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=74b5d5727bd204dc6159e0701a98c8f6ecfde055;hb=13e875628b49325490f5b187994ae3f6580a6da1;hpb=8b1c423b50e7c2801bc9c69772261940fc6a7854 diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 74b5d572..8a21eb2f 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -21,28 +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: -
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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`
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Some examples of monoids are:
@@ -60,8 +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