3 I really don't know much category theory. Just enough to put this
4 together. Also, this really is "put together." I haven't yet found an
5 authoritative source (that's accessible to a category theory beginner like
6 myself) that discusses the correspondence between the category-theoretic and
7 functional programming uses of these notions in enough detail to be sure that
8 none of the pieces here is mistaken.
9 In particular, it wasn't completely obvious how to map the polymorphism on the
10 programming theory side into the category theory. The way I accomplished this
11 may be more complex than it needs to be.
12 Also I'm bothered by the fact that our `<=<` operation is only partly defined
13 on our domain of natural transformations.
14 There are three additional points below that I wonder whether may be too
16 But all considered, this does seem to
17 me to be a reasonable way to put the pieces together. We very much welcome
18 feedback from anyone who understands these issues better, and will make
24 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:
28 for all s1, s2, s3 in S:
29 (i) s1⋆s2 etc are also in S
30 (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
31 (iii) z⋆s1 = s1 = s1⋆z
34 Some examples of monoids are:
36 * finite strings of an alphabet `A`, with <code>⋆</code> being concatenation and `z` being the empty string
37 * all functions <code>X→X</code> over a set `X`, with <code>⋆</code> being composition and `z` being the identity function over `X`
38 * 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**.
39 * if we let <code>⋆</code> be multiplication and `z` be 1, we get different monoids over the same sets as in the previous item.
43 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."
45 When a morphism `f` in category <b>C</b> has source `C1` and target `C2`, we'll write <code>f:C1→C2</code>.
47 To have a category, the elements and morphisms have to satisfy some constraints:
50 (i) the class of morphisms has to be closed under composition:
51 where f:C1→C2 and g:C2→C3, g ∘ f is also a
52 morphism of the category, which maps C1→C3.
54 (ii) composition of morphisms has to be associative
56 (iii) every element X of the category has to have an identity
57 morphism 1<sub>X</sub>, which is such that for every morphism f:C1→C2:
58 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
61 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.
63 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.
66 Some examples of categories are:
68 * 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.
70 * 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`.
72 * 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:
74 * 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)
75 * sets ordered by size (this illustrates it too)
77 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>.
82 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:
85 (i) associate with every element C1 of <b>C</b> an element F(C1) of <b>D</b>
87 (ii) associate with every morphism f:C1→C2 of <b>C</b> a morphism F(f):F(C1)→F(C2) of <b>D</b>
89 (iii) "preserve identity", that is, for every element C1 of <b>C</b>:
90 F of C1's identity morphism in <b>C</b> must be the identity morphism of F(C1) in <b>D</b>:
91 F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
93 (iv) "distribute over composition", that is for any morphisms f and g in <b>C</b>:
94 F(g ∘ f) = F(g) ∘ F(f)
97 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`.
99 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>.
101 I'll assert without proving that functor composition is associative.
105 Natural Transformation
106 ----------------------
107 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.
109 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:
112 for every morphism f:C1→C2 in <b>C</b>:
113 η[C2] ∘ G(f) = H(f) ∘ η[C1]
116 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)`.
119 How natural transformations compose:
121 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:
124 - <b>B</b> -+ +--- <b>C</b> --+ +---- <b>D</b> -----+ +-- <b>E</b> --
126 F: ------> G: ------> K: ------>
127 | | | | | η | | | ψ
129 | | H: ------> L: ------>
133 -----+ +--------+ +------------+ +-------
136 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>.
138 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>.
141 <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>:
144 φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1]
147 by naturalness of <code>φ</code>, is:
150 φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1]
153 by naturalness of <code>η</code>, is:
156 φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1]
159 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`:
162 (φ -v- η)[C2] ∘ G(f) = J(f) ∘ (φ -v- η)[C1]
165 An observation we'll rely on later: given the definitions of vertical composition and of how natural transformations compose with functors, it follows that:
168 ((φ -v- η) F) = ((φ F) -v- (η F))
171 I'll assert without proving that vertical composition is associative and has an identity, which we'll call "the identity transformation."
174 <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:
177 (φ -h- η)[C1] = L(η[C1]) ∘ ψ[G(C1)]
178 = ψ[H(C1)] ∘ K(η[C1])
181 Horizontal composition is also associative, and has the same identity as vertical composition.
187 In earlier days, these were also called "triples."
189 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.
191 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.
193 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)`.
195 We also need to designate for `M` a `join` transformation, which is a natural transformation from the (composite) functor `MM` to `M`.
197 These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion.
199 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.
202 γ <=< φ =def. ((join G') -v- (M γ) -v- φ)
205 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>.)
207 <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`.
