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=c45c948396c33d85882fc22dcf85cb8f20121396;hp=49cb8b6941202912b7a5cdfc5a46a2aa4369cee3;hb=b5951492496e5d63882d7bd59ec3568fe5df1185;hpb=0f832d9d2cb2df72b5996180cbf621cfe5709659
diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 49cb8b69..c45c9483 100644
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -8,74 +8,98 @@ together. Also, this really is "put together." I haven't yet found an
authoritative source (that's accessible to a category theory beginner like
myself) that discusses the correspondence between the category-theoretic and
functional programming uses of these notions in enough detail to be sure that
-none of the pieces here is misguided. In particular, it wasn't completely
-obvious how to map the polymorphism on the programming theory side into the
-category theory. And I'm bothered by the fact that our `<=<` operation is only
-partly defined on our domain of natural transformations. But this does seem to
-me to be the reasonable way to put the pieces together. We very much welcome
+none of the pieces here is mistaken.
+In particular, it wasn't completely obvious how to map the polymorphism on the
+programming theory side into the category theory. The way I accomplished this
+may be more complex than it needs to be.
+Also I'm bothered by the fact that our `<=<` operation is only partly defined
+on our domain of natural transformations.
+There are three additional points below that I wonder whether may be too
+cavalier.
+But all considered, this does seem to
+me to be a reasonable way to put the pieces together. We very much welcome
feedback from anyone who understands these issues better, and will make
corrections.
Monoids
-------
-A **monoid** is a structure `(S, *, z)` consisting of an associative binary operation `*` over some set `S`, which is closed under `*`, and which contains an identity element `z` for `*`. That is:
+A **monoid** is a structure (S,⋆,z)
consisting of an associative binary operation ⋆
over some set `S`, which is closed under ⋆
, and which contains an identity element `z` for ⋆
. That is:
- for all s1,s2,s3 in S:
- (i) s1*s2 etc are also in S
- (ii) (s1*s2)*s3 = s1*(s2*s3)
- (iii) z*s1 = s1 = s1*z
+
+
+ for all s1, s2, s3 in S: + (i) s1⋆s2 etc are also in S + (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3) + (iii) z⋆s1 = s1 = s1⋆z +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. +* 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.
Categories
----------
A **category** is a generalization of a monoid. A category consists of a class of **elements**, and a class of **morphisms** between those elements. Morphisms are sometimes also called maps or arrows. They are something like functions (and as we'll see below, given a set of functions they'll determine a category). However, a single morphism only maps between a single source element and a single target element. Also, there can be multiple distinct morphisms between the same source and target, so the identity of a morphism goes beyond its "extension."
-When a morphism `f` in category **C** has source `c1` and target `c2`, we'll write `f:c1->c2`.
+When a morphism `f` in category C has source `C1` and target `C2`, we'll write f:C1→C2
.
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 o 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 id[e], which is such that for every morphism f:c1->c2: id[c2] o f = f = f o id[c1]
++ (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 -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. + (iii) every element X of the category has to have an identity + morphism 1X, which is such that for every morphism f:C1→C2: + 1C2 ∘ f = f = f ∘ 1C1 +-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.) +These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `X` 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. Morphisms correspond to directed paths of length ≥ 0 in the graph. Some examples of categories are: -* Categories whose elements are sets and whose morphisms are functions between those sets. Here the source and target of a function are its domain and range, so distinct functions sharing a domain and range (e.g., sin and cos) are distinct morphisms between the same source and target elements. The identity morphism for any element/set is just the identity function for that set. +* Categories whose elements are sets and whose morphisms are functions between those sets. Here the source and target of a function are its domain and range, so distinct functions sharing a domain and range (e.g., `sin` and `cos`) are distinct morphisms between the same source and target elements. The identity morphism for any element/set is just the identity function for that set. -* 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`. +* any monoid
(S,⋆,z)
generates a category with a single element `Q`; this `Q` 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 `Q` 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 `Q` 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)
- Any pre-order `(S,<=)` generates a category whose elements are the members of `S` and which has only a single morphism between any two elements `s1` and `s2`, iff `s1<=s2`.
