+++ /dev/null
-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.
-
-
-Monoids
--------
-A **monoid** is a structure <code>(S,⋆,z)</code> consisting of an associative binary operation <code>⋆</code> over some set `S`, which is closed under <code>⋆</code>, and which contains an identity element `z` for <code>⋆</code>. That is:
-
-
-<pre>
- 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
-</pre>
-
-Some examples of monoids are:
-
-* finite strings of an alphabet `A`, with <code>⋆</code> being concatenation and `z` being the empty string
-* all functions <code>X→X</code> over a set `X`, with <code>⋆</code> being composition and `z` being the identity function over `X`
-* the natural numbers with <code>⋆</code> 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 <code>⋆</code> 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 <b>C</b> has source `C1` and target `C2`, we'll write <code>f:C1→C2</code>.
-
-To have a category, the elements and morphisms have to satisfy some constraints:
-
-<pre>
- (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<sub>X</sub>, which is such that for every morphism f:C1→C2:
- 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
-</pre>
-
-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 <code>(S,⋆,z)</code> 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 <code>s3=s1⋆s2</code>. The identity morphism for the (single) category element `Q` is the monoid's identity `z`.
-
-* a **preorder** is a structure <code>(S, ≤)</code> 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 <code>s1 ≤ s2</code> nor <code>s2 ≤ s1</code>). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that <code>s1 ≤ s2</code> and <code>s2 ≤ s1</code> 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 <code>(S,≤)</code> 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 <code>s1 ≤ s2</code>.
-
-
-Functors
---------
-A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category <b>C</b> to category <b>D</b> must:
-
-<pre>
- (i) associate with every element C1 of <b>C</b> an element F(C1) of <b>D</b>
-
- (ii) associate with every morphism f:C1→C2 of <b>C</b> a morphism F(f):F(C1)→F(C2) of <b>D</b>
-
- (iii) "preserve identity", that is, for every element C1 of <b>C</b>:
- F of C1's identity morphism in <b>C</b> must be the identity morphism of F(C1) in <b>D</b>:
- F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
-
- (iv) "distribute over composition", that is for any morphisms f and g in <b>C</b>:
- F(g ∘ f) = F(g) ∘ F(f)
-</pre>
-
-A functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism of <b>C</b> to itself is denoted `1C`.
-
-How functors compose: If `G` is a functor from category <b>C</b> to category <b>D</b>, and `K` is a functor from category <b>D</b> to category <b>E</b>, then `KG` is a functor which maps every element `C1` of <b>C</b> to element `K(G(C1))` of <b>E</b>, and maps every morphism `f` of <b>C</b> to morphism `K(G(f))` of <b>E</b>.
-
-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 <b>C</b> to category <b>D</b>, a natural transformation η between `G` and `H` is a family of morphisms <code>η[C1]:G(C1)→H(C1)</code> in <b>D</b> for each element `C1` of <b>C</b>. That is, <code>η[C1]</code> has as source `C1`'s image under `G` in <b>D</b>, and as target `C1`'s image under `H` in <b>D</b>. The morphisms in this family must also satisfy the constraint:
-
-<pre>
- for every morphism f:C1→C2 in <b>C</b>:
- η[C2] ∘ G(f) = H(f) ∘ η[C1]
-</pre>
-
-That is, the morphism via `G(f)` from `G(C1)` to `G(C2)`, and then via <code>η[C2]</code> to `H(C2)`, is identical to the morphism from `G(C1)` via <code>η[C1]</code> to `H(C1)`, and then via `H(f)` from `H(C1)` to `H(C2)`.
-
-
-How natural transformations compose:
-
-Consider four categories <b>B</b>, <b>C</b>, <b>D</b>, and <b>E</b>. Let `F` be a functor from <b>B</b> to <b>C</b>; `G`, `H`, and `J` be functors from <b>C</b> to <b>D</b>; and `K` and `L` be functors from <b>D</b> to <b>E</b>. 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:
-
-<pre>
- - <b>B</b> -+ +--- <b>C</b> --+ +---- <b>D</b> -----+ +-- <b>E</b> --
- | | | | | |
- F: ------> G: ------> K: ------>
- | | | | | η | | | ψ
- | | | | v | | v
- | | H: ------> L: ------>
- | | | | | φ | |
- | | | | v | |
- | | J: ------> | |
- -----+ +--------+ +------------+ +-------
-</pre>
-
-Then <code>(η F)</code> is a natural transformation from the (composite) functor `GF` to the composite functor `HF`, such that where `B1` is an element of category <b>B</b>, <code>(η F)[B1] = η[F(B1)]</code>---that is, the morphism in <b>D</b> that <code>η</code> assigns to the element `F(B1)` of <b>C</b>.
