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=fe12655bac9e24310213d1616279771202565bd0;hp=044d97332b942d86762e6ba2becf7dda514a8d2b;hb=2f14eb20bd2428f346f8d5c8caa35b67eb043096;hpb=06d07bead9501b9450d8c7a1c30926a83385c38a
diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 044d9733..fe12655b 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)
@@ -202,23 +202,29 @@ Let φ and γ be members of `T`, that is they
Since composition is associative I don't specify the order of composition on the rhs.
-In other words, `<=<` is a binary operator that takes us from two members φ and γ of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes its written φ >=> γ where that's the same as γ <=< φ.)
+In other words, `<=<` is a binary operator that takes us from two members φ and γ of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes it's written φ >=> γ where that's the same as γ <=< 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'`, where the latter = `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`.
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 φ is a transformation to some `MF'` and γ is a transformation 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:
+If φ is a natural transformation from `F` to `M(1C)` and γ 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')
= ((join -v- (M unit) -v- φ) G')
@@ -226,11 +232,13 @@ If φ is a natural transformation from `F` to `M(1C)` and γ is `(φ
= (join G') -v- (M (unit G')) -v- γ
??
= (unit G') <=< γ
+
-where as we said γ is a natural transformation from some `PG'` to `MG'`.
+where as we said γ 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:
+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:
+
γ = (φ G)
= ((φ <=< unit) G)
= (((join F') -v- (M φ) -v- unit) G)
@@ -238,8 +246,9 @@ Similarly, if φ is a natural transformation from `1C` to `MF'`, and γ
= ((join F'G) -v- (M (φ G)) -v- (unit G))
??
= γ <=< (unit G)
+
-where as we said γ is a natural transformation from `G` to some `MF'G`.
+where as we said γ is a natural transformation from `G` to some `MF'G`.
@@ -248,60 +257,99 @@ The standard category-theory presentation of the monad laws
-----------------------------------------------------------
In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`.
-(*
+
Let's remind ourselves of some principles:
- * composition of morphisms, functors, and natural compositions is associative
- * functors "distribute over composition", that is for any morphisms f and g in F's source category: F(g ∘ f) = F(g) ∘ F(f)
- * if η is a natural transformation from F to G, then for every f:C1→C2 in F and G's source category C: η[C2] ∘ F(f) = G(f) ∘ η[C1].
+* composition of morphisms, functors, and natural compositions is associative
+
+* functors "distribute over composition", that is for any morphisms `f` and `g` in `F`'s source category: F(g ∘ f) = F(g) ∘ F(f)
+
+* if η is a natural transformation from `F` to `G`, then for every f:C1→C2 in `F` and `G`'s source category C: η[C2] ∘ F(f) = G(f) ∘ η[C1].
Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
-Recall that join is a natural transformation from the (composite) functor MM to M. So for elements C1 in C, join[C1] will be a morphism from MM(C1) to M(C1). And for any morphism f:a→b in C:
+Recall that join is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` in C, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism f:C1→C2 in C:
- (1) join[b] ∘ MM(f) = M(f) ∘ join[a]
+
+ (1) join[C2] ∘ MM(f) = M(f) ∘ join[C1]
+
+
+Next, consider the composite transformation ((join MG') -v- (MM γ)).
+
+* γ is a transformation from `G` to `MG'`, and assigns elements `C1` in C a morphism γ\*: G(C1) → MG'(C1). (MM γ) is a transformation that instead assigns `C1` the morphism MM(γ\*).
+
+* `(join MG')` is a transformation from `MMMG'` to `MMG'` that assigns `C1` the morphism `join[MG'(C1)]`.
+
+Composing them:
+
+
+ (2) ((join MG') -v- (MM γ)) assigns to `C1` the morphism join[MG'(C1)] ∘ MM(γ*).
+
-Next, consider the composite transformation ((join MG') -v- (MM γ)).
- γ is a transformation from G to MG', and assigns elements C1 in C a morphism γ*: G(C1) → MG'(C1). (MM γ) is a transformation that instead assigns C1 the morphism MM(γ*).
- (join MG') is a transformation from MMMG' to MMG' that assigns C1 the morphism join[MG'(C1)].
- Composing them:
- (2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*).
+Next, consider the composite transformation ((M γ) -v- (join G)).
-Next, consider the composite transformation ((M γ) -v- (join G)).
+
(3) This assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
+
-So for every element C1 of C:
+So for every element `C1` of C:
+
+
((join MG') -v- (MM γ))[C1], by (2) is:
join[MG'(C1)] ∘ MM(γ*), which by (1), with f=γ*: G(C1)→MG'(C1) is:
M(γ*) ∘ join[G(C1)], which by 3 is:
((M γ) -v- (join G))[C1]
+
+
+So our **(lemma 1)** is:
+
+
+ ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), where γ is a transformation from G to MG'.
+
-So our (lemma 1) is: ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), where γ is a transformation from G to MG'.
+Next recall that unit is a natural transformation from `1C` to `M`. So for elements `C1` in C, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism f:a→b in C:
-Next recall that unit is a natural transformation from 1C to M. So for elements C1 in C, unit[C1] will be a morphism from C1 to M(C1). And for any morphism f:a→b in C:
+
(4) unit[b] ∘ f = M(f) ∘ unit[a]
+
+
+Next consider the composite transformation ((M γ) -v- (unit G)):
+
+
+ (5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
+
-Next consider the composite transformation ((M γ) -v- (unit G)). (5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
+Next consider the composite transformation ((unit MG') -v- γ).
-Next consider the composite transformation ((unit MG') -v- γ). (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
+
+ (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
+
So for every element C1 of C:
+
+
((M γ) -v- (unit G))[C1], by (5) =
M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*: G(C1)→MG'(C1) is:
unit[MG'(C1)] ∘ γ*, which by (6) =
((unit MG') -v- γ)[C1]
+
+
+So our **(lemma 2)** is:
-So our lemma (2) is: (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), where γ is a transformation from G to MG'.
+
+ (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), where γ is a transformation from G to MG'.
+
-Finally, we substitute ((join G') -v- (M γ) -v- φ) for γ <=< φ in the monad laws. For simplicity, I'll omit the "-v-".
+Finally, we substitute ((join G') -v- (M γ) -v- φ) for γ <=< φ in the monad laws. For simplicity, I'll omit the "-v-".
+
for all φ,γ,ρ in T, where φ is a transformation from F to MF', γ is a transformation from G to MG', R is a transformation from R to MR', and F'=G and G'=R:
(i) γ <=< φ etc are also in T
@@ -370,10 +418,12 @@ Finally, we substitute ((join G') -v- (M γ) -v- φ) for γ <=< &ph
which will in turn be true just in case:
(iii.2') (join (unit M)) = the identity transformation
+
Collecting the results, our monad laws turn out in this format to be:
+
when φ a transformation from F to MF', γ a transformation from F' to MG', ρ a transformation from G' to MR' all in T:
(i') ((join G') (M γ) φ) etc also in T
@@ -383,6 +433,7 @@ Collecting the results, our monad laws turn out in this format to be:
(iii.1') (join (M unit)) = the identity transformation
(iii.2')(join (unit M)) = the identity transformation
+