`(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⋆zSome 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 `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 1These 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_{E}, which is such that for every morphism f:C1→C2: 1_{C2}∘ f = f = f ∘ 1_{C1}

`(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 (i) associate with every element C1 ofA functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism ofCan element F(C1) ofD(ii) associate with every morphism f:C1→C2 ofCa morphism F(f):F(C1)→F(C2) ofD(iii) "preserve identity", that is, for every element C1 ofC: F of C1's identity morphism inCmust be the identity morphism of F(C1) inD: F(1_{C1}) = 1_{F(C1)}. (iv) "distribute over composition", that is for any morphisms f and g inC: F(g ∘ f) = F(g) ∘ F(f)

`η[C1]:G(C1)→H(C1)`

in `η[C1]`

has as source `C1`'s image under `G` in for every morphism f:C1→C2 inThat is, the morphism via `G(f)` from `G(C1)` to `G(C2)`, and then viaC: η[C2] ∘ G(f) = H(f) ∘ η[C1]

`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 -ThenB-+ +---C--+ +----D-----+ +--E-- | | | | | | F: -----→ G: -----→ K: -----→ | | | | | η | | | ψ | | | | v | | v | | H: -----→ L: -----→ | | | | | φ | | | | | | v | | | | J: -----→ | | -----+ +--------+ +------------+ +-------

`(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 `(η F)[B1] = η[F(B1)]`

---that is, the morphism in `η`

assigns to the element `F(B1)` of `(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 `(K η)[C1] = K(η[C1])`

---that is, the morphism in `η[C1]`

of `(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)[x]`

as: `φ[x] ∘ η[x]`

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