-
-And <code>(K η)</code> is a natural transformation from the (composite) functor `KG` to the (composite) functor `KH`, such that where `C1` is an element of category <b>C</b>, <code>(K η)[C1] = K(η[C1])</code>---that is, the morphism in <b>E</b> that `K` assigns to the morphism <code>η[C1]</code> of <b>D</b>.
-
-
-<code>(φ -v- η)</code> is a natural transformation from `G` to `J`; this is known as a "vertical composition". For any morphism <code>f:C1→C2</code> in <b>C</b>:
-
-<pre>
- φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1]
-</pre>
-
-by naturalness of <code>φ</code>, is:
-
-<pre>
- φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1]
-</pre>
-
-by naturalness of <code>η</code>, is:
-
-<pre>
- φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1]
-</pre>
-
-Hence, we can define <code>(φ -v- η)[\_]</code> as: <code>φ[\_] ∘ η[\_]</code> and rely on it to satisfy the constraints for a natural transformation from `G` to `J`:
-
-<pre>
- (φ -v- η)[C2] ∘ G(f) = J(f) ∘ (φ -v- η)[C1]
-</pre>
-
-An observation we'll rely on later: given the definitions of vertical composition and of how natural transformations compose with functors, it follows that:
-
-<pre>
- ((φ -v- η) F) = ((φ F) -v- (η F))
-</pre>
-
-I'll assert without proving that vertical composition is associative and has an identity, which we'll call "the identity transformation."
-
-
-<code>(ψ -h- η)</code> 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:
-
-<pre>
- (φ -h- η)[C1] = L(η[C1]) ∘ ψ[G(C1)]
- = ψ[H(C1)] ∘ K(η[C1])
-</pre>
-
-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 <b>C</b> to itself, along with some natural transformations, which we'll specify in a moment.
-
-Let `T` be a set of natural transformations <code>φ</code>, each being between some arbitrary endofunctor `F` on <b>C</b> and another functor which is the composite `MF'` of `M` and another arbitrary endofunctor `F'` on <b>C</b>. That is, for each element `C1` in <b>C</b>, <code>φ</code> 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, <code>φ</code> is a transformation from functor `F` to `MF'`, <code>γ</code> 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 <b>C</b> 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 <code>φ</code> and <code>γ</code> 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 <code>(M γ)</code> will also be a natural transformation, formed by composing the functor `M` with the natural transformation <code>γ</code>. 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')`, <code>(M γ)</code>, and <code>φ</code>, and abbreviate it as follows. Since composition is associative I don't specify the order of composition on the rhs.
-
-<pre>
- γ <=< φ =def. ((join G') -v- (M γ) -v- φ)
-</pre>
-
-In other words, `<=<` is a binary operator that takes us from two members <code>φ</code> and <code>γ</code> of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes it's written <code>φ >=> γ</code> where that's the same as <code>γ <=< φ</code>.)
-
-<code>φ</code> is a transformation from `F` to `MF'`, where the latter = `MG`; <code>(M γ)</code> is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite <code>γ <=< φ</code> 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:
-
-<pre>
- (T, <=<, unit) constitute a monoid
-</pre>
-
-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, <code>γ <=< φ</code> isn't fully defined on `T`, but only when <code>φ</code> is a transformation to some `MF'` and <code>γ</code> is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws must hold:
-
-<pre>
- (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)
-</pre>
-
-If <code>φ</code> is a natural transformation from `F` to `M(1C)` and <code>γ</code> is <code>(φ G')</code>, that is, a natural transformation from `FG'` to `MG'`, then we can extend (iii.1) as follows:
-
-<pre>
- 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
-</pre>
-
-where as we said <code>γ</code> is a natural transformation from some `FG'` to `MG'`.
-
-Similarly, if <code>ρ</code> is a natural transformation from `1C` to `MR'`, and <code>γ</code> is <code>(ρ G)</code>, that is, a natural transformation from `G` to `MR'G`, then we can extend (iii.2) as follows:
-
-<pre>
- 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)
-</pre>
-
-where as we said <code>γ</code> is a natural transformation from `G` to some `MR'G`.
