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 2467079..5183242 100644 (file)
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -476,34 +476,126 @@ Collecting the results, our monad laws turn out in this format to be:
 
 Getting to 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 sometimes called `return` and the monad laws are usually stated in terms of `unit`/`return` and an operation called `bind` which is interdefinable with `<=<` or with `join`.

 
 The base category <b>C</b> will have types as elements, and monadic functions as its morphisms. The source and target of a morphism will be the types of its argument and its result. (As always, there can be multiple distinct morphisms from the same source to the same target.)
 
 
 The base category <b>C</b> will have types as elements, and monadic functions as its morphisms. The source and target of a morphism will be the types of its argument and its result. (As always, there can be multiple distinct morphisms from the same source to the same target.)
 

-A monad M will consist of a mapping from types C1 to types M(C1), and a mapping from functions f:C1&rarr;C2 to functions M(f):M(C1)&rarr;M(C2). This is also known as "fmap f" or "liftM f" for M, and is called "function f lifted into the monad M." For example, where M is the list monad, M maps every type X into the type "list of Xs", and maps every function f:x&rarr;y into the function that maps [x1,x2...] to [y1,y2,...].
+A monad `M` will consist of a mapping from types `'t` to types `M('t)`, and a mapping from functions <code>f:C1&rarr;C2</code> to functions <code>M(f):M(C1)&rarr;M(C2)</code>. This is also known as <code>lift<sub>M</sub> f</code> for `M`, and is pronounced "function f lifted into the monad M." For example, where `M` is the list monad, `M` maps every type `'t` into the type `'t list`, and maps every function <code>f:x&rarr;y</code> into the function that maps `[x1,x2...]` to `[y1,y2,...]`.
+

 
 

+In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question.

 
 

+A "monadic value" is any member of a type M('t), for any type 't. For example, a list is a monadic value for the list monad. We can think of these monadic values as the result of applying some function <code>(&phi; : F('t) &rarr; M(F'('t)))</code> to an argument `a` of type `F('t)`.

 
 
 
 

-A natural transformation t assigns to each type C1 in <b>C</b> a morphism t[C1]: C1&rarr;M(C1) such that, for every f:C1&rarr;C2:
-       t[C2] &#8728; f = M(f) &#8728; t[C1]
+Let `'t` be a type variable, and `F` and `F'` be functors, and let `phi` be a polymorphic function that takes arguments of type `F('t)` and yields results of type `MF'('t)` in the monad `M`. An example with `M` being the list monad:

 
 

-The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2).
+<pre>
+       let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
+</pre>

 
 

+Here phi is defined when `'t = 't1*'t2`, `F('t1*'t2) = char * 't1 * 't2`, and `F'('t1 * 't2) = int * 't1 * 't2`.

 
 
 
 

