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
489 In category-theory presentations, you may see `unit` referred to as <code>η</code>, and `join` referred to as <code>μ</code>. Also, instead of the monad `(M, unit, join)`, you may sometimes see discussion of the "Kleisli triple" `(M, unit, =<<)`. Alternatively, `=<<` may be called <code>⋆</code>. These are interdefinable (see below).
492 Getting to the functional programming presentation of the monad laws
493 --------------------------------------------------------------------
494 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`.
496 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.)
498 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,...]`.
501 In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad."
503 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`:
506 let phi = fun ((_:char), x, y) -> [(1,x,y),(2,x,y)]
509 [-- 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. --]
512 Now where `gamma` is another function of type <code>F'('t) -> M(G'('t))</code>, we define:
515 gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a)
516 = ((join G') -v- (M gamma) -v- phi) a
523 gamma <=< phi = (fun a -> gamma =<< phi a)
526 `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`.
528 With these definitions, our monadic laws become:
532 Where phi is a polymorphic function of type F('t) -> M(F'('t))
533 gamma is a polymorphic function of type G('t) -> M(G'('t))
534 rho is a polymorphic function of type R('t) -> M(R'('t))
535 and F' = G and G' = R,
536 and a ranges over values of type F('t),
537 and b ranges over values of type G('t),
538 and c ranges over values of type G'('t):
540 (i) γ <=< φ is defined,
541 and is a natural transformation from F to MG'
543 (i'') fun a -> gamma =<< phi a is defined,
544 and is a function from type F('t) -> M(G'('t))
548 (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
550 (fun a -> (rho <=< gamma) =<< phi a) = (fun a -> rho =<< (gamma <=< phi) a)
551 (fun a -> (fun b -> rho =<< gamma b) =<< phi a) =
552 (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 whenever γ is a natural transformation from some FG' to MG'
561 (unit G') <=< gamma = gamma
562 whenever 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 whenever γ is a natural transformation from G to some MR'G
579 gamma = gamma <=< (unit G)
580 whenever 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`