From: Jim Pryor Date: Tue, 2 Nov 2010 12:07:13 +0000 (-0400) Subject: cat theory tweaks X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=c801371a433f178c07249aef38b74b7e38347280 cat theory tweaks Signed-off-by: Jim Pryor --- diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index dc9dc959..37976ea2 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -44,11 +44,11 @@ When a morphism `f` in category **C** has source `C1` and target `C2`, we'll wri To have a category, the elements and morphisms have to satisfy some constraints: -
-(i) the class of morphisms has to be closed under composition: where `f:C1->C2` and `g:C2->C3`, `g o f` is also a morphism of the category, which maps `C1->C3`.

-(ii) composition of morphisms has to be associative

-(iii) every element `E` of the category has to have an identity morphism 1E, which is such that for every morphism `f:C1->C2`: 1C2 o f = f = f o 1C1
-
+
+	(i) the class of morphisms has to be closed under composition: where `f:C1->C2` and `g:C2->C3`, `g o f` is also a morphism of the category, which maps `C1->C3`.
+	(ii) composition of morphisms has to be associative
+	(iii) every element `E` of the category has to have an identity morphism 1E, which is such that for every morphism `f:C1->C2`: 1C2 o f = f = f o 1C1
+
These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `E` and itself; they need not all be identity morphisms. Indeed from (iii) it follows that each element can have only a single identity morphism.