+ Any pre-order (S,≤)
generates a category whose elements are the members of `S` and which has only a single morphism between any two elements `s1` and `s2`, iff s1 ≤ s2
.
Functors
--------
-A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category **C** to category **D** must:
+A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category C to category D must:
+
++ (i) associate with every element C1 of C an element F(C1) of D - (i) associate with every element c1 of **C** an element F(c1) of **D** - (ii) associate with every morphism f:c1->c2 of **C** a morphism F(f):F(c1)->F(c2) of **D** - (iii) "preserve identity", that is, for every element c1 of **C**: F of c1's identity morphism in **C** must be the identity morphism of F(c1) in **D**: F(id[c1]) = id[F(c1)]. - (iv) "distribute over composition", that is for any morphisms f and g in **C**: F(g o f) = F(g) o F(f) + (ii) associate with every morphism f:C1→C2 of C a morphism F(f):F(C1)→F(C2) of D -A functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism of **C** to itself is denoted `1C`. + (iii) "preserve identity", that is, for every element C1 of C: + F of C1's identity morphism in C must be the identity morphism of F(C1) in D: + F(1C1) = 1F(C1). -How functors compose: If `G` is a functor from category **C** to category **D**, and `K` is a functor from category **D** to category **E**, then `KG` is a functor which maps every element `c1` of **C** to element `K(G(c1))` of **E**, and maps every morphism `f` of **C** to morphism `K(G(f))` of **E**. + (iv) "distribute over composition", that is for any morphisms f and g in C: + F(g ∘ f) = F(g) ∘ F(f) ++ +A functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism of C to itself is denoted `1C`. + +How functors compose: If `G` is a functor from category C to category D, and `K` is a functor from category D to category E, then `KG` is a functor which maps every element `C1` of C to element `K(G(C1))` of E, and maps every morphism `f` of C to morphism `K(G(f))` of E. I'll assert without proving that functor composition is associative. @@ -85,60 +109,77 @@ Natural Transformation ---------------------- So categories include elements and morphisms. Functors consist of mappings from the elements and morphisms of one category to those of another (or the same) category. **Natural transformations** are a third level of mappings, from one functor to another. -Where `G` and `H` are functors from category **C** to category **D**, a natural transformation η between `G` and `H` is a family of morphisms η[c1]:G(c1)->H(c1)` in **D** for each element `c1` of **C**. That is, η[c1]` has as source `c1`'s image under `G` in **D**, and as target `c1`'s image under `H` in **D**. The morphisms in this family must also satisfy the constraint: +Where `G` and `H` are functors from category C to category D, a natural transformation η between `G` and `H` is a family of morphisms
η[C1]:G(C1)→H(C1)
in D for each element `C1` of C. That is, η[C1]
has as source `C1`'s image under `G` in D, and as target `C1`'s image under `H` in D. The morphisms in this family must also satisfy the constraint:
- for every morphism f:c1->c2 in **C**: η[c2] o G(f) = H(f) o η[c1]
++ for every morphism f:C1→C2 in C: + η[C2] ∘ G(f) = H(f) ∘ η[C1] +-That is, the morphism via `G(f)` from `G(c1)` to `G(c2)`, and then via η[c2]` to `H(c2)`, is identical to the morphism from `G(c1)` via η[c1]` to `H(c1)`, and then via `H(f)` from `H(c1)` to `H(c2)`. +That is, the morphism via `G(f)` from `G(C1)` to `G(C2)`, and then via
η[C2]
to `H(C2)`, is identical to the morphism from `G(C1)` via η[C1]
to `H(C1)`, and then via `H(f)` from `H(C1)` to `H(C2)`.
