From 5b391a18cbbaa7234a3f84e47bb8cc8ac0babc01 Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Tue, 2 Nov 2010 08:24:59 -0400 Subject: [PATCH] cat theory tweaks Signed-off-by: Jim Pryor --- advanced_topics/monads_in_category_theory.mdwn | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 53a66112..66625766 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -19,22 +19,22 @@ corrections. Monoids ------- -A **monoid** is a structure `(S, *, z)` consisting of an associative binary operation `*` over some set `S`, which is closed under `*`, and which contains an identity element `z` for `*`. That is: +A **monoid** is a structure (S,⋆,z) consisting of an associative binary operation ⋆ over some set `S`, which is closed under ⋆, and which contains an identity element `z` for ⋆. That is: 
 	for all s1, s2, s3 in S:
-	(i) s1*s2 etc are also in S
-	(ii) (s1*s2)*s3 = s1*(s2*s3)
-	(iii) z*s1 = s1 = s1*z
+	(i) s1⋆s2 etc are also in S
+	(ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
+	(iii) z⋆s1 = s1 = s1⋆z
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` -* 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. +* 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` +* 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. Categories ---------- @@ -59,7 +59,7 @@ Some examples of categories are: * Categories whose elements are sets and whose morphisms are functions between those sets. Here the source and target of a function are its domain and range, so distinct functions sharing a domain and range (e.g., sin and cos) are distinct morphisms between the same source and target elements. The identity morphism for any element/set is just the identity function for that set. -* 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`. +* 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: -- 2.11.0