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=f05747f675b416fa5d5af01dee2d0f0278258874;hp=768b8542a6e22019ce98a8f09fda446bd537145c;hb=704abdbe4afe78e553738662a3cb0bb6944b54ff;hpb=636bebd66c03835ae35ec9b38b82485c4ae6460b
diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 768b8542..f05747f6 100644
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -33,8 +33,8 @@ Some examples of monoids are:
* finite strings of an alphabet `A`, with ⋆
being concatenation and `z` being the empty string
* all functions X→X
over a set `X`, with ⋆
being composition and `z` being the identity function over `X`
-* the natural numbers with ⋆
being plus and `z` being `0` (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.)
-* if we let ⋆
be multiplication and `z` be `1`, we get different monoids over the same sets as in the previous item.
+* the natural numbers with ⋆
being plus and `z` being 0 (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.
+* if we let ⋆
be multiplication and `z` be 1, we get different monoids over the same sets as in the previous item.
Categories
----------
@@ -58,7 +58,7 @@ To have a category, the elements and morphisms have to satisfy some constraints:
These 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.
-A good intuitive picture of a category is as a generalized directed graph, where the category elements are the graph's nodes, and there can be multiple directed edges between a given pair of nodes, and nodes can also have multiple directed edges to themselves. (Every node must have at least one such, which is that node's identity morphism.)
+A good intuitive picture of a category is as a generalized directed graph, where the category elements are the graph's nodes, and there can be multiple directed edges between a given pair of nodes, and nodes can also have multiple directed edges to themselves. Morphisms correspond to directed paths of length ≥ 0 in the graph.
Some examples of categories are:
@@ -67,7 +67,7 @@ Some examples of categories are:
* any monoid (S,⋆,z)
generates a category with a single element `x`; this `x` need not have any relation to `S`. The members of `S` play the role of *morphisms* of this category, rather than its elements. All of these morphisms are understood to map `x` to itself. The result of composing the morphism consisting of `s1` with the morphism `s2` is the morphism `s3`, where s3=s1⋆s2
. The identity morphism for the (single) category element `x` is the monoid's identity `z`.
-* a **preorder** is a structure (S, ≤)
consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `x`,`y` of `S` such that neither x≤y
nor y≤x
). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that s1≤s2
and s2≤s1
but `s1` and `s2` are not identical). Some examples:
+* a **preorder** is a structure (S, ≤)
consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `s1`,`s2` of `S` such that neither s1≤s2
nor s2≤s1
). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that s1≤s2
and s2≤s1
but `s1` and `s2` are not identical). Some examples:
* sentences ordered by logical implication ("p and p" implies and is implied by "p", but these sentences are not identical; so this illustrates a pre-order without anti-symmetry)
* sets ordered by size (this illustrates it too)
@@ -136,7 +136,7 @@ Then (η F)
is a natural transformation from the (composite) fun
And (K η)
is a natural transformation from the (composite) functor `KG` to the (composite) functor `KH`, such that where `C1` is an element of category C, (K η)[C1] = K(η[C1])
---that is, the morphism in E that `K` assigns to the morphism η[C1]
of D.
