Punctors

In programming and category theory, a functor is a structure that maps between categories. More specifically, a functor is a map between two categories, say C and D, that translates objects and morphisms in C to objects and morphisms in D, respectively. This translation must preserve the composition of morphisms and the identity morphisms. Think of a functor as a kind of structure-preserving transformation. It takes a structure in one category and gives you a corresponding structure in another, ensuring that the relationships within the original structure are maintained in the new one.

In functional programming, functors are often implemented as types that support a mapping operation. This operation, commonly called `map` or `fmap`, takes a function and a functor instance, applies the function to the value(s) inside the functor, and returns a new functor instance containing the result. For example, in Haskell, the `Maybe` type is a functor. If you have `Just 5` and you map the function `(+1)` over it, you get `Just 6`. If you have `Nothing`, mapping `(+1)` over it results in `Nothing`. This ability to apply a function inside a container without having to explicitly unwrap and rewrap the container is a key benefit of functors. They provide a consistent way to work with data wrapped in various contexts.