`⋆`

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.
+* 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
----------
@@ -51,28 +53,28 @@ To have a category, the elements and morphisms have to satisfy some constraints:
(ii) composition of morphisms has to be associative
- (iii) every element E of the category has to have an identity
- morphism 1`(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 `(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
@@ -136,7 +138,7 @@ Then `(η F)`

is a natural transformation from the (composite) fun
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 `(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 -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] @@ -186,245 +188,426 @@ In earlier days, these were also called "triples." A **monad** is a structure consisting of an (endo)functor `M` from some categoryCto 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 (variable) functor `F` and another functor which is the composite `MF'` of `M` and a (variable) functor `F'`. That is, for each element `C1` inC,`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. +Let `T` be a set of natural transformations`φ`

, each being between some arbitrary endofunctor `F` onCand another functor which is the composite `MF'` of `M` and another arbitrary endofunctor `F'` onC. That is, for each element `C1` inC,`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` forCto `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` forCto `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`. +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: +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)-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: +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.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) + (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: +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 - ?? + 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') <=< 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: +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 = (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 = (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 `MF'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' -The standard category-theory presentation of the monad laws ------------------------------------------------------------ + (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 -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 categoryC: η[C2] ∘ F(f) = G(f) ∘ η[C1]. +* 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 categoryC:`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 inC, join[C1] will be a morphism from MM(C1) to M(C1). And for any morphism f:a→b inC: +Recall that `join` is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` inC, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism`f:C1→C2`

inC: + ++ (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))`

. - (1) join[b] ∘ MM(f) = M(f) ∘ join[a] +*`γ`

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

.`(MM γ)`

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

. -Next, consider the composite transformation ((join MG') -v- (MM γ)). - γ is a transformation from G to MG', and assigns elements C1 inCa 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 MMMG' to MMG' that assigns C1 the morphism join[MG'(C1)]. - Composing them: +* `(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) This assigns to C1 the morphism M(γ*) ∘ join[G(C1)]. +Next, consider the composite transformation`((M γ) -v- (join G))`

: -So for every element C1 ofC: ++ (3) ((M γ) -v- (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)]. ++ +So for every element `C1` ofC: + +((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: + 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: -So our (lemma 1) is: ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), where γ is a transformation from G to MG'. ++ ((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` inC, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism`f:C1→C2`

inC: + ++ (4) unit[C2] ∘ f = M(f) ∘ unit[C1] ++Next, consider the composite transformation`((M γ) -v- (unit G))`

: -Next recall that unit is a natural transformation from 1C to M. So for elements C1 inC, unit[C1] will be a morphism from C1 to M(C1). And for any morphism f:a→b inC: - (4) unit[b] ∘ f = M(f) ∘ unit[a] ++ (5) ((M γ) -v- (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)]. +-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)`

: -Next consider the composite transformation ((unit MG') -v- γ). (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*. ++ (6) ((unit MG') -v- γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*. +So for every element C1 ofC: + +((M γ) -v- (unit G))[C1], by (5) = - M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*: G(C1)→MG'(C1) is: + 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'. - +So our **(lemma 2)** is: -Finally, we substitute ((join G') -v- (M γ) -v- φ) for γ <=< φ in the monad laws. For simplicity, I'll omit the "-v-". ++ (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), + where as we said γ is a natural transformation from G to MG'. +- 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 +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: - (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) + (i) γ <=< φ etc are also in T ==> - (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: + (i') ((join G') (M γ) φ) etc are also in T +- ((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) ++ (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 ρ,γ,φ 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 F') <=< φ = φ ++ (iii.1) (unit G') <=< γ = γ + when γ is a natural transformation from some FG' to MG' ==> - (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 + (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 φ just in case: + substituting in (iii.1), we get: + ((join G') ((M unit) G') γ) = γ - ((join F') (M unit F')) = the identity transformation, for any F' + 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) φ = φ <=< (unit F) ++ (iii.2) γ = γ <=< (unit G) + when γ is a natural transformation from G to some MR'G ==> - 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) + 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 φ just in case: + which is: + γ = (((join (unit M)) R'G) γ) - ((join F') (unit MF')) = the identity transformation, for any F' + [-- 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 φ a transformation from F to MF', γ a transformation from F' to MG', ρ a transformation from G' 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 G') (M γ) φ) 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 ++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). -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. +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 categoryCwill 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 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,...]. +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`lift`

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_{M}f`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)). -A natural transformation t assigns to each type C1 in+ +Ca morphism t[C1]: C1→M(C1) such that, for every f:C1→C2: - t[C2] ∘ f = M(f) ∘ t[C1] + (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 +-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 φ 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 φ = fun c → [(1,c), (2,c)] + Usually written reversed, and with `u` standing in for `gamma b`: -φ 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 φ, we'll work with (φ : C1 → M(int * C1)). This only accepts arguments of type C1. For generality, I'll talk of functions with the type (φ : 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 (φ : C1 → M(C1')) to an argument of type C1. + Usually written reversed: + `return b >>= gamma = gamma b` +