How natural transformations compose:
-Consider four categories **B**, **C**, **D**, and **E**. Let `F` be a functor from **B** to **C**; `G`, `H`, and `J` be functors from **C** to **D**; and `K` and `L` be functors from **D** to **E**. Let η be a natural transformation from `G` to `H`; φ be a natural transformation from `H` to `J`; and ψ be a natural transformation from `K` to `L`. Pictorally:
+Consider four categories B, C, D, and E. Let `F` be a functor from B to C; `G`, `H`, and `J` be functors from C to D; and `K` and `L` be functors from D to E. Let η be a natural transformation from `G` to `H`; φ be a natural transformation from `H` to `J`; and ψ be a natural transformation from `K` to `L`. Pictorally:
- - **B** -+ +--- **C** --+ +---- **D** -----+ +-- **E** --
++ - B -+ +--- C --+ +---- D -----+ +-- E -- | | | | | | F: ------> G: ------> K: ------> - | | | | | η | | | ψ + | | | | | η | | | ψ | | | | v | | v | | H: ------> L: ------> - | | | | | φ | | + | | | | | φ | | | | | | v | | | | J: ------> | | -----+ +--------+ +------------+ +------- +-Then `(η F)` is a natural transformation from the (composite) functor `GF` to the composite functor `HF`, such that where `b1` is an element of category **B**, `(η F)[b1] = η[F(b1)]`---that is, the morphism in **D** that η assigns to the element `F(b1)` of **C**. +Then
(η F)
is a natural transformation from the (composite) functor `GF` to the composite functor `HF`, such that where `B1` is an element of category B, (η F)[B1] = η[F(B1)]
---that is, the morphism in D that η
assigns to the element `F(B1)` of C.
-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**.
+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] o H(f) o η[c1] = φ[c2] o H(f) o η[c1]
++ φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1] +-by naturalness of φ, is: +by naturalness of
φ
, is:
- φ[c2] o H(f) o η[c1] = J(f) o φ[c1] o η[c1]
++ φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1] +-by naturalness of η, is: +by naturalness of
η
, is:
- φ[c2] o η[c2] o G(f) = J(f) o φ[c1] o η[c1]
++ φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1] +-Hence, we can define `(φ -v- η)[x]` as: φ[x] o 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] o G(f) = J(f) o (φ -v- η)[c1]
++ (φ -v- η)[C2] ∘ G(f) = J(f) ∘ (φ -v- η)[C1] +An observation we'll rely on later: given the definitions of vertical composition and of how natural transformations compose with functors, it follows that: +
((φ -v- η) F) = ((φ F) -v- (η F)) +I'll assert without proving that vertical composition is associative and has an identity, which we'll call "the identity transformation." -`(ψ -h- η)` is natural transformation from the (composite) functor `KG` to the (composite) functor `LH`; this is known as a "horizontal composition." It's trickier to define, but we won't be using it here. For reference: +
(ψ -h- η)
is natural transformation from the (composite) functor `KG` to the (composite) functor `LH`; this is known as a "horizontal composition." It's trickier to define, but we won't be using it here. For reference:
- (φ -h- η)[c1] = L(η[c1]) o ψ[G(c1)]
- = ψ[H(c1)] o K(η[c1])
++ (φ -h- η)[C1] = L(η[C1]) ∘ ψ[G(C1)] + = ψ[H(C1)] ∘ K(η[C1]) +Horizontal composition is also associative, and has the same identity as vertical composition. @@ -148,236 +189,425 @@ Monads ------ 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. +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 transformations `p`, each being between some (variable) functor `P` and another functor which is the composite `MP'` of `M` and a (variable) functor `P'`. That is, for each element `c1` in **C**, `p` assigns `c1` a morphism from element `P(c1)` to element `MP'(c1)`, satisfying the constraints detailed in the previous section. For different members of `T`, the relevant functors may differ; that is, `p` is a transformation from functor `P` to `MP'`, `q` is a transformation from functor `Q` to `MQ'`, and none of `P`, `P'`, `Q`, `Q'` need be the same. +Let `T` be a set of natural transformations
φ
, 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.
