Co-, contra-, and invariance
20 Oct 2019 ⇐ Notes archive(This is an entry in my technical notebook. There will likely be typos, mistakes, or wider logical leaps — the intent here is to “let others look over my shoulder while I figure things out.”)
Folks often refer to the component of a function’s “polarity.” “The input is in the negative position.” “The output is positive.”
And that made me wonder, is there a, well, neutral polarity?
Maybe that’s when a component is in both a negative and positive position, canceling one another out.
Let’s see what happens.
A -> ….
A is in a negative position? Check. Let’s add it to the positive spot.
A -> A.
This is our old friend, Endo! At the bottom of the file, I noticed imap and asked some folks what the “i” stood for. Turns out it’s short for, “invariant,” which reads nicely in that both co- and contravariance net out to invariance.
Pairing functor type, variance(s), and *map name:
- Functor, covariant,
map. - Functor, contravariant,
contramappullback. - Bifunctor, covariant (and I’m guessing contra-, maybe both working in the same direction is what matters?),
bimap. - Invariant functor, invariant (co- and contravariant in the same component),
imap. - Profunctor, co- and contravariant along two components,
dimap.