-In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question.
+Now where `gamma` is another function into monad `M` of type <code>F'('t) &rarr; MG'('t)</code>, we define:
+
+<pre>
+       gamma =<< phi a  =def. ((join G') -v- (M gamma)) (phi a)
+
+                        = ((join G') -v- (M gamma) -v- phi) a
+                                        = (gamma <=< phi) a
+</pre>
+
+Hence:
+
+<pre>
+       gamma <=< phi = fun a -> (gamma =<< phi a)
+</pre>
+
+`gamma =<< phi a` is called the operation of "binding" the function gamma to the monadic value `phi a`, and is usually written as `phi a >>= gamma`.
+
+With these definitions, our monadic laws become:
+
+
+<pre>
+       Where phi is a polymorphic function from type F('t) -> M F'('t)
+       and gamma is a polymorphic function from type G('t) -> M G' ('t)
+       and rho is a polymorphic function from type R('t) -> M R' ('t)
+       and F' = G and G' = R, 
+       and a ranges over values of type F('t) for some type 't,
+       and b ranges over values of type G('t):
+
+             (i) &gamma; <=< &phi; is defined,
+                         and is a natural transformation from F to MG'
+       ==>
+               (i'') fun a -> gamma =<< phi a is defined,
+                         and is a function from type F('t) -> M G' ('t)
+
+
+
+            (ii) (&rho; <=< &gamma;) <=< &phi;  =  &rho; <=< (&gamma; <=< &phi;)
+       ==>
+                         (fun a -> (rho <=< gamma) =<< phi a)  =  (fun a -> rho =<< (gamma <=< phi) a)
+                         (fun a -> (fun b -> rho =<< gamma b) =<< phi a)  =  (fun a -> rho =<< (gamma =<< phi a))
+
+          (ii'') (fun b -> rho =<< gamma b) =<< phi a  =  rho =<< (gamma =<< phi a)
+
+
+
+         (iii.1) (unit G') <=< &gamma;  =  &gamma;
+                 when &gamma; is a natural transformation from some FG' to MG'
+
+                         (unit G') <=< gamma  =  gamma
+                         when gamma is a function of type FQ'('t) -> M G'('t)
+
+                         fun b -> (unit G') =<< gamma b  =  gamma
+
+                         (unit G') =<< gamma b  =  gamma b
+
+                         As below, return will map arguments c of type G'('t)
+                         to the monadic value (unit G') b, of type M G'('t).
+
+       (iii.1'') return =<< gamma b  =  gamma b
+
+
+
+         (iii.2) &gamma;  =  &gamma; <=< (unit G)
+                 when &gamma; is a natural transformation from G to some MR'G
+       ==>
+                         gamma  =  gamma <=< (unit G)
+                         when gamma is a function of type G('t) -> M R' G('t)
+
+                         gamma  =  fun b -> gamma =<< ((unit G) b)
+
+                         Let return be a polymorphic function mapping arguments
+                         of any type 't to M('t). In particular, it maps arguments
+                         b of type G('t) to the monadic value (unit G) b, of
+                         type M G('t).
+
+                         gamma  =  fun b -> gamma =<< return b
+
+       (iii.2'') gamma b  =  gamma =<< return b
+</pre>
+
+Summarizing (ii''), (iii.1''), (iii.2''), these are the monadic laws as usually stated in the functional programming literature:
+
+*      `fun b -> rho =<< gamma b) =<< phi a  =  rho =<< (gamma =<< phi a)`
+
+       Usually written reversed, and with a monadic variable `u` standing in
+       for `phi a`:
+
+       `u >>= (fun b -> gamma b >>= rho)  =  (u >>= gamma) >>= rho`

 
 

-For an example of the latter, let &phi; be a function that takes arguments of some (schematic, polymorphic) type C1 and yields results of some (schematic, polymorphic) type M(C2). An example with M being the list monad, and C2 being the tuple type schema int * C1:
+*      `return =<< gamma b  =  gamma b`

 
 

-       let &phi; = fun c &rarr; [(1,c), (2,c)]
+       Usually written reversed, and with `u` standing in for `phi a`:

 
 

-&phi; is polymorphic: when you apply it to the int 0 you get a result of type "list of int * int": [(1,0), (2,0)]. When you apply it to the char 'e' you get a result of type "list of int * char": [(1,'e'), (2,'e')].
+       `u >>= return  =  u`

 
 

-However, to keep things simple, we'll work instead with functions whose type is settled. So instead of the polymorphic &phi;, we'll work with (&phi; : C1 &rarr; M(int * C1)). This only accepts arguments of type C1. For generality, I'll talk of functions with the type (&phi; : C1 &rarr; M(C1')), where we assume that C1' is a function of C1.
+*      `gamma b  =  gamma =<< return b`

 
 

-A "monadic value" is any member of a type M(C1), for any type C1. For example, a list is a monadic value for the list monad. We can think of these monadic values as the result of applying some function (&phi; : C1 &rarr; M(C1')) to an argument of type C1.
+       Usually written reversed:

 
 

+       return b >>= gamma  =  gamma b
+