-Let `p` and `q` be members of `T`, that is they are natural transformations from `P` to `MP'` and from `Q` to `MQ'`, respectively. Let them be such that `P' = Q`. Now `(M q)` will also be a natural transformation, formed by composing the functor `M` with the natural transformation `q`. Similarly, `(join Q')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `Q'`; it will transform the functor `MMQ'` to the functor `MQ'`. Now take the vertical composition of the three natural transformations `(join Q')`, `(M q)`, and `p`, and abbreviate it as follows:
-
- q <=< p =def. ((join Q') -v- (M q) -v- p)
+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. Since composition is associative I don't specify the order of composition on the rhs.
-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) +-In other words, `<=<` is a binary operator that takes us from two members `p` and `q` of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes its written `p >=> q` where that's the same as `q <=< p`.) +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
.)
-`p` is a transformation from `P` to `MP'` which = `MQ`; `(M q)` is a transformation from `MQ` to `MMQ'`; and `(join Q')` is a transformation from `MMQ'` to `MQ'`. So the composite `q <=< p` will be a transformation from `P` to `MQ'`, 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, 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 must hold:
+
++ (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) ρ = ρ <=< unit + (here ρ 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 `FG'` to `MG'`, then we can extend (iii.1) as follows:
+
++ γ = (φ G') + = ((unit <=< φ) G') + since unit is a natural transformation to M(1C), this is: + = (((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 is: + = (unit G') <=< γ +-That's it. (Well, perhaps we're cheating a bit, because `q <=< p` isn't fully defined on `T`, but only when `P` is a functor to `MP'` and `Q` is a functor from `P'`. But wherever `<=<` is defined, the monoid laws are satisfied: +where as we said
γ
is a natural transformation from some `FG'` to `MG'`.
- (i) q <=< p is also in T
- (ii) (r <=< q) <=< p = r <=< (q <=< p)
- (iii.1) unit <=< p = p (here p has to be a natural transformation to M(1C))
- (iii.2) p = p <=< unit (here p has to be a natural transformation from 1C)
+Similarly, if ρ
is a natural transformation from `1C` to `MR'`, and γ
is (ρ G)
, that is, a natural transformation from `G` to `MR'G`, then we can extend (iii.2) as follows:
-If `p` is a natural transformation from `P` to `M(1C)` and `q` is `(p Q')`, that is, a natural transformation from `PQ` to `MQ`, then we can extend (iii.1) as follows:
++ γ = (ρ G) + = ((ρ <=< unit) G) + = since ρ is a natural transformation to MR', this is: + = (((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 is: + = γ <=< (unit G) +- q = (p Q') - = ((unit <=< p) Q') - = ((join -v- (M unit) -v- p) Q') - = (join Q') -v- ((M unit) Q') -v- (p Q') - = (join Q') -v- (M (unit Q')) -v- q - ?? - = (unit Q') <=< q +where as we said
γ
is a natural transformation from `G` to some `MR'G`.
-where as we said `q` is a natural transformation from some `PQ'` to `MQ'`.
+Summarizing then, the monad laws can be expressed as:
-Similarly, if `p` is a natural transformation from `1C` to `MP'`, and `q` is `(p Q)`, that is, a natural transformation from `Q` to `MP'Q`, then we can extend (iii.2) as follows:
++ For all ρ, γ, φ in T for which ρ <=< γ and γ <=< φ are defined: - q = (p Q) - = ((p <=< unit) Q) - = (((join P') -v- (M p) -v- unit) Q) - = ((join P'Q) -v- ((M p) Q) -v- (unit Q)) - = ((join P'Q) -v- (M (p Q)) -v- (unit Q)) - ?? - = q <=< (unit Q) + (i) γ <=< φ etc are also in T -where as we said `q` is a natural transformation from `Q` to some `MP'Q`. + (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) + (iii.1) (unit G') <=< γ = γ + whenever γ is a natural transformation from some FG' to MG' + (iii.2) γ = γ <=< (unit G) + whenever γ is a natural transformation from G to some MR'G +-The standard category-theory presentation of the monad laws ------------------------------------------------------------ + +Getting to 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 `<=<`. -(* - P2. every element c1 of a category **C** has an identity morphism id[c1] such that for every morphism f:c1->c2 in **C**: id[c2] o f = f = f o id[c1]. - P3. functors "preserve identity", that is for every element c1 in F's source category: F(id[c1]) = id[F(c1)]. -*) + + +Let's remind ourselves of principles stated above: + +* 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)
-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 o f) = F(g) o 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] o F(f) = G(f) o η[c1].
