From: Jim Pryor Date: Tue, 2 Nov 2010 12:32:56 +0000 (-0400) Subject: cat theory tweaks X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=3ca93ad8ccdf0572c3b803b86ea68dd2ad21a5f2 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 6b2eb805..8c5f4cd9 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -80,10 +80,17 @@ Functors A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category C to category D must:
```-	(i) associate with every element C1 of C an element F(C1) of D
-	(ii) associate with every morphism f:C1→C2 of C a morphism F(f):F(C1)→F(C2) of D
-	(iii) "preserve identity", that is, for every element C1 of C: F of C1's identity morphism in C must be the identity morphism of F(C1) in D: F(1C1) = 1F(C1).
-	(iv) "distribute over composition", that is for any morphisms f and g in C: F(g ∘ f) = F(g) ∘ F(f)
+	  (i) associate with every element C1 of C an element F(C1) of D
+
+	 (ii) associate with every morphism f:C1→C2 of C a morphism
+	      F(f):F(C1)→F(C2) of D
+
+	(iii) "preserve identity", that is, for every element C1 of C:
+	      F of C1's identity morphism in C must be the identity morphism
+		  of F(C1) in D: F(1C1) = 1F(C1).
+
+	 (iv) "distribute over composition", that is for any morphisms f and g in C:
+	      F(g ∘ f) = F(g) ∘ F(f)
```
A functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism of C to itself is denoted `1C`.