-
-Summarizing then, the monad laws can be expressed as:
-
-<pre>
- 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
-</pre>
-
-
-
-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 <b>C</b> has an identity morphism 1<sub>C1</sub> such that for every morphism f:C1→C2 in <b>C</b>: 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>.
- P3. functors "preserve identity", that is for every element C1 in F's source category: F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
--->
-
-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: <code>F(g ∘ f) = F(g) ∘ F(f)</code>
-
-* if <code>η</code> is a natural transformation from `G` to `H`, then for every <code>f:C1→C2</code> in `G` and `H`'s source category <b>C</b>: <code>η[C2] ∘ G(f) = H(f) ∘ η[C1]</code>.
-
-* <code>(η F)[X] = η[F(X)]</code>
-
-* <code>(K η)[X] = K(η[X])</code>
-
-* <code>((φ -v- η) F) = ((φ F) -v- (η F))</code>
-
-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 <b>C</b>, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism <code>f:C1→C2</code> in <b>C</b>:
-
-<pre>
- (1) join[C2] ∘ MM(f) = M(f) ∘ join[C1]
-</pre>
-
-Next, let <code>γ</code> be a transformation from `G` to `MG'`, and
- consider the composite transformation <code>((join MG') -v- (MM γ))</code>.
-
-* <code>γ</code> assigns elements `C1` in <b>C</b> a morphism <code>γ\*:G(C1) → MG'(C1)</code>. <code>(MM γ)</code> is a transformation that instead assigns `C1` the morphism <code>MM(γ\*)</code>.
-
-* `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.
-
-Composing them:
-
-<pre>
- (2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*).
-</pre>
-
-Next, consider the composite transformation <code>((M γ) -v- (join G))</code>:
-
-<pre>
- (3) ((M γ) -v- (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
-</pre>
-
-So for every element `C1` of <b>C</b>:
-
-<pre>
- ((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]
-</pre>
-
-So our **(lemma 1)** is:
-
-<pre>
- ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)),
- where as we said γ is a natural transformation from G to MG'.
-</pre>
-
-
-Next recall that `unit` is a natural transformation from `1C` to `M`. So for elements `C1` in <b>C</b>, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism <code>f:C1→C2</code> in <b>C</b>:
-
-<pre>
- (4) unit[C2] ∘ f = M(f) ∘ unit[C1]
-</pre>
-
-Next, consider the composite transformation <code>((M γ) -v- (unit G))</code>:
-
-<pre>
- (5) ((M γ) -v- (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
-</pre>
-
-Next, consider the composite transformation <code>((unit MG') -v- γ)</code>:
-
-<pre>
- (6) ((unit MG') -v- γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
-</pre>
-
-So for every element C1 of <b>C</b>:
-
-<pre>
- ((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]
-</pre>
-
-So our **(lemma 2)** is:
-
-<pre>
- (((M γ) -v- (unit G)) = ((unit MG') -v- γ)),
- where as we said γ is a natural transformation from G to MG'.
-</pre>
-
-
-Finally, we substitute <code>((join G') -v- (M γ) -v- φ)</code> for <code>γ <=< φ</code> in the monad laws. For simplicity, I'll omit the "-v-".
-
-<pre>
- 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
-</pre>
-
-<pre>
- (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))
-</pre>
-
-<pre>
- (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
-</pre>
-
-<pre>
- (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
-</pre>
-
-
-Collecting the results, our monad laws turn out in this format to be:
-
-<pre>
- 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
-</pre>
-
-
-
-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 <b>C</b> 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 <code>f:C1→C2</code> to functions <code>M(f):M(C1)→M(C2)</code>. This is also known as <code>lift<sub>M</sub> f</code> 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 <code>f:x→y</code> 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`:
-
-<pre>
- let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
-</pre>
-
-
-
-Now where `gamma` is another function of type <code>F'('t) → M(G'('t))</code>, we define:
-
-<pre>
- gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a)
- = ((join G') -v- (M gamma) -v- phi) a
- = (gamma <=< phi) a
-</pre>
-
-Hence:
-
-<pre>
- gamma <=< phi = fun a -> (gamma =<< phi a)
-</pre>
-
-`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:
-
-
-<pre>
- 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))
-</pre>
-
-<pre>
- (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)
-</pre>
-
-<pre>
- (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
-</pre>
-
-<pre>
- (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
-</pre>
-
-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`
-
-