X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=inline;f=advanced_topics%2Fmonads_in_category_theory.mdwn;h=29b6feb3e250ad195f8884215570afa5fe639ecf;hb=c4eb20ae862369e97cadef43183d0663f3eddd11;hp=3e34f85b8e2841bacb7675190bd3e30ed77b425d;hpb=7026bf51a9211af0586d30d49b61ed187f897a79;p=lambda.git diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 3e34f85b..29b6feb3 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -121,13 +121,13 @@ Consider four categories B, C, D, and E. Let `F` be
- B -+ +--- C --+ +---- D -----+ +-- E -- | | | | | | - F: -----→ G: -----→ K: -----→ - | | | | | η | | | ψ + F: ------> G: ------> K: ------> + | | | | | η | | | ψ | | | | v | | v - | | H: -----→ L: -----→ - | | | | | φ | | + | | H: ------> L: ------> + | | | | | φ | | | | | | v | | - | | J: -----→ | | + | | J: ------> | | -----+ +--------+ +------------+ +-------@@ -154,7 +154,7 @@ by naturalness of
η
, is:
φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1]
-Hence, we can define (φ -v- η)[x]
as: φ[x] ∘ η[x]
and rely on it to satisfy the constraints for a natural transformation from `G` to `J`:
+Hence, we can define (φ -v- η)[\_]
as: φ[\_] ∘ η[\_]
and rely on it to satisfy the constraints for a natural transformation from `G` to `J`:
(φ -v- η)[C2] ∘ G(f) = J(f) ∘ (φ -v- η)[C1] @@ -173,7 +173,7 @@ I'll assert without proving that vertical composition is associative and has an(φ -h- η)[C1] = L(η[C1]) ∘ ψ[G(C1)] - = ψ[H(C1)] ∘ K(η[C1]) + = ψ[H(C1)] ∘ K(η[C1])Horizontal composition is also associative, and has the same identity as vertical composition.