+* if η
is a natural transformation from `G` to `H`, then for every f:C1→C2
in `G` and `H`'s source category C: η[C2] ∘ G(f) = H(f) ∘ η[C1]
.
+* (η F)[X] = η[F(X)]
+
+* (K η)[X] = K(η[X])
+
+* ((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 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] o MM(f) = M(f) o join[a]
++ (1) join[C2] ∘ MM(f) = M(f) ∘ join[C1] +-Next, consider the composite transformation ((join MQ') -v- (MM q)). - q is a transformation from Q to MQ', and assigns elements c1 in **C** a morphism q*: Q(c1) -> MQ'(c1). (MM q) is a transformation that instead assigns c1 the morphism MM(q*). - (join MQ') is a transformation from MMMQ' to MMQ' that assigns c1 the morphism join[MQ'(c1)]. - Composing them: - (2) ((join MQ') -v- (MM q)) assigns to c1 the morphism join[MQ'(c1)] o MM(q*). +Next, let
γ
be a transformation from `G` to `MG'`, and
+ consider the composite transformation ((join MG') -v- (MM γ))
.
-Next, consider the composite transformation ((M q) -v- (join Q)).
- (3) This assigns to c1 the morphism M(q*) o join[Q(c1)].
+* γ
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\*)
.
-So for every element c1 of **C**:
- ((join MQ') -v- (MM q))[c1], by (2) is:
- join[MQ'(c1)] o MM(q*), which by (1), with f=q*: Q(c1)->MQ'(c1) is:
- M(q*) o join[Q(c1)], which by 3 is:
- ((M q) -v- (join Q))[c1]
+* `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.
-So our (lemma 1) is: ((join MQ') -v- (MM q)) = ((M q) -v- (join Q)), where q is a transformation from Q to MQ'.
+Composing them:
++ (2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*). +-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] o f = M(f) o unit[a] +Next, consider the composite transformation
((M γ) -v- (join G))
:
-Next consider the composite transformation ((M q) -v- (unit Q)). (5) This assigns to c1 the morphism M(q*) o unit[Q(c1)].
++ (3) ((M γ) -v- (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)]. +-Next consider the composite transformation ((unit MQ') -v- q). (6) This assigns to c1 the morphism unit[MQ'(c1)] o q*. +So for every element `C1` of C: -So for every element c1 of **C**: - ((M q) -v- (unit Q))[c1], by (5) = - M(q*) o unit[Q(c1)], which by (4), with f=q*: Q(c1)->MQ'(c1) is: - unit[MQ'(c1)] o q*, which by (6) = - ((unit MQ') -v- q)[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: + M(γ*) ∘ join[G(C1)], which by 3 is: + ((M γ) -v- (join G))[C1] +-So our lemma (2) is: (((M q) -v- (unit Q)) = ((unit MQ') -v- q)), where q is a transformation from Q to MQ'. +So our **(lemma 1)** is: +
+ ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), + where as we said γ is a natural transformation from G to MG'. +-Finally, we substitute ((join Q') -v- (M q) -v- p) for q <=< p in the monad laws. For simplicity, I'll omit the "-v-". - for all p,q,r in T, where p is a transformation from P to MP', q is a transformation from Q to MQ', R is a transformation from R to MR', and P'=Q and Q'=R: +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:C1→C2
in C:
- (i) q <=< p etc are also in T
- ==>
- (i') ((join Q') (M q) p) etc are also in T
++ (4) unit[C2] ∘ f = M(f) ∘ unit[C1] ++Next, consider the composite transformation
((M γ) -v- (unit G))
:
- (ii) (r <=< q) <=< p = r <=< (q <=< p)
- ==>
- (r <=< q) is a transformation from Q to MR', so:
- (r <=< q) <=< p becomes: (join R') (M (r <=< q)) p
- which is: (join R') (M ((join R') (M r) q)) p
- substituting in (ii), and helping ourselves to associativity on the rhs, we get:
++ (5) ((M γ) -v- (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)]. ++ +Next, consider the composite transformation
((unit MG') -v- γ)
:
+
