many kernel algebraic typeclasses should be Invariant

[johnynek]

If you have Invariant[M] and an isomorphism (or bijection) A => B, B => A then you can go between M[A] <=> M[B]. This can be very useful when combined with shapeless which can generate iso morphisms between case classes and tuples, etc...

However, Semigroup, Semilattice, Monoid, Group, etc... are left out of that as far as I can see.

It would be nice to have instances for those in object Invariant so they could be used without import.

[kailuowang]

Shall we schedule this in RC1?

[johnynek]

I would like to. I hate when we have obvious instances for our own types that we don't ship.

[LukaJCB]

Hey @johnynek, can you elaborate a bit on what you mean by this? :)
Do you mean instances like Invariant[Group] etc.?
We do already have InvariantMonoidal which does extend Invariant, so we can already do

Monoid[A].imap(aToB)(bToA)

I don't think we have these for any of the other kernel classes apart from Semigroup and Monoid though.

[johnynek]

Concretely:

1. we usually put instances for cats owned types into companions. For Monoid we have an import. Should not need that. We should just put that into Invariant or InvariantMonodial object.
2. we are missing many instances: e.g. all the ones I mentioned above. I'm not sure all are InvariantMonoidal. Monoid is, but some of the others may not (pure may or may not work for all cases).

[LukaJCB]

I like the idea of having everything in object Invariant, but we'd break the API, no?
We could do something like object InvariantInstances extends MonoidInstances with SemigroupInstances, WDYT?

[johnynek]

I think these are so rarely used, I don't mind breaking the API to make it consistent current cats practice before 1.0

[LukaJCB]

I'll submit a PR soon :)

[LukaJCB]

Fixed by #1914 :)