209 Now we can specify the "monad laws" governing a monad as follows:
212 (T, <=<, unit) constitute a monoid
215 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:
218 (i) γ <=< φ is also in T
220 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
222 (iii.1) unit <=< φ = φ
223 (here φ has to be a natural transformation to M(1C))
225 (iii.2) ρ = ρ <=< unit
226 (here ρ has to be a natural transformation from 1C)
229 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:
233 = ((unit <=< φ) G')
234 since unit is a natural transformation to M(1C), this is:
235 = (((join 1C) -v- (M unit) -v- φ) G')
236 = (((join 1C) G') -v- ((M unit) G') -v- (φ G'))
237 = ((join (1C G')) -v- (M (unit G')) -v- γ)
238 = ((join G') -v- (M (unit G')) -v- γ)
239 since (unit G') is a natural transformation to MG', this is:
240 = (unit G') <=< γ
243 where as we said <code>γ</code> is a natural transformation from some `FG'` to `MG'`.
245 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:
249 = ((ρ <=< unit) G)
250 = since ρ is a natural transformation to MR', this is:
251 = (((join R') -v- (M ρ) -v- unit) G)
252 = (((join R') G) -v- ((M ρ) G) -v- (unit G))
253 = ((join (R'G)) -v- (M (ρ G)) -v- (unit G))
254 since γ = (ρ G) is a natural transformation to MR'G, this is:
255 = γ <=< (unit G)
258 where as we said <code>γ</code> is a natural transformation from `G` to some `MR'G`.
260 Summarizing then, the monad laws can be expressed as:
263 For all ρ, γ, φ in T for which ρ <=< γ and γ <=< φ are defined:
265 (i) γ <=< φ etc are also in T
267 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
269 (iii.1) (unit G') <=< γ = γ
270 whenever γ is a natural transformation from some FG' to MG'
272 (iii.2) γ = γ <=< (unit G)
273 whenever γ is a natural transformation from G to some MR'G
278 Getting to the standard category-theory presentation of the monad laws
279 ----------------------------------------------------------------------
280 In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`.
283 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>.
284 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>.
287 Let's remind ourselves of principles stated above:
289 * composition of morphisms, functors, and natural compositions is associative
291 * 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>
293 * 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>.
295 * <code>(η F)[X] = η[F(X)]</code>
297 * <code>(K η)[X] = K(η[X])</code>
299 * <code>((φ -v- η) F) = ((φ F) -v- (η F))</code>
301 Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
304 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>:
307 (1) join[C2] ∘ MM(f) = M(f) ∘ join[C1]
310 Next, let <code>γ</code> be a transformation from `G` to `MG'`, and
311 consider the composite transformation <code>((join MG') -v- (MM γ))</code>.
313 * <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>.
315 * `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.
320 (2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*).
323 Next, consider the composite transformation <code>((M γ) -v- (join G))</code>:
326 (3) ((M γ) -v- (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
329 So for every element `C1` of <b>C</b>:
332 ((join MG') -v- (MM γ))[C1], by (2) is:
333 join[MG'(C1)] ∘ MM(γ*), which by (1), with f=γ*:G(C1)→MG'(C1) is:
334 M(γ*) ∘ join[G(C1)], which by 3 is:
335 ((M γ) -v- (join G))[C1]
338 So our **(lemma 1)** is:
341 ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)),
342 where as we said γ is a natural transformation from G to MG'.
346 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>:
349 (4) unit[C2] ∘ f = M(f) ∘ unit[C1]
352 Next, consider the composite transformation <code>((M γ) -v- (unit G))</code>:
355 (5) ((M γ) -v- (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
358 Next, consider the composite transformation <code>((unit MG') -v- γ)</code>:
361 (6) ((unit MG') -v- γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
364 So for every element C1 of <b>C</b>:
367 ((M γ) -v- (unit G))[C1], by (5) =
368 M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*:G(C1)→MG'(C1) is:
369 unit[MG'(C1)] ∘ γ*, which by (6) =
370 ((unit MG') -v- γ)[C1]
373 So our **(lemma 2)** is:
376 (((M γ) -v- (unit G)) = ((unit MG') -v- γ)),
377 where as we said γ is a natural transformation from G to MG'.
381 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-".
