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1 | /*! |
2 | @file | |
3 | Forward declares `boost::hana::Comonad`. | |
4 | ||
5 | @copyright Louis Dionne 2013-2016 | |
6 | Distributed under the Boost Software License, Version 1.0. | |
7 | (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt) | |
8 | */ | |
9 | ||
10 | #ifndef BOOST_HANA_FWD_CONCEPT_COMONAD_HPP | |
11 | #define BOOST_HANA_FWD_CONCEPT_COMONAD_HPP | |
12 | ||
13 | #include <boost/hana/config.hpp> | |
14 | ||
15 | ||
16 | BOOST_HANA_NAMESPACE_BEGIN | |
17 | // Note: We use a multiline C++ comment because there's a double backslash | |
18 | // symbol in the documentation (for LaTeX), which triggers | |
19 | // warning: multi-line comment [-Wcomment] | |
20 | // on GCC. | |
21 | ||
22 | /*! | |
23 | @ingroup group-concepts | |
24 | @defgroup group-Comonad Comonad | |
25 | The `Comonad` concept represents context-sensitive computations and | |
26 | data. | |
27 | ||
28 | Formally, the Comonad concept is dual to the Monad concept. | |
29 | But unless you're a mathematician, you don't care about that and it's | |
30 | fine. So intuitively, a Comonad represents context sensitive values | |
31 | and computations. First, Comonads make it possible to extract | |
32 | context-sensitive values from their context with `extract`. | |
33 | In contrast, Monads make it possible to wrap raw values into | |
34 | a given context with `lift` (from Applicative). | |
35 | ||
36 | Secondly, Comonads make it possible to apply context-sensitive values | |
37 | to functions accepting those, and to return the result as a | |
38 | context-sensitive value using `extend`. In contrast, Monads make | |
39 | it possible to apply a monadic value to a function accepting a normal | |
40 | value and returning a monadic value, and to return the result as a | |
41 | monadic value (with `chain`). | |
42 | ||
43 | Finally, Comonads make it possible to wrap a context-sensitive value | |
44 | into an extra layer of context using `duplicate`, while Monads make | |
45 | it possible to take a value with an extra layer of context and to | |
46 | strip it with `flatten`. | |
47 | ||
48 | Whereas `lift`, `chain` and `flatten` from Applicative and Monad have | |
49 | signatures | |
50 | \f{align*}{ | |
51 | \mathtt{lift}_M &: T \to M(T) \\ | |
52 | \mathtt{chain} &: M(T) \times (T \to M(U)) \to M(U) \\ | |
53 | \mathtt{flatten} &: M(M(T)) \to M(T) | |
54 | \f} | |
55 | ||
56 | `extract`, `extend` and `duplicate` from Comonad have signatures | |
57 | \f{align*}{ | |
58 | \mathtt{extract} &: W(T) \to T \\ | |
59 | \mathtt{extend} &: W(T) \times (W(T) \to U) \to W(U) \\ | |
60 | \mathtt{duplicate} &: W(T) \to W(W(T)) | |
61 | \f} | |
62 | ||
63 | Notice how the "arrows" are reversed. This symmetry is essentially | |
64 | what we mean by Comonad being the _dual_ of Monad. | |
65 | ||
66 | @note | |
67 | The [Typeclassopedia][1] is a nice Haskell-oriented resource for further | |
68 | reading about Comonads. | |
69 | ||
70 | ||
71 | Minimal complete definition | |
72 | --------------------------- | |
73 | `extract` and (`extend` or `duplicate`) satisfying the laws below. | |
74 | A `Comonad` must also be a `Functor`. | |
75 | ||
76 | ||
77 | Laws | |
78 | ---- | |
79 | For all Comonads `w`, the following laws must be satisfied: | |
80 | @code | |
81 | extract(duplicate(w)) == w | |
82 | transform(duplicate(w), extract) == w | |
83 | duplicate(duplicate(w)) == transform(duplicate(w), duplicate) | |
84 | @endcode | |
85 | ||
86 | @note | |
87 | There are several equivalent ways of defining Comonads, and this one | |
88 | is just one that was picked arbitrarily for simplicity. | |
89 | ||
90 | ||
91 | Refined concept | |
92 | --------------- | |
93 | 1. Functor\n | |
94 | Every Comonad is also required to be a Functor. At first, one might think | |
95 | that it should instead be some imaginary concept CoFunctor. However, it | |
96 | turns out that a CoFunctor is the same as a `Functor`, hence the | |
97 | requirement that a `Comonad` also is a `Functor`. | |
98 | ||
99 | ||
100 | Concrete models | |
101 | --------------- | |
102 | `hana::lazy` | |
103 | ||
104 | [1]: https://wiki.haskell.org/Typeclassopedia#Comonad | |
105 | ||
106 | */ | |
107 | template <typename W> | |
108 | struct Comonad; | |
109 | BOOST_HANA_NAMESPACE_END | |
110 | ||
111 | #endif // !BOOST_HANA_FWD_CONCEPT_COMONAD_HPP |