1 **Don't try to read this yet!!! Many substantial edits are still in process.
6 I really don't know much category theory. Just enough to put this
7 together. Also, this really is "put together." I haven't yet found an
8 authoritative source (that's accessible to a category theory beginner like
9 myself) that discusses the correspondence between the category-theoretic and
10 functional programming uses of these notions in enough detail to be sure that
11 none of the pieces here is mistaken.
12 In particular, it wasn't completely obvious how to map the polymorphism on the
13 programming theory side into the category theory. The way I accomplished this
14 may be more complex than it needs to be.
15 Also I'm bothered by the fact that our `<=<` operation is only partly defined
16 on our domain of natural transformations.
17 There are three additional points below that I wonder whether may be too
19 But all considered, this does seem to
20 me to be a reasonable way to put the pieces together. We very much welcome
21 feedback from anyone who understands these issues better, and will make
27 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:
31 for all s1, s2, s3 in S:
32 (i) s1⋆s2 etc are also in S
33 (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
34 (iii) z⋆s1 = s1 = s1⋆z
37 Some examples of monoids are:
39 * finite strings of an alphabet `A`, with <code>⋆</code> being concatenation and `z` being the empty string
40 * all functions <code>X→X</code> over a set `X`, with <code>⋆</code> being composition and `z` being the identity function over `X`
41 * 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**.
42 * if we let <code>⋆</code> be multiplication and `z` be 1, we get different monoids over the same sets as in the previous item.
46 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."
48 When a morphism `f` in category <b>C</b> has source `C1` and target `C2`, we'll write <code>f:C1→C2</code>.
50 To have a category, the elements and morphisms have to satisfy some constraints:
53 (i) the class of morphisms has to be closed under composition:
54 where f:C1→C2 and g:C2→C3, g ∘ f is also a
55 morphism of the category, which maps C1→C3.
57 (ii) composition of morphisms has to be associative
59 (iii) every element X of the category has to have an identity
60 morphism 1<sub>X</sub>, which is such that for every morphism f:C1→C2:
61 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
64 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.
66 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.
69 Some examples of categories are:
71 * 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.
73 * 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`.
75 * 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:
77 * 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)
78 * sets ordered by size (this illustrates it too)
80 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>.
85 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:
88 (i) associate with every element C1 of <b>C</b> an element F(C1) of <b>D</b>
90 (ii) associate with every morphism f:C1→C2 of <b>C</b> a morphism F(f):F(C1)→F(C2) of <b>D</b>
92 (iii) "preserve identity", that is, for every element C1 of <b>C</b>:
93 F of C1's identity morphism in <b>C</b> must be the identity morphism of F(C1) in <b>D</b>:
94 F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
96 (iv) "distribute over composition", that is for any morphisms f and g in <b>C</b>:
97 F(g ∘ f) = F(g) ∘ F(f)
100 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`.
102 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>.
104 I'll assert without proving that functor composition is associative.
108 Natural Transformation
109 ----------------------
110 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.
112 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:
115 for every morphism f:C1→C2 in <b>C</b>:
116 η[C2] ∘ G(f) = H(f) ∘ η[C1]
119 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)`.
122 How natural transformations compose:
124 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:
127 - <b>B</b> -+ +--- <b>C</b> --+ +---- <b>D</b> -----+ +-- <b>E</b> --
129 F: ------> G: ------> K: ------>
130 | | | | | η | | | ψ
132 | | H: ------> L: ------>
136 -----+ +--------+ +------------+ +-------
139 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>.
141 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>.
144 <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>:
147 φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1]
150 by naturalness of <code>φ</code>, is:
153 φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1]
156 by naturalness of <code>η</code>, is:
159 φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1]
162 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`:
165 (φ -v- η)[C2] ∘ G(f) = J(f) ∘ (φ -v- η)[C1]
168 An observation we'll rely on later: given the definitions of vertical composition and of how natural transformations compose with functors, it follows that:
171 ((φ -v- η) F) = ((φ F) -v- (η F))
174 I'll assert without proving that vertical composition is associative and has an identity, which we'll call "the identity transformation."
177 <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:
180 (φ -h- η)[C1] = L(η[C1]) ∘ ψ[G(C1)]
181 = ψ[H(C1)] ∘ K(η[C1])
184 Horizontal composition is also associative, and has the same identity as vertical composition.
190 In earlier days, these were also called "triples."
192 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.
194 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.
196 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)`.
198 We also need to designate for `M` a `join` transformation, which is a natural transformation from the (composite) functor `MM` to `M`.
200 These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion.
202 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.
205 γ <=< φ =def. ((join G') -v- (M γ) -v- φ)
208 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>.)
210 <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`.
