b>C</b>, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism <code>f:C1&rarr;C2</code> in <b>C</b>:

 
 <pre>
 
 <pre>

-       (4) unit[b] &#8728; f = M(f) &#8728; unit[a]
+       (4) unit[C2] &#8728; f = M(f) &#8728; unit[C1]

 </pre>
 
 </pre>
 

-Next consider the composite transformation <code>((M &gamma;) -v- (unit G))</code>:
+Next, consider the composite transformation <code>((M &gamma;) -v- (unit G))</code>:

 
 <pre>
 
 <pre>

-       (5) This assigns to C1 the morphism M(&gamma;*) &#8728; unit[G(C1)].
+       (5) ((M &gamma;) -v- (unit G)) assigns to C1 the morphism M(&gamma;*) &#8728; unit[G(C1)].

 </pre>
 
 </pre>
 

-Next consider the composite transformation <code>((unit MG') -v- &gamma;)</code>.
+Next, consider the composite transformation <code>((unit MG') -v- &gamma;)</code>:

 
 <pre>
 
 <pre>

-       (6) This assigns to C1 the morphism unit[MG'(C1)] &#8728; &gamma;*.
+       (6) ((unit MG') -v- &gamma;) assigns to C1 the morphism unit[MG'(C1)] &#8728; &gamma;*.

 </pre>
 
 So for every element C1 of <b>C</b>:
 
 <pre>
        ((M &gamma;) -v- (unit G))[C1], by (5) =
 </pre>
 
 So for every element C1 of <b>C</b>:
 
 <pre>
        ((M &gamma;) -v- (unit G))[C1], by (5) =

-       M(&gamma;*) &#8728; unit[G(C1)], which by (4), with f=&gamma;*: G(C1)&rarr;MG'(C1) is:
+       M(&gamma;*) &#8728; unit[G(C1)], which by (4), with f=&gamma;*:G(C1)&rarr;MG'(C1) is:

        unit[MG'(C1)] &#8728; &gamma;*, which by (6) =
        ((unit MG') -v- &gamma;)[C1]
 </pre>
        unit[MG'(C1)] &#8728; &gamma;*, which by (6) =
        ((unit MG') -v- &gamma;)[C1]
 </pre>


@@ -363,46 +371,54 @@ So for every element C1 of <b>C</b>:
 So our **(lemma 2)** is:
 
 <pre>
 So our **(lemma 2)** is:
 
 <pre>

-       (((M &gamma;) -v- (unit G))  =  ((unit MG') -v- &gamma;)), where &gamma; is a transformation from G to MG'.
+       (((M &gamma;) -v- (unit G))  =  ((unit MG') -v- &gamma;)),
+       where as we said &gamma; is a natural transformation from G to MG'.

 </pre>
 
 
 Finally, we substitute <code>((join G') -v- (M &gamma;) -v- &phi;)</code> for <code>&gamma; &lt;=&lt; &phi;</code> in the monad laws. For simplicity, I'll omit the "-v-".
 
 <pre>
 </pre>
 
 
 Finally, we substitute <code>((join G') -v- (M &gamma;) -v- &phi;)</code> for <code>&gamma; &lt;=&lt; &phi;</code> in the monad laws. For simplicity, I'll omit the "-v-".
 
 <pre>

-       for all &phi;,&gamma;,&rho; in T, where &phi; is a transformation from F to MF', &gamma; is a transformation from G to MG', R is a transformation from R to MR', and F'=G and G'=R:
+       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) &gamma; <=< &phi; etc are also in T
+            (i) &gamma; <=< &phi; etc are also in T

        ==>
        ==>

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

 
 
 
 

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

        ==>
        ==>

-                (&rho; <=< &gamma;) is a transformation from G to MR', so:
-                       (&rho; <=< &gamma;) <=< &phi; becomes: (join R') (M (&rho; <=< &gamma;)) &phi;
-                                                       which is: (join R') (M ((join R') (M &rho;) &gamma;)) &phi;
-                       substituting in (ii), and helping ourselves to associativity on the rhs, we get:
+                   (&rho; <=< &gamma;) is a transformation from G to MR', so
+                       (&rho; <=< &gamma;) <=< &phi; becomes: ((join R') (M (&rho; <=< &gamma;)) &phi;)
+                                                       which is: ((join R') (M ((join R') (M &rho;) &gamma;)) &phi;)
+
+                       similarly, &rho; <=< (&gamma; <=< &phi;) is:
+                                                       ((join R') (M &rho;) ((join G') (M &gamma;) &phi;))

