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=7b52c3a6fe699400241714549176eb154ada8439;hp=46c575cccc8671e46f1595239f43d0399792ba95;hb=50e06b4a50c0fcdc84f5cc94773316935871ceb1;hpb=b40dafe674003107ac10de2a66c3679b50dd9db2
diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 46c575cc..7b52c3a6 100644
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -24,8 +24,8 @@ A **monoid** is a structure (S,⋆,z)
consisting of an associat
for all s1, s2, s3 in S: - (i) s1⋆s2 etc are also in S - (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3) + (i) s1⋆s2 etc are also in S + (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3) (iii) z⋆s1 = s1 = s1⋆z@@ -45,13 +45,15 @@ 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 ∘ 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 ∘ 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 + morphism 1E, which is such that for every morphism f:C1→C2: + 1C2 ∘ f = f = f ∘ 1C1These 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.