`(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 + (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. +* 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`.
+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 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-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. +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. (Every node must have at least one such, which is that node's 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. +* 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`. +* any monoid_{E}, which is such that for every morphism `f:C1->C2`: 1_{C2}o f = f = f o 1_{C1}+ (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 1_{X}, 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 `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 `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:
+* 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`.
+ 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:
+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 of+ +A 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-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`. + (ii) associate with every morphism f:C1→C2 ofCa morphism F(f):F(C1)→F(C2) ofD-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**. + (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 in-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)`. +That 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 **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:
+Consider four categories + --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**. +ThenB-+ +---C--+ +----D-----+ +--E-- | | | | | | F: ------> G: ------> K: ------> - | | | | | η | | | ψ + | | | | | η | | | ψ | | | | v | | v | | H: ------> L: ------> - | | | | | φ | | + | | | | | 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". For any morphism `f:C1→C2`

in + φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1] +-by naturalness of φ, is: +by naturalness of

`φ`

, is:
- φ[C2] o H(f) o η[C1] = J(f) o φ[C1] o η[C1]
++ φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1] +-by naturalness of η, is: +by naturalness of

`η`

, is:
- φ[C2] o η[C2] o G(f) = J(f) o φ[C1] o η[C1]
++ φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ 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`: +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] o G(f) = J(f) o (φ -v- η)[C1]
++ (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: +

`(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])
++ (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. @@ -153,236 +186,425 @@ 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. +A **monad** is a structure consisting of an (endo)functor `M` from some category

`φ`

, each being between some arbitrary endofunctor `F` on `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)`.
+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. Since composition is associative I don't specify the order of composition on the rhs.
-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 `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`.) +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`

.)
-`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`.
+`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)) -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: + (iii.2) ρ = ρ <=< unit + (here ρ has to be a natural transformation from 1C) +- (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

`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:
-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:
++ 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 +- 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

`γ`

is a natural transformation from some `FG'` to `MG'`.
-where as we said `q` is a natural transformation from some `PQ'` to `MQ'`.
+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:
-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:
++ 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) +- 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

`γ`

is a natural transformation from `G` to some `MR'G`.
-where as we said `q` is a natural transformation from `Q` to some `MP'Q`.
+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 +-The standard category-theory presentation of the monad laws ------------------------------------------------------------ + + +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 `<=<`. -(* - P2. every element C1 of a category **C** has an identity morphism 1

`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 `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:a->b in **C**:
+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, let

`γ`

be a transformation from `G` to `MG'`, and
+ consider the composite transformation `((join MG') -v- (MM γ))`

.
-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*).
+* `γ`

assigns elements `C1` in `γ\*:G(C1) → MG'(C1)`

. `(MM γ)`

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

.
-Next, consider the composite transformation ((M q) -v- (join Q)).
- (3) This assigns to C1 the morphism M(q*) o join[Q(C1)].
+* `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(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]
+Composing them:
+
++ (2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*). +-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, 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)]. +-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] +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] +-Next consider the composite transformation ((unit MQ') -v- q). (6) This assigns to C1 the morphism unit[MQ'(C1)] o q*. +So our **(lemma 1)** is: -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] +

+ ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), + where as we said γ is a natural transformation from G to MG'. +-So our lemma (2) is: (((M q) -v- (unit Q)) = ((unit MQ') -v- 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

`f:C1→C2`

in + (4) unit[C2] ∘ f = M(f) ∘ unit[C1] +- 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: +Next, consider the composite transformation

`((M γ) -v- (unit G))`

:
- (i) q <=< p etc are also in T
- ==>
- (i') ((join Q') (M q) p) etc are also in T
++ (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)`

:
- (ii) (r <=< q) <=< p = r <=< (q <=< p)
++ (6) ((unit MG') -v- γ) 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 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 ==> - (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: + (i') ((join G') (M γ) φ) etc are also in T +- ((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) +

+ (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) - which will be true for all r,q,p just in case: + similarly, ρ <=< (γ <=< φ) is: + ((join R') (M ρ) ((join G') (M γ) φ)) - ((join R') (M join R')) = ((join R') (join MR')), for any R'. + 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) - which will in turn be true just in case: + [-- Are the next two steps too cavalier? --] - (ii') (join (M join)) = (join (join M)) + 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 P') <=< p = p +

+ (iii.1) (unit G') <=< γ = γ + when γ is a natural transformation from some FG' to MG' ==> - (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 + (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) - which will be true for all p just in case: + substituting in (iii.1), we get: + ((join G') ((M unit) G') γ) = γ - ((join P') (M unit P')) = the identity transformation, for any P' + which is: + (((join (M unit)) G') γ) = γ - which will in turn be true just in case: + [-- Are the next two steps too cavalier? --] - (iii.1') (join (M unit) = the identity transformation + 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) p = p <=< (unit P) +

+ (iii.2) γ = γ <=< (unit G) + when γ is a natural transformation from G to some MR'G ==> - 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) + 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 will be true for all p just in case: + which is: + γ = (((join (unit M)) R'G) γ) - ((join P') (unit MP')) = the identity transformation, for any P' + [-- Are the next two steps too cavalier? --] - which will in turn be true just in case: + 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: - 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: +

+ 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 Q') (M q) p) etc also in T + (i') ((join G') (M γ) φ) etc also in T - (ii') (join (M join)) = (join (join M)) + (ii') (join (M join)) = (join (join M)) (iii.1') (join (M unit)) = the identity transformation - (iii.2')(join (unit M)) = the identity transformation + (iii.2') (join (unit M)) = the identity transformation ++ + + +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

`f:C1→C2`

to functions `M(f):M(C1)→M(C2)`

. This is also known as `lift`_{M} 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)] +-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.) +Now where `gamma` is another function of type

`F'('t) → M(G'('t))`

, we define:
-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,...].
++ 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: -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] +

+ 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), + 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 + when γ is a natural transformation from some FG' to MG' + ==> + (unit G') <=< gamma = gamma + when 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) + when γ is a natural transformation from G to some MR'G + ==> + gamma = gamma <=< (unit G) + when 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 +-The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2). +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`: -In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question. + `u >>= (fun b -> gamma b >>= rho) = (u >>= gamma) >>= rho` -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: +* `return =<< gamma b = gamma b` - let p = fun c -> [(1,c), (2,c)] + Usually written reversed, and with `u` standing in for `gamma b`: -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')]. + `u >>= return = u` -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. +* `gamma b = gamma =<< return b` -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. + Usually written reversed: + `return b >>= gamma = gamma b` +