From 7b00a5a3dc3f7208f67ed5c87faf22b351e14b0c Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Tue, 2 Nov 2010 09:06:40 -0400 Subject: [PATCH] cat theory tweaks Signed-off-by: Jim Pryor --- advanced_topics/monads_in_category_theory.mdwn | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 044d9733..a6165e9d 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -194,7 +194,7 @@ We also need to designate for `M` a "join" transformation, which is a natural tr These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion. -Let `φ` and `γ` be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now `(M γ)` will also be a natural transformation, formed by composing the functor `M` with the natural transformation `γ`. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, `(M γ)`, and `φ`, and abbreviate it as follows: +Let `φ` and `γ` be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now `(M γ)` will also be a natural transformation, formed by composing the functor `M` with the natural transformation `γ`. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, `(M γ)`, and `φ`, and abbreviate it as follows:
``` 	γ <=< φ  =def.  ((join G') -v- (M γ) -v- φ)
@@ -204,20 +204,20 @@ Since composition is associative I don't specify the order of composition on the

In other words, `<=<` is a binary operator that takes us from two members `φ` and `γ` of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes its written `φ >=> γ` where that's the same as `γ <=< φ`.)

-φ is a transformation from `F` to `MF'` which = `MG`; `(M γ)` is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite γ <=< φ will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`.
+`φ` is a transformation from `F` to `MF'` which = `MG`; `(M γ)` is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite `γ <=< φ` will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`.

Now we can specify the "monad laws" governing a monad as follows:

(T, <=<, unit) constitute a monoid

-That's it. (Well, perhaps we're cheating a bit, because γ <=< φ isn't fully defined on `T`, but only when `F` is a functor to `MF'` and `G` is a functor from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied:
+That's it. Well, there may be a wrinkle here. I don't know whether the definition of a monoid requires the operation to be defined for every pair in its set. In the present case, `γ <=< φ` isn't fully defined on `T`, but only when `F` is a functor to `MF'` and `G` is a functor from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied:

-	(i) γ <=< φ is also in T
-	(ii) (ρ <=< γ) <=< φ  =  ρ <=< (γ <=< φ)
+	    (i) γ <=< φ is also in T
+	   (ii) (ρ <=< γ) <=< φ  =  ρ <=< (γ <=< φ)
(iii.1) unit <=< φ  =  φ                 (here φ has to be a natural transformation to M(1C))
(iii.2)                φ  =  φ <=< unit  (here φ has to be a natural transformation from 1C)

-If φ is a natural transformation from `F` to `M(1C)` and γ is `(φ G')`, that is, a natural transformation from `PG` to `MG`, then we can extend (iii.1) as follows:
+If `φ` is a natural transformation from `F` to `M(1C)` and `γ` is `(φ G')`, that is, a natural transformation from `FG` to `MG`, then we can extend (iii.1) as follows:

γ = (φ G')
= ((unit <=< φ) G')
@@ -227,7 +227,7 @@ If φ is a natural transformation from `F` to `M(1C)` and γ is `(φ
??
= (unit G') <=< γ

-where as we said γ is a natural transformation from some `PG'` to `MG'`.
+where as we said `γ` is a natural transformation from some `FG'` to `MG'`.

Similarly, if φ is a natural transformation from `1C` to `MF'`, and γ is `(φ G)`, that is, a natural transformation from `G` to `MF'G`, then we can extend (iii.2) as follows:

--
2.11.0

```