++ (6) ((unit MG') -v- γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*. ++ +So for every element C1 of C: - ((join R') (M ((join R') (M r) q)) p) = ((join R') (M r) (join Q') (M q) p) - --------------------- - which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields: - ------------------------ - ((join R') (M join R') (MM r) (M q) p) = ((join R') (M r) (join Q') (M q) p) - --------------- - which by lemma 1, with r a transformation from Q' to MR', yields: - ----------------- - ((join R') (M join R') (MM r) (M q) p) = ((join R') (join MR') (MM r) (M q) p) +
+ ((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] +- which will be true for all r,q,p just in case: +So our **(lemma 2)** is: - ((join R') (M join R')) = ((join R') (join MR')), for any R'. +
+ (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), + where as we said γ is a natural transformation from G to MG'. +- which will in turn be true just in case: - (ii') (join (M join)) = (join (join M)) +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 ρ, γ, φ in T, + where φ is a transformation from F to MF', + γ is a transformation from G to MG', + ρ is a transformation from R to MR', + and F'=G and G'=R: - (iii.1) (unit P') <=< p = p + (i) γ <=< φ etc are also in T + ==> + (i') ((join G') (M γ) φ) etc are also in T ++ +
+ (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) + ==> + (ρ <=< γ) is a transformation from G to MR', so + (ρ <=< γ) <=< φ becomes: ((join R') (M (ρ <=< γ)) φ) + which is: ((join R') (M ((join R') (M ρ) γ)) φ) + + similarly, ρ <=< (γ <=< φ) is: + ((join R') (M ρ) ((join G') (M γ) φ)) + + substituting these into (ii), and helping ourselves to associativity on the rhs, we get: + ((join R') (M ((join R') (M ρ) γ)) φ) = ((join R') (M ρ) (join G') (M γ) φ) + + which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields: + ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (M ρ) (join G') (M γ) φ) + + which by lemma 1, with ρ a transformation from G' to MR', yields: + ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (join MR') (MM ρ) (M γ) φ) + + [-- Are the next two steps too cavalier? --] + + which will be true for all ρ, γ, φ only when: + ((join R') (M join R')) = ((join R') (join MR')), for any R' + + which will in turn be true when: + (ii') (join (M join)) = (join (join M)) ++ +
+ (iii.1) (unit G') <=< γ = γ + when γ is a natural transformation from some FG' to MG' ==> - (unit P') is a transformation from P' to MP', so: - (unit P') <=< p becomes: (join P') (M unit P') p - which is: (join P') (M unit P') p - substituting in (iii.1), we get: - ((join P') (M unit P') p) = p + (unit G') is a transformation from G' to MG', so: + (unit G') <=< γ becomes: ((join G') (M (unit G')) γ) + which is: ((join G') ((M unit) G') γ) - which will be true for all p just in case: + substituting in (iii.1), we get: + ((join G') ((M unit) G') γ) = γ - ((join P') (M unit P')) = the identity transformation, for any P' + which is: + (((join (M unit)) G') γ) = γ - which will in turn be true just in case: + [-- Are the next two steps too cavalier? --] - (iii.1') (join (M unit) = the identity transformation + which will be true for all γ just in case: + for any G', ((join (M unit)) G') = the identity transformation + which will in turn be true just in case: + (iii.1') (join (M unit)) = the identity transformation +- (iii.2) p = p <=< (unit P) +
+ (iii.2) γ = γ <=< (unit G) + when γ is a natural transformation from G to some MR'G ==> - p is a transformation from P to MP', so: - unit <=< p becomes: (join P') (M p) unit - substituting in (iii.2), we get: - p = ((join P') (M p) (unit P)) - -------------- - which by lemma (2), yields: - ------------ - p = ((join P') ((unit MP') p) + γ <=< (unit G) becomes: ((join R'G) (M γ) (unit G)) + + substituting in (iii.2), we get: + γ = ((join R'G) (M γ) (unit G)) + + which by lemma 2, yields: + γ = (((join R'G) ((unit MR'G) γ) - which will be true for all p just in case: + which is: + γ = (((join (unit M)) R'G) γ) - ((join P') (unit MP')) = the identity transformation, for any P' + [-- Are the next two steps too cavalier? --] - which will in turn be true just in case: + which will be true for all γ just in case: + for any R'G, ((join (unit M)) R'G) = the identity transformation + 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 p a transformation from P to MP', q a transformation from P' to MQ', r a transformation from Q' to MR' all in T: +