384 For all ρ, γ, φ in T,
385 where φ is a transformation from F to MF',
386 γ is a transformation from G to MG',
387 ρ is a transformation from R to MR',
390 (i) γ <=< φ etc are also in T
392 (i') ((join G') (M γ) φ) etc are also in T
396 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
398 (ρ <=< γ) is a transformation from G to MR', so
399 (ρ <=< γ) <=< φ becomes: ((join R') (M (ρ <=< γ)) φ)
400 which is: ((join R') (M ((join R') (M ρ) γ)) φ)
402 similarly, ρ <=< (γ <=< φ) is:
403 ((join R') (M ρ) ((join G') (M γ) φ))
405 substituting these into (ii), and helping ourselves to associativity on the rhs, we get:
406 ((join R') (M ((join R') (M ρ) γ)) φ) = ((join R') (M ρ) (join G') (M γ) φ)
408 which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields:
409 ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (M ρ) (join G') (M γ) φ)
411 which by lemma 1, with ρ a transformation from G' to MR', yields:
412 ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (join MR') (MM ρ) (M γ) φ)
414 [-- Are the next two steps too cavalier? --]
416 which will be true for all ρ, γ, φ only when:
417 ((join R') (M join R')) = ((join R') (join MR')), for any R'
419 which will in turn be true when:
420 (ii') (join (M join)) = (join (join M))
424 (iii.1) (unit G') <=< γ = γ
425 when γ is a natural transformation from some FG' to MG'
427 (unit G') is a transformation from G' to MG', so:
428 (unit G') <=< γ becomes: ((join G') (M (unit G')) γ)
429 which is: ((join G') ((M unit) G') γ)
431 substituting in (iii.1), we get:
432 ((join G') ((M unit) G') γ) = γ
435 (((join (M unit)) G') γ) = γ
437 [-- Are the next two steps too cavalier? --]
439 which will be true for all γ just in case:
440 for any G', ((join (M unit)) G') = the identity transformation
442 which will in turn be true just in case:
443 (iii.1') (join (M unit)) = the identity transformation
447 (iii.2) γ = γ <=< (unit G)
448 when γ is a natural transformation from G to some MR'G
450 γ <=< (unit G) becomes: ((join R'G) (M γ) (unit G))
452 substituting in (iii.2), we get:
453 γ = ((join R'G) (M γ) (unit G))
455 which by lemma 2, yields:
456 γ = (((join R'G) ((unit MR'G) γ)
459 γ = (((join (unit M)) R'G) γ)
461 [-- Are the next two steps too cavalier? --]
463 which will be true for all γ just in case:
464 for any R'G, ((join (unit M)) R'G) = the identity transformation
466 which will in turn be true just in case:
467 (iii.2') (join (unit M)) = the identity transformation
471 Collecting the results, our monad laws turn out in this format to be:
474 For all ρ, γ, φ in T,
475 where φ is a transformation from F to MF',
476 γ is a transformation from G to MG',
477 ρ is a transformation from R to MR',
480 (i') ((join G') (M γ) φ) etc also in T
482 (ii') (join (M join)) = (join (join M))
484 (iii.1') (join (M unit)) = the identity transformation
486 (iii.2') (join (unit M)) = the identity transformation
491 Getting to the functional programming presentation of the monad laws
492 --------------------------------------------------------------------
493 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`.
495 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.)
497 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,...]`.
500 In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad."
502 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`:
505 let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
510 Now where `gamma` is another function of type <code>F'('t) → M(G'('t))</code>, we define:
513 gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a)
514 = ((join G') -v- (M gamma) -v- phi) a
521 gamma <=< phi = fun a -> (gamma =<< phi a)
524 `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`.
526 With these definitions, our monadic laws become:
530 Where phi is a polymorphic function of type F('t) -> M(F'('t))
531 gamma is a polymorphic function of type G('t) -> M(G'('t))
532 rho is a polymorphic function of type R('t) -> M(R'('t))
533 and F' = G and G' = R,
534 and a ranges over values of type F('t),
535 b ranges over values of type G('t),
536 and c ranges over values of type G'('t):
538 (i) γ <=< φ is defined,
539 and is a natural transformation from F to MG'
541 (i'') fun a -> gamma =<< phi a is defined,
542 and is a function from type F('t) -> M(G'('t))
546 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
548 (fun a -> (rho <=< gamma) =<< phi a) = (fun a -> rho =<< (gamma <=< phi) a)
549 (fun a -> (fun b -> rho =<< gamma b) =<< phi a) = (fun a -> rho =<< (gamma =<< phi a))
551 (ii'') (fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)
555 (iii.1) (unit G') <=< γ = γ
556 when γ is a natural transformation from some FG' to MG'
558 (unit G') <=< gamma = gamma
559 when gamma is a function of type F(G'('t)) -> M(G'('t))
561 fun b -> (unit G') =<< gamma b = gamma
563 (unit G') =<< gamma b = gamma b
565 Let return be a polymorphic function mapping arguments of any
566 type 't to M('t). In particular, it maps arguments c of type
567 G'('t) to the monadic value (unit G') c, of type M(G'('t)).
569 (iii.1'') return =<< gamma b = gamma b
573 (iii.2) γ = γ <=< (unit G)
574 when γ is a natural transformation from G to some MR'G
576 gamma = gamma <=< (unit G)
577 when gamma is a function of type G('t) -> M(R'(G('t)))
579 gamma = fun b -> gamma =<< (unit G) b
581 As above, return will map arguments b of type G('t) to the
582 monadic value (unit G) b, of type M(G('t)).
584 gamma = fun b -> gamma =<< return b
586 (iii.2'') gamma b = gamma =<< return b
589 Summarizing (ii''), (iii.1''), (iii.2''), these are the monadic laws as usually stated in the functional programming literature:
591 * `fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)`
593 Usually written reversed, and with a monadic variable `u` standing in
596 `u >>= (fun b -> gamma b >>= rho) = (u >>= gamma) >>= rho`
598 * `return =<< gamma b = gamma b`
600 Usually written reversed, and with `u` standing in for `gamma b`:
604 * `gamma b = gamma =<< return b`
606 Usually written reversed:
608 `return b >>= gamma = gamma b`