From: Jim Pryor Date: Tue, 2 Nov 2010 02:02:28 +0000 (-0400) Subject: tweak cat theory X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=57439ef74347e5ca607d9b717f20e888f879ef87;hp=3a4225f3e0df9583725eac59035ffd6edb53d888 tweak cat theory Signed-off-by: Jim Pryor --- diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 6ebcd8d6..173e9868 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -11,7 +11,7 @@ functional programming uses of these notions in enough detail to be sure that none of the pieces here is misguided. In particular, it wasn't completely obvious how to map the polymorphism on the programming theory side into the category theory. And I'm bothered by the fact that our `<=<` operation is only -partly defined on our domain of natural transformations. But these do seem to +partly defined on our domain of natural transformations. But this does seem to me to be the reasonable way to put the pieces together. We very much welcome feedback from anyone who understands these issues better, and will make corrections. @@ -176,7 +176,7 @@ That's it. (Well, perhaps we're cheating a bit, because `q <=< p` isn't fully de (i) q <=< p is also in T (ii) (r <=< q) <=< p = r <=< (q <=< p) - (iii.1) unit <=< p = p (here p has to be a natural transformation to M(1C)) + (iii.1) unit <=< p = p (here p has to be a natural transformation to M(1C)) (iii.2) p = p <=< unit (here p has to be a natural transformation from 1C) If `p` is a natural transformation from `P` to `M(1C)` and `q` is `(p Q')`, that is, a natural transformation from `PQ` to `MQ`, then we can extend (iii.1) as follows: @@ -186,6 +186,7 @@ If `p` is a natural transformation from `P` to `M(1C)` and `q` is `(p Q')`, that = ((join -v- (M unit) -v- p) Q') = (join Q') -v- ((M unit) Q') -v- (p Q') = (join Q') -v- (M (unit Q')) -v- q + ?? = (unit Q') <=< q where as we said `q` is a natural transformation from some `PQ'` to `MQ'`. @@ -194,9 +195,10 @@ Similarly, if `p` is a natural transformation from `1C` to `MP'`, and `q` is `(p q = (p Q) = ((p <=< unit) Q) - = (((join P') (M p) unit) Q) - = ((join P'Q) ((M p) Q) (unit Q)) - = ? + = (((join P') -v- (M p) -v- unit) Q) + = ((join P'Q) -v- ((M p) Q) -v- (unit Q)) + = ((join P'Q) -v- (M (p Q)) -v- (unit Q)) + ?? = q <=< (unit Q) where as we said `q` is a natural transformation from `Q` to some `MP'Q`. @@ -206,7 +208,7 @@ where as we said `q` is a natural transformation from `Q` to some `MP'Q`. 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 `<=<`. (* P2. every element c1 of a category C has an identity morphism id[c1] such that for every morphism f:c1->c2 in C: id[c2] o f = f = f o id[c1].