What you’re looking for is Kleisli category. It was originally developed to show that every monad can be constructed from two adjoint functors. The problem is that Haskell Functor is not a generic functor, it’s an endo-functor in the Haskell category. So we need something different (AFAIK) to represent functors between other categories: {-# LANGUAGE … Read more
As Phillip JF mentioned, comonoids are interesting to talk about in substructural logics. Let’s talk about linear lambda calculus. This is much like your normal typed lambda calculus except that every variable must be used exactly once. To get a feel, let’s count linear functions of given types, i.e. a -> a has exactly one … Read more
This probably isn’t the answer you’re looking for, but here you go anyways: A really crooked way of looking at monads & co. One way of looking at abstract concepts like these is to link them with basic concepts, such as ordinary list processing operations. Then, you could say that, A category generalizes the (.) … Read more
This question has some good answers: Monads as adjunctions More to the point, Derek Elkins’ “Calculating Monads with Category Theory” article in TMR #13 should have the sort of constructions you’re looking for: http://www.haskell.org/wikiupload/8/85/TMR-Issue13.pdf Finally, and perhaps this is really the closest to what you’re looking for, you can go straight to the source and … Read more
Does he mean, the datatype String and Int are monoid. No, neither String nor Int are monoids. A monoid is a 3-tuple (S, ⊕, e) where ⊕ is a binary operator ⊕ : S×S → S, such that for all elements a, b, c∈S it holds that (a⊕b)⊕c=a⊕(b⊕c), and e∈S is an “identity element” such … Read more
I’m with @Rhymoid on this one, I believe all Monads have two (!!) transformers. My construction is a bit different, and far less complete. I’d like to be able to take this sketch into a proof, but I think I’m either missing the skills/intuition and/or it may be quite involved. Due to Kleisli, every monad … Read more
Edit: Just for fun, I’m going to do this right. Original answer preserved below The current adjunction code for category-extras now is in the adjunctions package: http://hackage.haskell.org/package/adjunctions I’m just going to work through the state monad explicitly and simply. This code uses Data.Functor.Compose from the transformers package, but is otherwise self-contained. An adjunction between f … Read more
Sharon Curtis and Shin-Cheng Mu have a Functional Pearl using zygomorphisms to find maximally dense segments (a generalization of maximum segment sums). Zygomorphisms are seemingly a good fit for sliding window problems once you are accustomed to them. http://www.iis.sinica.edu.tw/~scm/2010/functional-pearl-maximally-dense-segments/ I’d nominate the authors for extra credit as they’ve avoided the use of the fixed-point Mu … Read more
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, rings, monoids and so on. Most of the time, these things are introduced in terms of sets, but since we’re among friends, I’ll talk about Haskell types instead. (I … Read more
Here’s an even simpler answer: A Monad is something that “computes” when monadic context is collapsed by join :: m (m a) -> m a (recalling that >>= can be defined as x >>= y = join (fmap y x)). This is how Monads carry context through a sequential chain of computations: because at each … Read more