To use deriving Functor you must enable the DeriveFunctor language pragma and apply it to a polymorphic type which has a covariant final type variable—in other words, a type which admits a valid Functor instance. It’ll then derive the “obvious” Functor instance. There’s been some concern in the past that the derived instance is not … Read more
For one, the functor can contain internal state; a state that is valid for this invocation of the function object only. You could add static variables to your function, but those would be used for any invocation of the function. Second, the compiler can inline calls to the functor; it cannot do the same for … Read more
What profunctors lack compared to arrows is the ability to compose them. If we add composition, will we get an arrow? MONOIDS This is exactly the question tackled in section 6 of “Notions of Computation as Monoids,” which unpacks a result from the (rather dense) “Categorical semantics for arrows”. “Notions” is a great paper because … Read more
Here’s the functor class: class Functor f where fmap :: (a -> b) -> f a -> f b Note that “f” by itself is a type constructor because it’s applied to a type variable in the fmap line. Here are some examples to make this clear: Type constructors: IO Maybe Either String Types: IO … Read more
A function object is just that. Something which is both an object and a function. Aside: calling a function object a “functor” is a serious abuse of the term: a different kind of “functors” are a central concept in mathematics, and one that has a direct role in computer science (see “Haskell Functors”). The term … Read more
You may want to clarify whether you’re asking about “functors in Haskell”, or Functors. It’s not always clear what category is being assumed when Category Theory terms are used in Haskell. But yes, the default assumption is Hask, which is taken to be the category of Haskell types with functions as morphisms. In that case, … Read more
Let me explain my understanding without going into category theory: Functors and monads both provide some tool to wrapped input, returning a wrapped output. Functor = unit + map (i.e. the tool) where, unit = something which takes raw input and wraps it inside a small context. map = the tool which takes a function … Read more
The “neat alternative presentation” for Applicative is based on the following two equivalencies pure a = fmap (const a) unit unit = pure () ff <*> fa = fmap (\(f,a) -> f a) $ ff ** fa fa ** fb = pure (,) <*> fa <*> fb The trick to get this “neat alternative presentation” … Read more
A lot of the difference comes down to pointers and addressing. In C++ยน pointers to functions and pointers to objects are strictly separate kinds of things. C++ requires that you can convert a pointer to any object type into a pointer to void, then convert it back to the original type, and the result will … Read more
There’s no fundamental limitation of .NET that stops functors from being implemented in F#. True, they can’t be represented directly in .NET metadata, but neither can other F# language features like union types. Compilers for languages with functors (e.g., Standard ML, OCaml) have a pass called defunctorize; it works just like C++ template expansion, in … Read more