I know that the
Applicative class is described in category theory as a "lax monoidal functor" but I've never heard the term "lax" before, and the
nlab page on lax functor a bunch of stuff I don't recognize at all, re: bicategories and things that I didn't know we cared about in Haskell. If it is actually about bicategories, can someone give me a plebian view of what that means? Otherwise, what is "lax" doing in this name?
Let's switch to the monoidal view of
unit :: () -> f () mult :: (f s, f t) -> f (s, t) pure :: x -> f x pure x = fmap (const x) (unit ()) (<*>) :: f (s -> t) -> f s -> f t ff <*> fs = fmap (uncurry ($)) (mult (ff, fs))
For a strict monoidal functor,
mult must be isomorphisms. The impact of "lax" is to drop that requirement.
E.g., (up to the usual naivete)
(->) a is strict-monoidal, but
 is only lax-monoidal.