From b40dafe674003107ac10de2a66c3679b50dd9db2 Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Tue, 2 Nov 2010 08:27:07 -0400 Subject: [PATCH] cat theory tweaks Signed-off-by: Jim Pryor --- advanced_topics/monads_in_category_theory.mdwn | 12 ++++++++---- 1 file changed, 8 insertions(+), 4 deletions(-) diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 66625766..46c575cc 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -32,7 +32,7 @@ A **monoid** is a structure (S,⋆,z) consisting of an associat Some examples of monoids are: * finite strings of an alphabet `A`, with ⋆ being concatenation and `z` being the empty string -* all functions `X→X` over a set `X`, with ⋆ being composition and `z` being the identity function over `X` +* all functions X→X over a set `X`, with ⋆ being composition and `z` being the identity function over `X` * the natural numbers with ⋆ being plus and `z` being `0` (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.) * if we let ⋆ be multiplication and `z` be `1`, we get different monoids over the same sets as in the previous item. @@ -40,14 +40,18 @@ Categories ---------- A **category** is a generalization of a monoid. A category consists of a class of **elements**, and a class of **morphisms** between those elements. Morphisms are sometimes also called maps or arrows. They are something like functions (and as we'll see below, given a set of functions they'll determine a category). However, a single morphism only maps between a single source element and a single target element. Also, there can be multiple distinct morphisms between the same source and target, so the identity of a morphism goes beyond its "extension." -When a morphism `f` in category C has source `C1` and target `C2`, we'll write `f:C1→C2`. +When a morphism `f` in category C has source `C1` and target `C2`, we'll write f:C1→C2. 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 ∘ f is also a morphism of the category, which maps C1→C3.
+	(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
-	(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
+	(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
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. -- 2.11.0