212 Now we can specify the "monad laws" governing a monad as follows:
215 (T, <=<, unit) constitute a monoid
218 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:
221 (i) γ <=< φ is also in T
223 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
225 (iii.1) unit <=< φ = φ
226 (here φ has to be a natural transformation to M(1C))
228 (iii.2) ρ = ρ <=< unit
229 (here ρ has to be a natural transformation from 1C)
232 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:
236 = ((unit <=< φ) G')
237 since unit is a natural transformation to M(1C), this is:
238 = (((join 1C) -v- (M unit) -v- φ) G')
239 = (((join 1C) G') -v- ((M unit) G') -v- (φ G'))
240 = ((join (1C G')) -v- (M (unit G')) -v- γ)
241 = ((join G') -v- (M (unit G')) -v- γ)
242 since (unit G') is a natural transformation to MG', this is:
243 = (unit G') <=< γ
246 where as we said <code>γ</code> is a natural transformation from some `FG'` to `MG'`.
248 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:
252 = ((ρ <=< unit) G)
253 = since ρ is a natural transformation to MR', this is:
254 = (((join R') -v- (M ρ) -v- unit) G)
255 = (((join R') G) -v- ((M ρ) G) -v- (unit G))
256 = ((join (R'G)) -v- (M (ρ G)) -v- (unit G))
257 since γ = (ρ G) is a natural transformation to MR'G, this is:
258 = γ <=< (unit G)
261 where as we said <code>γ</code> is a natural transformation from `G` to some `MR'G`.
263 Summarizing then, the monad laws can be expressed as:
266 For all ρ, γ, φ in T for which ρ <=< γ and γ <=< φ are defined:
268 (i) γ <=< φ etc are also in T
270 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
272 (iii.1) (unit G') <=< γ = γ
273 whenever γ is a natural transformation from some FG' to MG'
275 (iii.2) γ = γ <=< (unit G)
276 whenever γ is a natural transformation from G to some MR'G
281 Getting to the standard category-theory presentation of the monad laws
282 ----------------------------------------------------------------------
283 In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`.
286 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>.
287 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>.
290 Let's remind ourselves of principles stated above:
292 * composition of morphisms, functors, and natural compositions is associative
294 * 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>
296 * 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>.
298 * <code>(η F)[X] = η[F(X)]</code>
300 * <code>(K η)[X] = K(η[X])</code>
302 * <code>((φ -v- η) F) = ((φ F) -v- (η F))</code>
304 Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
307 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>:
310 (1) join[C2] ∘ MM(f) = M(f) ∘ join[C1]
313 Next, let <code>γ</code> be a transformation from `G` to `MG'`, and
314 consider the composite transformation <code>((join MG') -v- (MM γ))</code>.
316 * <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>.
318 * `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.
323 (2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*).
326 Next, consider the composite transformation <code>((M γ) -v- (join G))</code>:
329 (3) ((M γ) -v- (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
332 So for every element `C1` of <b>C</b>:
335 ((join MG') -v- (MM γ))[C1], by (2) is:
336 join[MG'(C1)] ∘ MM(γ*), which by (1), with f=γ*:G(C1)→MG'(C1) is:
337 M(γ*) ∘ join[G(C1)], which by 3 is:
338 ((M γ) -v- (join G))[C1]
341 So our **(lemma 1)** is:
344 ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)),
345 where as we said γ is a natural transformation from G to MG'.
349 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>:
352 (4) unit[C2] ∘ f = M(f) ∘ unit[C1]
355 Next, consider the composite transformation <code>((M γ) -v- (unit G))</code>:
358 (5) ((M γ) -v- (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
361 Next, consider the composite transformation <code>((unit MG') -v- γ)</code>:
364 (6) ((unit MG') -v- γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
367 So for every element C1 of <b>C</b>:
370 ((M γ) -v- (unit G))[C1], by (5) =
371 M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*:G(C1)→MG'(C1) is:
372 unit[MG'(C1)] ∘ γ*, which by (6) =
373 ((unit MG') -v- γ)[C1]
376 So our **(lemma 2)** is:
379 (((M γ) -v- (unit G)) = ((unit MG') -v- γ)),
380 where as we said γ is a natural transformation from G to MG'.
384 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-".
