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=a6165e9deae3a47b9401f946e8c3087d85a33cdb;hp=044d97332b942d86762e6ba2becf7dda514a8d2b;hb=7b00a5a3dc3f7208f67ed5c87faf22b351e14b0c;hpb=06d07bead9501b9450d8c7a1c30926a83385c38a
diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 044d9733..a6165e9d 100644
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -194,7 +194,7 @@ We also need to designate for `M` a "join" transformation, which is a natural tr
These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion.
-Let φ
and γ
be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now `(M γ)` will also be a natural transformation, formed by composing the functor `M` with the natural transformation γ
. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, (M γ)
, and φ
, and abbreviate it as follows:
+Let φ
and γ
be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now (M γ)
will also be a natural transformation, formed by composing the functor `M` with the natural transformation γ
. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, (M γ)
, and φ
, and abbreviate it as follows:
γ <=< φ =def. ((join G') -v- (M γ) -v- φ) @@ -204,20 +204,20 @@ Since composition is associative I don't specify the order of composition on the In other words, `<=<` is a binary operator that takes us from two membersφ
andγ
of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes its writtenφ >=> γ
where that's the same asγ <=< φ
.) -φ is a transformation from `F` to `MF'` which = `MG`; `(M γ)` is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite γ <=< φ will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`. +φ
is a transformation from `F` to `MF'` which = `MG`;(M γ)
is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the compositeγ <=< φ
will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`. Now we can specify the "monad laws" governing a monad as follows: (T, <=<, unit) constitute a monoid -That's it. (Well, perhaps we're cheating a bit, because γ <=< φ isn't fully defined on `T`, but only when `F` is a functor to `MF'` and `G` is a functor from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied: +That's it. Well, there may be a wrinkle here. I don't know whether the definition of a monoid requires the operation to be defined for every pair in its set. In the present case,γ <=< φ
isn't fully defined on `T`, but only when `F` is a functor to `MF'` and `G` is a functor from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied: - (i) γ <=< φ is also in T - (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) + (i) γ <=< φ is also in T + (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) (iii.1) unit <=< φ = φ (here φ has to be a natural transformation to M(1C)) (iii.2) φ = φ <=< unit (here φ has to be a natural transformation from 1C) -If φ is a natural transformation from `F` to `M(1C)` and γ is `(φ G')`, that is, a natural transformation from `PG` to `MG`, then we can extend (iii.1) as follows: +Ifφ
is a natural transformation from `F` to `M(1C)` andγ
is(φ G')
, that is, a natural transformation from `FG` to `MG`, then we can extend (iii.1) as follows: γ = (φ G') = ((unit <=< φ) G') @@ -227,7 +227,7 @@ If φ is a natural transformation from `F` to `M(1C)` and γ is `(φ ?? = (unit G') <=< γ -where as we said γ is a natural transformation from some `PG'` to `MG'`. +where as we saidγ
is a natural transformation from some `FG'` to `MG'`. Similarly, if φ is a natural transformation from `1C` to `MF'`, and γ is `(φ G)`, that is, a natural transformation from `G` to `MF'G`, then we can extend (iii.2) as follows: