**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. (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(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". We will rely later on this, where f:C1→C2:
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])