In real world application I noticed a pattern that could be generalized to something like: purescript: class Profunctor p <= Zero p where pzero :: forall a b. p a b -- such that `forall f g. di...

I'm trying to understand what Monoid is from a category theory perspective, but I'm a bit confused with the notation used to describe it. Here is Wikipedia: In category theory, a monoid (or monoid...

It is known that natural transformations with type signature a -> a must be identity functions. This follows from the Yoneda lemma but can be also derived directly. This question asks for the sa...

I've been experimenting with monoids and Distributives lately, and I think I've found something of interest (described in my answer) - are these already known structures? (I've been unable to find ...

The free monoids are often being regarded as "list monoids". Yet, I am interested in other possible structures which might give us free monoids. Firstly, let us go over the definition of ...

Edward Kmett writes on his blog that using the Co newtype (from the kan-extensions package), it's possible to derive a Monad from any Comonad. I'd like to learn how to mechanically do this for any ...

The codensity monad on a type constructor f is defined by: newtype C f a = C { unC ∷ forall r. (a → f r) → f r } It is well known that C f is a monad for any type constructor f (not necessarily c...

Across programming languages, I've encountered similar composite types with different names: Optional / Maybe Any Variant / Sum Record / Product People often use the term vocabulary type, yet...

In many cases, it isn't clear to me what is to be gained by combining two monads with a transformer rather than using two separate monads. Obviously, using two separate monads is a hassle and can i...

From categorical point of view, functor is pair of two maps (one between objects and another between arrows of categories), following some axioms. I have assumed, what every Functor instance is si...

I am busy reading Bartosz Milewski's Category Theory book for programmers and I'm struggling with the depiction of non-identity morphisms when moving between describing a monoid as a set and a mono...

While learning about the Yoneda lemma, I came across the following encoding of the underlying natural isomorphism in Haskell: forward :: Functor f => (forall r . (a -> r) -> f r) -> f a...

I've been on a bit of a "distilling everything to its fundamentals" kick lately, and I've been unable to find clear theoretical reasons for how the Traversable typeclass is defined, only ...

I started to learn Prolog and I just read that the atom at the beginning of an structure is usually called functor. I'm also familiar with the term functor from Category Theory and Functional Progr...

In his paper Generics for the Masses Hinze reviews encoding of data type. Starting from Nat data Nat :: ⋆ where Zero :: Nat Succ :: Nat → Nat It can be viewed as an initial algebra NatF Nat -&g...

In the past 2 years, I was interested in using free monad to helping me to solve practical software engineering problem. And came up my own construction of free monad using some elementary category...

On page 12 of One Monad to Prove Them All, it is written that "a prominent example [of container] is the list data type. A list can be represented by the length of the list and a function mapp...

Various recursion scheme boil down to specific instantiation of refold refold :: Functor s => (s b -> b) -> (a -> s a) -> a -> b refold f g = go where go a = f (fmap go (g a)) Wh...

(Disclaimer: I'm not 100% sure how codatatype works, especially when not referring to terminal algebras). Consider the "category of types", something like Hask but with whatever adjustmen...

As I understand it, one way to express that something is a free monoid is with a class like this: class (Foldable s, forall a. Monoid (s a)) => Sequence s where singleton :: a -> s a and th...

The Cartesian class from the constrained-category project is for categories, products of objects in which are objects in the same category yet again. I wonder about the classes Cartesian extends: c...

As a math student, the first thing I did when I learned about monads in Haskell was check that they really were monads in the sense I knew about. But then I learned about monad transformers and tho...

I understand that many of the names in Haskell are inspired by category theory terminology, and I'm trying to understand exactly where the analogy begins and ends. The Category Hask I already know ...

I watched this video on the composite pattern, where the main example is how to use the pattern as a mean to generate HTML code from a tree structure describing a todo list where each item can be i...

I've read about monoid homomorphism from Monoid Morphisms, Products, and Coproducts and could not understand 100%. The author says (emphasis original): The length function maps from String to ...