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/*! @file Forward declares `boost::hana::Comonad`. Copyright Louis Dionne 2013-2022 Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt) */ #ifndef BOOST_HANA_FWD_CONCEPT_COMONAD_HPP #define BOOST_HANA_FWD_CONCEPT_COMONAD_HPP #include namespace boost { namespace hana { // Note: We use a multiline C++ comment because there's a double backslash // symbol in the documentation (for LaTeX), which triggers // warning: multi-line comment [-Wcomment] // on GCC. /*! @ingroup group-concepts @defgroup group-Comonad Comonad The `Comonad` concept represents context-sensitive computations and data. Formally, the Comonad concept is dual to the Monad concept. But unless you're a mathematician, you don't care about that and it's fine. So intuitively, a Comonad represents context sensitive values and computations. First, Comonads make it possible to extract context-sensitive values from their context with `extract`. In contrast, Monads make it possible to wrap raw values into a given context with `lift` (from Applicative). Secondly, Comonads make it possible to apply context-sensitive values to functions accepting those, and to return the result as a context-sensitive value using `extend`. In contrast, Monads make it possible to apply a monadic value to a function accepting a normal value and returning a monadic value, and to return the result as a monadic value (with `chain`). Finally, Comonads make it possible to wrap a context-sensitive value into an extra layer of context using `duplicate`, while Monads make it possible to take a value with an extra layer of context and to strip it with `flatten`. Whereas `lift`, `chain` and `flatten` from Applicative and Monad have signatures \f{align*}{ \mathtt{lift}_M &: T \to M(T) \\ \mathtt{chain} &: M(T) \times (T \to M(U)) \to M(U) \\ \mathtt{flatten} &: M(M(T)) \to M(T) \f} `extract`, `extend` and `duplicate` from Comonad have signatures \f{align*}{ \mathtt{extract} &: W(T) \to T \\ \mathtt{extend} &: W(T) \times (W(T) \to U) \to W(U) \\ \mathtt{duplicate} &: W(T) \to W(W(T)) \f} Notice how the "arrows" are reversed. This symmetry is essentially what we mean by Comonad being the _dual_ of Monad. @note The [Typeclassopedia][1] is a nice Haskell-oriented resource for further reading about Comonads. Minimal complete definition --------------------------- `extract` and (`extend` or `duplicate`) satisfying the laws below. A `Comonad` must also be a `Functor`. Laws ---- For all Comonads `w`, the following laws must be satisfied: @code extract(duplicate(w)) == w transform(duplicate(w), extract) == w duplicate(duplicate(w)) == transform(duplicate(w), duplicate) @endcode @note There are several equivalent ways of defining Comonads, and this one is just one that was picked arbitrarily for simplicity. Refined concept --------------- 1. Functor\n Every Comonad is also required to be a Functor. At first, one might think that it should instead be some imaginary concept CoFunctor. However, it turns out that a CoFunctor is the same as a `Functor`, hence the requirement that a `Comonad` also is a `Functor`. Concrete models --------------- `hana::lazy` [1]: https://wiki.haskell.org/Typeclassopedia#Comonad */ template struct Comonad; }} // end namespace boost::hana #endif // !BOOST_HANA_FWD_CONCEPT_COMONAD_HPP