+++ /dev/null
-**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:
-
-
-<pre>
- 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
-</pre>
-
-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:
-
-<pre>
- (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 1<sub>E</sub>, which is such that for every morphism `f:C1->C2`: 1<sub>C2</sub> o f = f = f o 1<sub>C1</sub>
-</pre>
-
-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.)
-
-
-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 `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:
-
- * 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(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
- (iv) "distribute over composition", that is for any morphisms f and g in **C**: F(g o f) = F(g) o 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] o G(f) = H(f) o 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". We will rely later on this, where `f:C1->C2`:
-
- φ[C2] o H(f) o η[C1] = φ[C2] o H(f) o η[C1]
-
-by naturalness of φ, is:
-
- φ[C2] o H(f) o η[C1] = J(f) o φ[C1] o η[C1]
-
-by naturalness of η, is:
-
- φ[C2] o η[C2] o G(f) = J(f) o φ[C1] o 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`:
-
- (φ -v- η)[C2] o G(f) = J(f) o (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]) o ψ[G(C1)]
- = ψ[H(C1)] o 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 `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.
-
-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 `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)
-
-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 `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`.)
-
-`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`.
-
-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 `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:
-
- (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)
-
-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:
-
- 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 `q` is a natural transformation from some `PQ'` to `MQ'`.
-
-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:
-
- 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)
-
-where as we said `q` is a natural transformation from `Q` to some `MP'Q`.
-
-
-
-
-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 1<sub>C1</sub> such that for every morphism f:C1->C2 in **C**: 1<sub>C2</sub> o f = f = f o 1<sub>C1</sub>.
- P3. functors "preserve identity", that is for every element C1 in F's source category: F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
-*)
-
-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].
-
-
-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**:
-
- (1) join[b] o MM(f) = M(f) o join[a]
-
-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, consider the composite transformation ((M q) -v- (join Q)).
- (3) This assigns to C1 the morphism M(q*) o join[Q(C1)].
-
-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]
-
-So our (lemma 1) is: ((join MQ') -v- (MM q)) = ((M q) -v- (join Q)), where q is a transformation from Q to MQ'.
-
-
-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 q) -v- (unit Q)). (5) This assigns to C1 the morphism M(q*) o unit[Q(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**:
- ((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]
-
-So our lemma (2) is: (((M q) -v- (unit Q)) = ((unit MQ') -v- q)), where q is a transformation from Q to MQ'.
-
-
-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:
-
- (i) q <=< p etc are also in T
- ==>
- (i') ((join Q') (M q) p) etc are also in T
-
-
- (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:
-
- ((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)
-
- which will be true for all r,q,p just in case:
-
- ((join R') (M join R')) = ((join R') (join MR')), for any R'.
-
- which will in turn be true just in case:
-
- (ii') (join (M join)) = (join (join M))
-
-
- (iii.1) (unit P') <=< p = p
- ==>
- (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
-
- which will be true for all p just in case:
-
- ((join P') (M unit P')) = the identity transformation, for any P'
-
- which will in turn be true just in case:
-
- (iii.1') (join (M unit) = the identity transformation
-
-
- (iii.2) p = p <=< (unit P)
- ==>
- 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)
-
- which will be true for all p just in case:
-
- ((join P') (unit MP')) = the identity transformation, for any P'
-
- 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:
-
- (i') ((join Q') (M q) p) 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
-
-
-
-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.
-
-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] o f = M(f) o 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 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:
-
- let p = fun c -> [(1,c), (2,c)]
-
-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')].
-
-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.
-
-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.
-
-