+ For all ρ, γ, φ in T, + where φ is a transformation from F to MF', + γ is a transformation from G to MG', + ρ is a transformation from R to MR', + and F'=G and G'=R: - (i') ((join Q') (M q) p) etc also in T + (i') ((join G') (M γ) φ) etc also in T - (ii') (join (M join)) = (join (join M)) + (ii') (join (M join)) = (join (join M)) (iii.1') (join (M unit)) = the identity transformation - (iii.2')(join (unit M)) = the identity transformation + (iii.2') (join (unit M)) = the identity transformation ++ + + +Getting to the functional programming presentation of the monad laws +-------------------------------------------------------------------- +In functional programming, `unit` is sometimes called `return` and the monad laws are usually stated in terms of `unit`/`return` and an operation called `bind` which is interdefinable with `<=<` or with `join`. + +The base category C will have types as elements, and monadic functions as its morphisms. The source and target of a morphism will be the types of its argument and its result. (As always, there can be multiple distinct morphisms from the same source to the same target.) + +A monad `M` will consist of a mapping from types `'t` to types `M('t)`, and a mapping from functions
f:C1→C2
to functions M(f):M(C1)→M(C2)
. This is also known as liftM f
for `M`, and is pronounced "function f lifted into the monad M." For example, where `M` is the list monad, `M` maps every type `'t` into the type `'t list`, and maps every function f:x→y
into the function that maps `[x1,x2...]` to `[y1,y2,...]`.
+
+
+In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad."
+
+A "monadic value" is any member of a type `M('t)`, for any type `'t`. For example, any `int list` is a monadic value for the list monad. We can think of these monadic values as the result of applying some function `phi`, whose type is `F('t)->M(F'('t))`. `'t` here is any collection of free type variables, and `F('t)` and `F'('t)` are types parameterized on `'t`. An example, with `M` being the list monad, `'t` being `('t1,'t2)`, `F('t1,'t2)` being `char * 't1 * 't2`, and `F'('t1,'t2)` being `int * 't1 * 't2`:
+
++ let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)] ++ + + +Now where `gamma` is another function of type
F'('t) → M(G'('t))
, we define:
++ gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a) + = ((join G') -v- (M gamma) -v- phi) a + = (gamma <=< phi) a ++Hence: -7. The functional programming presentation of the monad laws ------------------------------------------------------------- -In functional programming, unit is usually called "return" and the monad laws are usually stated in terms of return and an operation called "bind" which is interdefinable with <=< or with join. +
+ gamma <=< phi = fun a -> (gamma =<< phi a) +-Additionally, whereas in category-theory one works "monomorphically", in functional programming one usually works with "polymorphic" functions. +`gamma =<< phi a` is called the operation of "binding" the function gamma to the monadic value `phi a`, and is usually written as `phi a >>= gamma`. -The base category **C** will have types as elements, and monadic functions as its morphisms. The source and target of a morphism will be the types of its argument and its result. (As always, there can be multiple distinct morphisms from the same source to the same target.) +With these definitions, our monadic laws become: -A monad M will consist of a mapping from types c1 to types M(c1), and a mapping from functions f:c1->c2 to functions M(f):M(c1)->M(c2). This is also known as "fmap f" or "liftM f" for M, and is called "function f lifted into the monad M." For example, where M is the list monad, M maps every type X into the type "list of Xs", and maps every function f:x->y into the function that maps [x1,x2...] to [y1,y2,...]. +
