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
The key to understanding applicative functors is to figure out what structure they preserve. Regular functors preserve the basic categorical structure: they map objects and morphisms between categories, and they preserve the laws of the category (associativity and identity). But a category may have more structure. For instance, it may allow the definition of mappings … Read more
Monad transformers aren’t exceedingly mathematically pleasant. However, we can get nice (co)products from free monads, and, more generally, ideal monads: See Ghani and Uustalu’s “Coproducts of Ideal Monads”: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.4.2698
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
liftA, liftM, fmap, and . should all be the same function, and they must be if they satisfy the functor law: fmap id = id However, this is not checked by Haskell. Now for Applicative. It’s possible for ap and <*> to be distinct for some functors simply because there could be more than one … 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
tl;dr the proposed candidate is not quite an equaliser, but its irrelevant counterpart is The candidate for an equaliser in Agda looks good. So let’s just try it. We’ll need some basic kit. Here are my refusenik ASCII dependent pair type and homogeneous intensional equality. record Sg (S : Set)(T : S -> Set) : … Read more
(N.B. this combines a bit from both mine and @Gabriel’s comments above.) It’s possible for every inhabitant of the Fixed point of a Functor to be infinite, i.e. let x = (Fix (Id x)) in x === (Fix (Id (Fix (Id …)))) is the only inhabitant of Fix Identity. Free differs immediately from Fix in … Read more
It’s become clear that this answer is wrong. I’m leaving it here to preserve valuable discussion in the comments until someone formulates a correct answer. Consider the power set in Set. If we have a function f : S -> T, we can form f’ : PS S -> PS T by f’ X = … Read more
Every applicative yields an arrow and every arrow yields an applicative, but they are not equivalent. If you have an arrow arr and a morphism arr a b it does not follow that you can generate a morphism arr o (a \to b) that replicates its functionality. Thus if you round trip through applicative you … Read more