From: Jim Pryor Date: Tue, 2 Nov 2010 12:16:56 +0000 (-0400) Subject: cat theory tweaks X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=5bda5b5a2df9af3e0b46179854b370819f14ced8 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 a5e6f979..f26e2cd6 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -45,7 +45,7 @@ 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: where f:C1→C2 and g:C2→C3, g o 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 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
@@ -61,12 +61,12 @@ Some examples of categories are: * any monoid `(S,*,z)` generates a category with a single element `x`; this `x` need not have any relation to `S`. The members of `S` play the role of *morphisms* of this category, rather than its elements. All of these morphisms are understood to map `x` to itself. The result of composing the morphism consisting of `s1` with the morphism `s2` is the morphism `s3`, where `s3=s1*s2`. The identity morphism for the (single) category element `x` is the monoid's identity `z`. -* a **preorder** is a structure `(S, <=)` consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `x`,`y` of `S` such that neither `x<=y` nor `y<=x`). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that `s1<=s2` and `s2<=s1` but `s1` and `s2` are not identical). Some examples: +* a **preorder** is a structure `(S, ≤)` consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `x`,`y` of `S` such that neither `x≤y` nor `y≤x`). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that `s1≤s2` and `s2≤s1` but `s1` and `s2` are not identical). Some examples: * sentences ordered by logical implication ("p and p" implies and is implied by "p", but these sentences are not identical; so this illustrates a pre-order without anti-symmetry) * sets ordered by size (this illustrates it too) - Any pre-order `(S,<=)` generates a category whose elements are the members of `S` and which has only a single morphism between any two elements `s1` and `s2`, iff `s1<=s2`. + Any pre-order (S,≤) generates a category whose elements are the members of `S` and which has only a single morphism between any two elements `s1` and `s2`, iff s1≤s2. Functors