`(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)[\_]`

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

`φ`

, each being between some (variable) functor `F` and another functor which is the composite `MF'` of `M` and a (variable) functor `F'`. That is, for each element `C1` in `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 `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:
γ <=< φ =def. ((join G') -v- (M γ) -v- φ)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

`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 are satisfied:
(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) φ = φ <=< unit (here φ 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 -v- (M unit) -v- φ) G') = (join G') -v- ((M unit) G') -v- (φ G') = (join G') -v- (M (unit G')) -v- γ ?? = (unit G') <=< γwhere as we said

`γ`

is a natural transformation from some `FG'` to `MG'`.
Similarly, if `φ`

is a natural transformation from `1C` to `MF'`, and `γ`

is `(φ G)`

, that is, a natural transformation from `G` to `MF'G`, then we can extend (iii.2) as follows:
γ = (φ G) = ((φ <=< unit) G) = (((join F') -v- (M φ) -v- unit) G) = ((join F'G) -v- ((M φ) G) -v- (unit G)) = ((join F'G) -v- (M (φ G)) -v- (unit G)) ?? = γ <=< (unit G)where as we said

`γ`

is a natural transformation from `G` to some `MF'G`.
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 `F` to `G`, then for every `f:C1→C2`

in `F` and `G`'s source category `η[C2] ∘ F(f) = G(f) ∘ 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 `f:C1→C2`

in (1) join[C2] ∘ MM(f) = M(f) ∘ join[C1]Next, consider the composite transformation

`((join MG') -v- (MM γ))`

.
* `γ`

is a transformation from `G` to `MG'`, and assigns elements `C1` in `γ\*: G(C1) → MG'(C1)`

. `(MM γ)`

is a transformation that instead assigns `C1` the morphism `MM(γ\*)`

.
* `(join MG')` is a transformation from `MMMG'` to `MMG'` that assigns `C1` the morphism `join[MG'(C1)]`.
Composing them:
(2)Next, consider the composite transformation`((join MG') -v- (MM γ))`

assigns to `C1` the morphism`join[MG'(C1)] ∘ MM(γ*)`

.

`((M γ) -v- (join G))`

.
(3) This assigns to C1 the morphism M(γ*) ∘ join[G(C1)].So for every element `C1` of

((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 γ is a transformation from G to MG'.Next recall that unit is a natural transformation from `1C` to `M`. So for elements `C1` in

`f:a→b`

in (4) unit[b] ∘ f = M(f) ∘ unit[a]Next consider the composite transformation

`((M γ) -v- (unit G))`

:
(5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].Next consider the composite transformation

`((unit MG') -v- γ)`

.
(6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.So for every element C1 of

((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 γ is a 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 φ,γ,ρ in T, where φ is a transformation from F to MF', γ is a transformation from G to MG', R 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 substituting in (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 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 F') <=< φ = φ ==> (unit F') is a transformation from F' to MF', so: (unit F') <=< φ becomes: (join F') (M unit F') φ which is: (join F') (M unit F') φ substituting in (iii.1), we get: ((join F') (M unit F') φ) = φ which will be true for all φ just in case: ((join F') (M unit F')) = the identity transformation, for any F' which will in turn be true just in case: (iii.1') (join (M unit) = the identity transformation (iii.2) φ = φ <=< (unit F) ==> φ is a transformation from F to MF', so: unit <=< φ becomes: (join F') (M φ) unit substituting in (iii.2), we get: φ = ((join F') (M φ) (unit F)) -------------- which by lemma (2), yields: ------------ φ = ((join F') ((unit MF') φ) which will be true for all φ just in case: ((join F') (unit MF')) = the identity transformation, for any F' which will in turn be true just in case: (iii.2') (join (unit M)) = the identity transformationCollecting the results, our monad laws turn out in this format to be: when φ a transformation from F to MF', γ a transformation from F' to MG', ρ a transformation from G' to MR' all in T: (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 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