 
 

-            ((join R') (M ((join R') (M &rho;) &gamma;)) &phi;) = ((join R') (M &rho;) (join G') (M &gamma;) &phi;)
-                     ---------------------
+                       substituting these into (ii), and helping ourselves to associativity on the rhs, we get:
+               ((join R') (M ((join R') (M &rho;) &gamma;)) &phi;) = ((join R') (M &rho;) (join G') (M &gamma;) &phi;)
+    

                        which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields:
                        which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields:

-                    ------------------------
-            ((join R') (M join R') (MM &rho;) (M &gamma;) &phi;) = ((join R') (M &rho;) (join G') (M &gamma;) &phi;)
-                                                             ---------------
+               ((join R') (M join R') (MM &rho;) (M &gamma;) &phi;) = ((join R') (M &rho;) (join G') (M &gamma;) &phi;)
+  

                        which by lemma 1, with &rho; a transformation from G' to MR', yields:
                        which by lemma 1, with &rho; a transformation from G' to MR', yields:

-                                                             -----------------
-            ((join R') (M join R') (MM &rho;) (M &gamma;) &phi;) = ((join R') (join MR') (MM &rho;) (M &gamma;) &phi;)
+               ((join R') (M join R') (MM &rho;) (M &gamma;) &phi;) = ((join R') (join MR') (MM &rho;) (M &gamma;) &phi;)

 
 

-                       which will be true for all &rho;,&gamma;,&phi; just in case:
+                       which will be true for all &rho;,&gamma;,&phi; only when:
+               ((join R') (M join R')) = ((join R') (join MR')), for any R'.

 
 

-             ((join R') (M join R')) = ((join R') (join MR')), for any R'.
+                       which will in turn be true when:
+      (ii') (join (M join)) = (join (join M))

 
 

-                       which will in turn be true just in case:
-
-       (ii') (join (M join)) = (join (join M))

 
 
 
 

+        (iii.1) (unit G') <=< &gamma;  =  &gamma;
+                when &gamma; is a natural transformation from some FG' to MG'

        (iii.1) (unit F') <=< &phi;  =  &phi;
        ==>
                        (unit F') is a transformation from F' to MF', so:
        (iii.1) (unit F') <=< &phi;  =  &phi;
        ==>
                        (unit F') is a transformation from F' to MF', so:


@@ -420,6 +436,10 @@ Finally, we substitute <code>((join G') -v- (M &gamma;) -v- &phi;)</code> for <c
        (iii.1') (join (M unit) = the identity transformation
 
 
        (iii.1') (join (M unit) = the identity transformation
 
 

+
+
+        (iii.2)                     &gamma;  =  &gamma; <=< (unit G)
+                when &gamma; is a natural transformation from G to some MR'G

        (iii.2) &phi;  =  &phi; <=< (unit F)
        ==>
                        &phi; is a transformation from F to MF', so:
        (iii.2) &phi;  =  &phi; <=< (unit F)
        ==>
                        &phi; is a transformation from F to MF', so:


@@ -457,8 +477,8 @@ Collecting the results, our monad laws turn out in this format to be:
 
 
 
 
 
 

-7. The functional programming presentation of the monad laws
-------------------------------------------------------------
+Getting to the functional programming presentation of the monad laws
+--------------------------------------------------------------------

 In functional programming, unit is usually called "return" and the monad laws are usually stated in terms of return and an operation called "bind" which is interdefinable with <=< or with join.
 
 Additionally, whereas in category-theory one works "monomorphically", in functional programming one usually works with "polymorphic" functions.
 In functional programming, unit is usually called "return" and the monad laws are usually stated in terms of return and an operation called "bind" which is interdefinable with <=< or with join.
 
 Additionally, whereas in category-theory one works "monomorphically", in functional programming one usually works with "polymorphic" functions.