387 For all ρ, γ, φ in T,
388 where φ is a transformation from F to MF',
389 γ is a transformation from G to MG',
390 ρ is a transformation from R to MR',
393 (i) γ <=< φ etc are also in T
395 (i') ((join G') (M γ) φ) etc are also in T
399 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
401 (ρ <=< γ) is a transformation from G to MR', so
402 (ρ <=< γ) <=< φ becomes: ((join R') (M (ρ <=< γ)) φ)
403 which is: ((join R') (M ((join R') (M ρ) γ)) φ)
405 similarly, ρ <=< (γ <=< φ) is:
406 ((join R') (M ρ) ((join G') (M γ) φ))
408 substituting these into (ii), and helping ourselves to associativity on the rhs, we get:
409 ((join R') (M ((join R') (M ρ) γ)) φ) = ((join R') (M ρ) (join G') (M γ) φ)
411 which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields:
412 ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (M ρ) (join G') (M γ) φ)
414 which by lemma 1, with ρ a transformation from G' to MR', yields:
415 ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (join MR') (MM ρ) (M γ) φ)
417 [-- Are the next two steps too cavalier? --]
419 which will be true for all ρ, γ, φ only when:
420 ((join R') (M join R')) = ((join R') (join MR')), for any R'
422 which will in turn be true when:
423 (ii') (join (M join)) = (join (join M))
427 (iii.1) (unit G') <=< γ = γ
428 when γ is a natural transformation from some FG' to MG'
430 (unit G') is a transformation from G' to MG', so:
431 (unit G') <=< γ becomes: ((join G') (M (unit G')) γ)
432 which is: ((join G') ((M unit) G') γ)
434 substituting in (iii.1), we get:
435 ((join G') ((M unit) G') γ) = γ
438 (((join (M unit)) G') γ) = γ
440 [-- Are the next two steps too cavalier? --]
442 which will be true for all γ just in case:
443 for any G', ((join (M unit)) G') = the identity transformation
445 which will in turn be true just in case:
446 (iii.1') (join (M unit)) = the identity transformation
450 (iii.2) γ = γ <=< (unit G)
451 when γ is a natural transformation from G to some MR'G
453 γ <=< (unit G) becomes: ((join R'G) (M γ) (unit G))
455 substituting in (iii.2), we get:
456 γ = ((join R'G) (M γ) (unit G))
458 which by lemma 2, yields:
459 γ = (((join R'G) ((unit MR'G) γ)
462 γ = (((join (unit M)) R'G) γ)
464 [-- Are the next two steps too cavalier? --]
466 which will be true for all γ just in case:
467 for any R'G, ((join (unit M)) R'G) = the identity transformation
469 which will in turn be true just in case:
470 (iii.2') (join (unit M)) = the identity transformation
474 Collecting the results, our monad laws turn out in this format to be:
477 For all ρ, γ, φ in T,
478 where φ is a transformation from F to MF',
479 γ is a transformation from G to MG',
480 ρ is a transformation from R to MR',
483 (i') ((join G') (M γ) φ) etc also in T
485 (ii') (join (M join)) = (join (join M))
487 (iii.1') (join (M unit)) = the identity transformation
489 (iii.2') (join (unit M)) = the identity transformation
494 Getting to the functional programming presentation of the monad laws
495 --------------------------------------------------------------------
496 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`.
498 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.)
500 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,...]`.
503 In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad."
505 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`:
508 let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
513 Now where `gamma` is another function of type <code>F'('t) → M(G'('t))</code>, we define:
516 gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a)
517 = ((join G') -v- (M gamma) -v- phi) a
524 gamma <=< phi = fun a -> (gamma =<< phi a)
527 `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`.
529 With these definitions, our monadic laws become:
533 Where phi is a polymorphic function of type F('t) -> M(F'('t))
534 gamma is a polymorphic function of type G('t) -> M(G'('t))
535 rho is a polymorphic function of type R('t) -> M(R'('t))
536 and F' = G and G' = R,
537 and a ranges over values of type F('t),
538 b ranges over values of type G('t),
539 and c ranges over values of type G'('t):
541 (i) γ <=< φ is defined,
542 and is a natural transformation from F to MG'
544 (i'') fun a -> gamma =<< phi a is defined,
545 and is a function from type F('t) -> M(G'('t))
549 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
551 (fun a -> (rho <=< gamma) =<< phi a) = (fun a -> rho =<< (gamma <=< phi) a)
552 (fun a -> (fun b -> rho =<< gamma b) =<< phi a) = (fun a -> rho =<< (gamma =<< phi a))
554 (ii'') (fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)
558 (iii.1) (unit G') <=< γ = γ
559 when γ is a natural transformation from some FG' to MG'
561 (unit G') <=< gamma = gamma
562 when gamma is a function of type F(G'('t)) -> M(G'('t))
564 fun b -> (unit G') =<< gamma b = gamma
566 (unit G') =<< gamma b = gamma b
568 Let return be a polymorphic function mapping arguments of any
569 type 't to M('t). In particular, it maps arguments c of type
570 G'('t) to the monadic value (unit G') c, of type M(G'('t)).
572 (iii.1'') return =<< gamma b = gamma b
576 (iii.2) γ = γ <=< (unit G)
577 when γ is a natural transformation from G to some MR'G
579 gamma = gamma <=< (unit G)
580 when gamma is a function of type G('t) -> M(R'(G('t)))
582 gamma = fun b -> gamma =<< (unit G) b
584 As above, return will map arguments b of type G('t) to the
585 monadic value (unit G) b, of type M(G('t)).
587 gamma = fun b -> gamma =<< return b
589 (iii.2'') gamma b = gamma =<< return b
592 Summarizing (ii''), (iii.1''), (iii.2''), these are the monadic laws as usually stated in the functional programming literature:
594 * `fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)`
596 Usually written reversed, and with a monadic variable `u` standing in
599 `u >>= (fun b -> gamma b >>= rho) = (u >>= gamma) >>= rho`
601 * `return =<< gamma b = gamma b`
603 Usually written reversed, and with `u` standing in for `gamma b`:
607 * `gamma b = gamma =<< return b`
609 Usually written reversed:
611 `return b >>= gamma = gamma b`