X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=advanced_topics%2Fmonads_in_category_theory.mdwn;h=8a21eb2f803576198bbf70c1e1b18e29f8c341fb;hp=0c139c6b1b5901b3b69c0578747b23b8a3814409;hb=13e875628b49325490f5b187994ae3f6580a6da1;hpb=9eef3614ecbf18d8f4713ec5c8eec4674ef65c4a diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 0c139c6b..8a21eb2f 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -21,12 +21,13 @@ 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: -
+
+
 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`
-
+ Some examples of monoids are: @@ -44,8 +45,8 @@ When a morphism `f` in category **C** has source `C1` and target `C2`, we'll wri 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`.
-(ii) composition of morphisms has to be associative
+(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