# Co-, contra-, and invariance

20 Oct 2019 ⇐ Notes archive(This is an entry in my technical diary. 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 functor’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,
`contramap`

`pullback`

. - 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`

.