category-theory

Functors can be covariant and contravariant. Can this covariant/contravariant duality also be applied to monads? Something like: class Monad m where return :: a -> m a (>>=) :: m a -> (a -> m b) -> m b class ContraMonad m where return :: a -> m a contrabind :: m a -> (b -> m a) -> m b Does ContraMonad class make sense? Any examples? Well, of course, it's possible to define it, but I doubt it would be of any use. There is a popular saying that "monad is just a monoid in a category of endofunctors". What it means is, first of all, that we have a category of endofunctors (meaning, (covariant)

I was able to map Functor's definition from category theory to Haskell's definition in the following way: since objects of Hask are types, the functor F maps every type a of Hask to the new type F a by, roughly saying, prepending "F " to it. maps every morphism a -> b of Hask to the new morphism F a -> F b using fmap :: (a -> b) -> (f a -> f b) . So far, so good. Now I get to the Applicative , and can't find any mention of such a concept in textbooks. By looking at what it adds to Functor , ap :: f (a -> b) -> f a -> f b , I tried to come up with my own definition. First, I noticed that since

So we have Category of Hask, where: Types are the objects of the category Functions are the morphisms from object to object in the category. Similarly for Functor we have: a Type constructor as the mapping of objects from one category to another fmap for the mapping of morphisms from one category to another. Now, when we write program we basically transform values (not types) and it seems that the Category of Hask doesn't talk about values at all. I tried to fit values in the whole equation and came up with the following observation: Each Type is a category itself. Ex: Int is a category of all

I have heard the term "coalgebras" several times in functional programming and PLT circles, especially when the discussion is about objects, comonads, lenses, and such. Googling this term gives pages that give mathematical description of these structures which is pretty much incomprehensible to me. Can anyone please explain what coalgebras mean in the context of programming, what is their significance, and how they relate to objects and comonads? Algebras I think the place to start would be to understand the idea of an algebra . This is just a generalization of algebraic structures like groups

For an uncertainty-propagating Approximate type , I'd like to have instances for Functor through Monad . This however doesn't work because I need a vector space structure on the contained types, so it must actually be restricted versions of the classes. As there still doesn't seem to be a standard library for those (or is there? please point me. There's rmonad , but it uses * rather than Constraint as the context kind, which seems just outdated to me), I wrote my own version for the time being. It all works easy for Functor class CFunctor f where type CFunctorCtxt f a :: Constraint cfmap ::

As described this question/answers , Functor instances are uniquely determined, if they exists. For lists, there are two well know Applicative instances: [] and ZipList . So Applicative isn't unique (see also Can GHC derive Functor and Applicative instances for a monad transformer? and Why is there no -XDeriveApplicative extension? ). However, ZipList needs infinite lists, as its pure repeats a given element indefinitely. Are there other, perhaps better examples of data structures that have at least two Applicative instances? Are there any such examples that only involve finite data structures