## 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 mistaken. In particular, it wasn't completely obvious how to map the polymorphism on the programming theory side into the category theory. The way I accomplished this may be more complex than it needs to be. Also I'm bothered by the fact that our <=< operation is only partly defined on our domain of natural transformations. There are three additional points below that I wonder whether may be too cavalier. But all considered, this does seem to me to be a reasonable way to put the pieces together. We very much welcome feedback from anyone who understands these issues better, and will make corrections.

Thanks Wren Thornton for helpful comments on these notes (not yet incorporated).

## 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 X of the category has to have an identity
morphism 1X, 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 X 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 Q; this Q 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 Q 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 Q 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.

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')
since unit is a natural transformation to M(1C), this is:
= (((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 is:
= (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)
= since ρ is a natural transformation to MR', this is:
= (((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 is:
= 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
whenever γ is a natural transformation from some FG' to MG'

(iii.2)                     γ  =  γ <=< (unit G)
whenever 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 principles stated above:

• 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)[X] = η[F(X)]

• (K η)[X] = K(η[X])

• ((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)

[-- Are the next two steps too cavalier? --]

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)
which is: ((join G') ((M unit) G') γ)

substituting in (iii.1), we get:
((join G') ((M unit) G') γ) = γ

which is:
(((join (M unit)) G') γ) = γ

[-- Are the next two steps too cavalier? --]

which will be true for all γ just in case:
for any G', ((join (M unit)) G') = the identity transformation

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
==>
γ <=< (unit G) becomes: ((join R'G) (M γ) (unit G))

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 is:
γ = (((join (unit M)) R'G) γ)

[-- Are the next two steps too cavalier? --]

which will be true for all γ just in case:
for any R'G, ((join (unit M)) R'G) = the identity transformation

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


In category-theory presentations, you may see unit referred to as η, and join referred to as μ. Also, instead of the monad (M, unit, join), you may sometimes see discussion of the "Kleisli triple" (M, unit, =<<). Alternatively, =<< may be called ⋆. These are interdefinable (see below).

## Getting to the functional programming presentation of the monad laws

In functional programming, unit is sometimes called return and the monad laws are usually stated in terms of unit/return and an operation called bind which is interdefinable with <=< or with join.

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 't to types M('t), and a mapping from functions f:C1→C2 to functions M(f):M(C1)→M(C2). This is also known as liftM f for M, and is pronounced "function f lifted into the monad M." For example, where M is the List monad, M maps every type 't into the type 't list, and maps every function f:x→y into the function that maps [x1,x2...] to [y1,y2,...].

In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad."

A "monadic value" is any member of a type M('t), for any type 't. For example, any int list is a monadic value for the List monad. We can think of these monadic values as the result of applying some function phi, whose type is F('t) -> M(F'('t)). 't here is any collection of free type variables, and F('t) and F'('t) are types parameterized on 't. An example, with M being the List monad, 't being ('t1,'t2), F('t1,'t2) being char * 't1 * 't2, and F'('t1,'t2) being int * 't1 * 't2:

    let phi = fun ((_:char), x, y) -> [(1,x,y),(2,x,y)]


[-- I intentionally chose this polymorphic function because simpler ways of mapping the polymorphic monad operations from functional programming onto the category theory notions can't accommodate it. We have all the F, MF' (unit G') and so on in order to be able to be handle even phis like this. --]

Now where gamma is another function of type F'('t) -> M(G'('t)), we define:

    gamma =<< phi a  =def. ((join G') -v- (M gamma)) (phi a)
= ((join G') -v- (M gamma) -v- phi) a
= (gamma <=< phi) a


Hence:

    gamma <=< phi = (fun a -> gamma =<< phi a)


gamma =<< phi a is called the operation of "binding" the function gamma to the monadic value phi a, and is usually written as phi a >>= gamma.

With these definitions, our monadic laws become:

    Where phi is a polymorphic function of type F('t) -> M(F'('t))
gamma is a polymorphic function of type G('t) -> M(G'('t))
rho is a polymorphic function of type R('t) -> M(R'('t))
and F' = G and G' = R,
and a ranges over values of type F('t),
and b ranges over values of type G('t),
and c ranges over values of type G'('t):

(i) γ <=< φ is defined,
and is a natural transformation from F to MG'
==>
(i'') fun a -> gamma =<< phi a is defined,
and is a function from type F('t) -> M(G'('t))

         (ii) (ρ <=< γ) <=< φ  =  ρ <=< (γ <=< φ)
==>
(fun a -> (rho <=< gamma) =<< phi a)  =  (fun a -> rho =<< (gamma <=< phi) a)
(fun a -> (fun b -> rho =<< gamma b) =<< phi a)  =
(fun a -> rho =<< (gamma =<< phi a))

(ii'') (fun b -> rho =<< gamma b) =<< phi a  =  rho =<< (gamma =<< phi a)

      (iii.1) (unit G') <=< γ  =  γ
whenever γ is a natural transformation from some FG' to MG'
==>
(unit G') <=< gamma  =  gamma
whenever gamma is a function of type F(G'('t)) -> M(G'('t))

(fun b -> (unit G') =<< gamma b)  =  gamma

(unit G') =<< gamma b  =  gamma b

Let return be a polymorphic function mapping arguments of any
type 't to M('t). In particular, it maps arguments c of type
G'('t) to the monadic value (unit G') c, of type M(G'('t)).

(iii.1'') return =<< gamma b  =  gamma b

      (iii.2) γ  =  γ <=< (unit G)
whenever γ is a natural transformation from G to some MR'G
==>
gamma  =  gamma <=< (unit G)
whenever gamma is a function of type G('t) -> M(R'(G('t)))

gamma  =  (fun b -> gamma =<< (unit G) b)

As above, return will map arguments b of type G('t) to the
monadic value (unit G) b, of type M(G('t)).

gamma  =  (fun b -> gamma =<< return b)

(iii.2'') gamma b  =  gamma =<< return b


Summarizing (ii''), (iii.1''), (iii.2''), these are the monadic laws as usually stated in the functional programming literature:

• (fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)

Usually written reversed, and with a monadic variable u standing in for phi a:

u >>= (fun b -> gamma b >>= rho) = (u >>= gamma) >>= rho

• return =<< gamma b = gamma b

Usually written reversed, and with u standing in for gamma b:

u >>= return = u

• gamma b = gamma =<< return b

Usually written reversed:

return b >>= gamma = gamma b