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-**Don't try to read this yet!!! Many substantial edits are still in process.
-Will be ready soon.**
-
-Caveats
--------
-I really don't know much category theory. Just enough to put this
-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
-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:
-
-
-
- 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.
-
-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
.
-
-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 ∘ 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 1E, which is such that for every morphism f:C1→C2: - 1C2 ∘ f = f = f ∘ 1C1 -- -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. 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. - -* 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 `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
.
-
-
-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:
-
-- (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(1C1) = 1F(C1). - - (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. - - - -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:
-
-- 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)`.
-
-
-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:
-
-- - B -+ +--- C --+ +---- D -----+ +-- E -- - | | | | | | - F: ------> G: ------> K: ------> - | | | | | η | | | ψ - | | | | v | | v - | | H: ------> L: ------> - | | | | | 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.
-
-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". For any morphism f:C1→C2
in C:
-
-- φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1] -- -by naturalness of
φ
, is:
-
-- φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1] -- -by naturalness of
η
, is:
-
-- φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1] -- -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] -- -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:
-
-- (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. - - - -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. - -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)`.
-
-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 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.
-
-- γ <=< φ =def. ((join G') -v- (M γ) -v- 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'`, 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') - = (((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') <=< γ -- -where as we said
γ
is a natural transformation from some `FG'` to `MG'`.
-
-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:
-
-- 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 said
γ
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 -- - - -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 `<=<`. - - - -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 `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)[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 morphism f:C1→C2
in C:
-
-- (1) join[C2] ∘ MM(f) = M(f) ∘ join[C1] -- -Next, let
γ
be a transformation from `G` to `MG'`, and
- consider the composite transformation ((join MG') -v- (MM γ))
.
-
-* 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\*)
.
-
-* `(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*). -- -Next, consider the composite transformation
((M γ) -v- (join G))
:
-
-- (3) ((M γ) -v- (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)]. -- -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 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 morphism
f:C1→C2
in C:
-
-- (4) unit[C2] ∘ f = M(f) ∘ unit[C1] -- -Next, consider the composite transformation
((M γ) -v- (unit G))
:
-
-- (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: - -
- ((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: - -
- (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), - where as we said γ is a natural 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-".
-
-- 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) 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) - - 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 G') is a transformation from G' to MG', so: - (unit G') <=< γ becomes: ((join G') (M unit G') γ) - - substituting in (iii.1), we get: - ((join G') (M unit G') γ) = γ - - which will be true for all γ just in case: - ((join G') (M unit G')) = the identity transformation, for any G' - - which will in turn be true just in case: - (iii.1') (join (M unit) = the identity transformation - - - - - (iii.2) γ = γ <=< (unit G) - when γ is a natural transformation from G to some MR'G - ==> - unit <=< γ becomes: ((join R'G) (M γ) unit) - - 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 γ just in case: - ((join R'G) (unit MR'G)) = the identity transformation, for any R'G - - 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: - -
- 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 G') (M γ) φ) etc also in T - - (ii') (join (M join)) = (join (join M)) - - (iii.1') (join (M unit)) = 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 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. - -Additionally, whereas in category-theory one works "monomorphically", in functional programming one usually works with "polymorphic" functions. - -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 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,...]. - - - - -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] ∘ f = M(f) ∘ t[C1] - -The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2). - - - -In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question. - -For an example of the latter, let φ 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: - - let φ = fun c → [(1,c), (2,c)] - -φ 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')]. - -However, to keep things simple, we'll work instead with functions whose type is settled. So instead of the polymorphic φ, we'll work with (φ : C1 → M(int * C1)). This only accepts arguments of type C1. For generality, I'll talk of functions with the type (φ : C1 → M(C1')), where we assume that C1' is a function of C1. - -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 (φ : C1 → M(C1')) to an argument of type C1. - -