+ Where phi is a polymorphic function of type F('t) -> M(F'('t)) + gamma is a polymorphic function of type G('t) -> M(G'('t)) + rho is a polymorphic function of type R('t) -> M(R'('t)) + and F' = G and G' = R, + and a ranges over values of type F('t), + b ranges over values of type G('t), + and c ranges over values of type G'('t): + + (i) γ <=< φ is defined, + and is a natural transformation from F to MG' + ==> + (i'') fun a -> gamma =<< phi a is defined, + and is a function from type F('t) -> M(G'('t)) ++ +
+ (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) + ==> + (fun a -> (rho <=< gamma) =<< phi a) = (fun a -> rho =<< (gamma <=< phi) a) + (fun a -> (fun b -> rho =<< gamma b) =<< phi a) = (fun a -> rho =<< (gamma =<< phi a)) + + (ii'') (fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a) ++ +
+ (iii.1) (unit G') <=< γ = γ + when γ is a natural transformation from some FG' to MG' + ==> + (unit G') <=< gamma = gamma + when gamma is a function of type F(G'('t)) -> M(G'('t)) + + fun b -> (unit G') =<< gamma b = gamma + + (unit G') =<< gamma b = gamma b + + Let return be a polymorphic function mapping arguments of any + type 't to M('t). In particular, it maps arguments c of type + G'('t) to the monadic value (unit G') c, of type M(G'('t)). + + (iii.1'') return =<< gamma b = gamma b ++ +
+ (iii.2) γ = γ <=< (unit G) + when γ is a natural transformation from G to some MR'G + ==> + gamma = gamma <=< (unit G) + when gamma is a function of type G('t) -> M(R'(G('t))) + + gamma = fun b -> gamma =<< (unit G) b + As above, return will map arguments b of type G('t) to the + monadic value (unit G) b, of type M(G('t)). + gamma = fun b -> gamma =<< return b -A natural transformation t assigns to each type c1 in **C** a morphism t[c1]: c1->M(c1) such that, for every f:c1->c2: - t[c2] o f = M(f) o t[c1] + (iii.2'') gamma b = gamma =<< return b +-The composite morphisms said here to be identical are morphisms from the type c1 to the type M(c2). +Summarizing (ii''), (iii.1''), (iii.2''), these are the monadic laws as usually stated in the functional programming literature: +* `fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)` + Usually written reversed, and with a monadic variable `u` standing in + for `phi a`: -In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question. + `u >>= (fun b -> gamma b >>= rho) = (u >>= gamma) >>= rho` -For an example of the latter, let p be a function that takes arguments of some (schematic, polymorphic) type c1 and yields results of some (schematic, polymorphic) type M(c2). An example with M being the list monad, and c2 being the tuple type schema int * c1: +* `return =<< gamma b = gamma b` - let p = fun c -> [(1,c), (2,c)] + Usually written reversed, and with `u` standing in for `gamma b`: -p is polymorphic: when you apply it to the int 0 you get a result of type "list of int * int": [(1,0), (2,0)]. When you apply it to the char 'e' you get a result of type "list of int * char": [(1,'e'), (2,'e')]. + `u >>= return = u` -However, to keep things simple, we'll work instead with functions whose type is settled. So instead of the polymorphic p, we'll work with (p : c1 -> M(int * c1)). This only accepts arguments of type c1. For generality, I'll talk of functions with the type (p : c1 -> M(c1')), where we assume that c1' is a function of c1. +* `gamma b = gamma =<< return b` -A "monadic value" is any member of a type M(c1), for any type c1. For example, a list is a monadic value for the list monad. We can think of these monadic values as the result of applying some function (p : c1 -> M(c1')) to an argument of type c1. + Usually written reversed: + `return b >>= gamma = gamma b` +