> When you're talking about invertible functions, the idea you're probably reaching for is an isomorphism -- that is, we want the function to have certain nice properties on top of just being a map from a -> b with an inverse map from b -> a.

The usual meaning of 'f is invertible' is that it is both left- and right-invertible, thus making it bijective: see first bullet in [1].

Here you're alluding to f being merely left-invertible, something I don't see mentioned in OP.

> You also want the function to be a bijection, which is captured in the notion of an isomorphism.

I'm reminded of a reddit convo where the idea was tossed out that semigroups should always be promoted to monoids [2].

I argued no. I also cited a case where a supposedly nicer monoid causes more problems for a ghc hacker than the true semigroup [3].

Having structure is nice. And sometimes we just have to work with what's given to us.

Category theory calls a /monomorphism/ something that's strictly weaker than left-invertible. An arrow that's (additionally) left-invertible corresponds to a /split mono/.

Hence in order of _decreasing_ niceness: Iso, Split mono, Mono. As research uncovers more interesting phenomena, this sequence will continuing growing to the right.

We can't always impose that niceness because that nukes whatever we're studying. So we gotta respect the situation. And given lemons, make lemonade.


[1] http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection#Bijection

[2] http://www.reddit.com/r/haskell/comments/1ou06l/improving_applicative_donotation/ccvtqot?context=1

[3] http://www.reddit.com/r/haskell/comments/1ou06l/improving_applicative_donotation/ccy4n2d