From: Jim Pryor Date: Tue, 2 Nov 2010 12:29:02 +0000 (-0400) Subject: cat theory tweaks X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=50e06b4a50c0fcdc84f5cc94773316935871ceb1;ds=sidebyside 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 1590619a..7b52c3a6 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -24,8 +24,8 @@ A **monoid** is a structure `(S,⋆,z)` consisting of an associat
``` 	for all s1, s2, s3 in S:
-	(i) s1⋆s2 etc are also in S
-	(ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
+	  (i) s1⋆s2 etc are also in S
+	 (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
(iii) z⋆s1 = s1 = s1⋆z
```
@@ -45,15 +45,15 @@ When a morphism `f` in category C has source `C1` and target `C2`, we'll To have a category, the elements and morphisms have to satisfy some constraints:
```-	(i)   the class of morphisms has to be closed under composition:
+	  (i) the class of morphisms has to be closed under composition:
where f:C1→C2 and g:C2→C3, g ∘ f is also a
morphism of the category, which maps C1→C3.

-	(ii)  composition of morphisms has to be associative
+	 (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 ∘ f = f = f ∘ 1C1
+	      morphism 1E, which is such that for every morphism f:C1→C2:
+	      1C2 ∘ f = f = f ∘ 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.