-(φ -v- η)
is a natural transformation from `G` to `J`; this is known as a "vertical composition". We will rely later on this, where f:C1→C2
:
+(φ -v- η)
is a natural transformation from `G` to `J`; this is known as a "vertical composition". For any morphism f:C1→C2
in C:
φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1] @@ -186,70 +186,90 @@ In earlier days, these were also called "triples." A **monad** is a structure consisting of an (endo)functor `M` from some category C to itself, along with some natural transformations, which we'll specify in a moment. -Let `T` be a set of natural transformationsφ
, each being between some (variable) functor `F` and another functor which is the composite `MF'` of `M` and a (variable) functor `F'`. That is, for each element `C1` in C,φ
assigns `C1` a morphism from element `F(C1)` to element `MF'(C1)`, satisfying the constraints detailed in the previous section. For different members of `T`, the relevant functors may differ; that is,φ
is a transformation from functor `F` to `MF'`,γ
is a transformation from functor `G` to `MG'`, and none of `F`, `F'`, `G`, `G'` need be the same. +Let `T` be a set of natural transformationsφ
, each being between some arbitrary endofunctor `F` on C and another functor which is the composite `MF'` of `M` and another arbitrary endofunctor `F'` on C. That is, for each element `C1` in C,φ
assigns `C1` a morphism from element `F(C1)` to element `MF'(C1)`, satisfying the constraints detailed in the previous section. For different members of `T`, the relevant functors may differ; that is,φ
is a transformation from functor `F` to `MF'`,γ
is a transformation from functor `G` to `MG'`, and none of `F`, `F'`, `G`, `G'` need be the same. -One of the members of `T` will be designated the "unit" transformation for `M`, and it will be a transformation from the identity functor `1C` for C to `M(1C)`. So it will assign to `C1` a morphism from `C1` to `M(C1)`. +One of the members of `T` will be designated the `unit` transformation for `M`, and it will be a transformation from the identity functor `1C` for C to `M(1C)`. So it will assign to `C1` a morphism from `C1` to `M(C1)`. -We also need to designate for `M` a "join" transformation, which is a natural transformation from the (composite) functor `MM` to `M`. +We also need to designate for `M` a `join` transformation, which is a natural transformation from the (composite) functor `MM` to `M`. These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion. -Letφ
andγ
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 transformationγ
. 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)
, andφ
, and abbreviate it as follows: +Letφ
andγ
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 transformationγ
. 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)
, andφ
, and abbreviate it as follows. Since composition is associative I don't specify the order of composition on the rhs.γ <=< φ =def. ((join G') -v- (M γ) -v- φ)-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 membersφ
andγ
of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes it's writtenφ >=> γ
where that's the same asγ <=< φ
.)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 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, 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 whenφ
is a transformation to some `MF'` andγ
is a transformation 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 whenφ
is a transformation to some `MF'` andγ
is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws must hold:- (i) γ `<=<` φ is also in T + (i) γ <=< φ is also in T - (ii) (ρ `<=<` γ) `<=<` φ = ρ `<=<` (γ `<=<` φ) + (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) - (iii.1) unit `<=<` φ = φ (here φ has to be a natural transformation to M(1C)) + (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) + (iii.2) ρ = ρ <=< unit + (here ρ has to be a natural transformation from 1C)-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: +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') - = ((join -v- (M unit) -v- φ) G') - = (join G') -v- ((M unit) G') -v- (φ G') - = (join G') -v- (M (unit G')) -v- γ - ?? - = (unit G') `<=<` γ + = ((unit <=< φ) G') + = (((join 1C) -v- (M unit) -v- φ) G') + = (((join 1C) G') -v- ((M unit) G') -v- (φ G')) + = ((join (1C G')) -v- (M (unit G')) -v- γ) + = ((join G') -v- (M (unit G')) -v- γ) + since (unit G') is a natural transformation to MG', + this satisfies the definition for <=<: + = (unit G') <=< γwhere as we saidγ
