concept transform in category c++

This is an excerpt from Manning's book Functional Programming in C++. Login to get full access to this book.

transform—What the code does

A class template F is a functor if it has a transform (or map) function defined on it (see figure 10.1). The transform function takes an instance f of a type F<T1> and a function t: T1 → T2, and it returns a value of type F<T2>. This function can have multiple forms, so we’ll use the pipe notation from chapter 7 for clarity.

Figure 10.2 The main rule the transform function needs to obey is that composing two transforms—one that lifts function f and another that lifts function g—needs to have the same effect as having a single transformation with the composition of f and g.

This looks much like the std::transform algorithm and view::transform from the ranges library. That’s not accidental: generic collections from the STL and ranges are functors. They’re all wrapper types that have a well-behaved transform function defined on them. It’s important to note that the other direction doesn’t hold: not all functors are collections or ranges.

You can—and this is a more common way to define monads. You can say that a monad M is a wrapper type that has a constructor (a function that constructs an instance of M<T> from a value of type T) and an mbind function (it’s usually called just bind, but we’ll use this name so it doesn’t get confused with std::bind), which is a composition of transform and join:

construct : T → M<T>
mbind     : (M<T1>, T1 → M<T2>) → M<T2>

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It’s easy to show that all monads are functors. It’s trivial to implement transform by using mbind and construct.