is a natural transformation from some `FG'` to `MG'`. -Similarly, ifφ
is a natural transformation from `1C` to `MF'`, andγ
is(φ G)
, that is, a natural transformation from `G` to `MF'G`, then we can extend (iii.2) as follows: +Similarly, ifρ
is a natural transformation from `1C` to `MR'`, andγ
is(ρ G)
, that is, a natural transformation from `G` to `MR'G`, then we can extend (iii.2) as follows:- γ = (φ G) - = ((φ `<=<` unit) G) - = (((join F') -v- (M φ) -v- unit) G) - = ((join F'G) -v- ((M φ) G) -v- (unit G)) - = ((join F'G) -v- (M (φ G)) -v- (unit G)) - ?? - = γ `<=<` (unit G) + γ = (ρ G) + = ((ρ <=< unit) G) + = (((join R') -v- (M ρ) -v- unit) G) + = (((join R') G) -v- ((M ρ) G) -v- (unit G)) + = ((join (R'G)) -v- (M (ρ G)) -v- (unit G)) + since γ = (ρ G) is a natural transformation to MR'G, + this satisfies the definition <=<: + = γ <=< (unit G)-where as we saidγ
is a natural transformation from `G` to some `MF'G`. +where as we saidγ
is a natural transformation from `G` to some `MR'G`. + +Summarizing then, the monad laws can be expressed as: + ++ For all ρ, γ, φ in T for which ρ <=< γ and γ <=< φ are defined: + + (i) γ <=< φ etc are also in T + (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) + + (iii.1) (unit G') <=< γ = γ + when γ is a natural transformation from some FG' to MG' + + (iii.2) γ = γ <=< (unit G) + when γ is a natural transformation from G to some MR'G +@@ -268,40 +288,47 @@ Let's remind ourselves of some principles: * functors "distribute over composition", that is for any morphisms `f` and `g` in `F`'s source category:F(g ∘ f) = F(g) ∘ F(f)
-* ifη
is a natural transformation from `F` to `G`, then for everyf:C1→C2
in `F` and `G`'s source category C:η[C2] ∘ F(f) = G(f) ∘ η[C1]
. +* ifη
is a natural transformation from `G` to `H`, then for everyf:C1→C2
in `G` and `H`'s source category C:η[C2] ∘ G(f) = H(f) ∘ η[C1]
. + +*(η F)[E] = η[F(E)]
+ +*(K η)[E} = K(η[E])
+ +*((φ -v- η) F) = ((φ F) -v- (η F))
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 morphismf:C1→C2
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 morphismf:C1→C2
in C:(1) join[C2] ∘ MM(f) = M(f) ∘ join[C1]-Next, consider the composite transformation((join MG') -v- (MM γ))
. +Next, letγ
be a transformation from `G` to `MG'`, and + consider the composite transformation((join MG') -v- (MM γ))
. -*γ
is a transformation from `G` to `MG'`, and assigns elements `C1` in C a morphismγ\*: G(C1) → MG'(C1)
.(MM γ)
is a transformation that instead assigns `C1` the morphismMM(γ\*)
. +*γ
assigns elements `C1` in C a morphismγ\*: G(C1) → MG'(C1)
.(MM γ)
is a transformation that instead assigns `C1` the morphismMM(γ\*)
. -* `(join MG')` is a transformation from `MMMG'` to `MMG'` that assigns `C1` the morphism `join[MG'(C1)]`. +* `(join MG')` is a transformation from `MM(MG')` to `M(MG')` 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*). + (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:- (3) This assigns to C1 the morphism M(γ*) ∘ join[G(C1)]. + (3) Consider the composite transformationSo for every element `C1` of C:((M γ) -v- (join G))
. This assigns to C1 the morphism M(γ*) ∘ join[G(C1)].((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: + 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]@@ -309,33 +336,34 @@ So for every element `C1` of C: So our **(lemma 1)** is:- ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), where γ is a transformation from G to MG'. + ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), + where as we said γ is a natural 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 morphismf: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 morphismf:C1→C2
in C:- (4) unit[b] ∘ f = M(f) ∘ unit[a] + (4) unit[C2] ∘ f = M(f) ∘ unit[C1]-Next consider the composite transformation((M γ) -v- (unit G))
: +Next:- (5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)]. + (5) Consider the composite transformation ((M γ) -v- (unit G)). This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].-Next consider the composite transformation((unit MG') -v- γ)
. +Next:- (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*. + (6) Consider the composite transformation ((unit MG') -v- γ). 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: + 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]@@ -343,7 +371,8 @@ So for every element C1 of C: 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 as we said γ is a natural transformation from G to MG'.