It’s another name-choosing exercise that I’m vacillating about, but here’s today’s terminology for the dual to Applicative functors. Remember we had the lax monoidal presentation
Well, switch from products to coproducts and flip round those arrows and you’ve got
A Decisive f gives us some sort of certainty with respect to its parameter. If you think of f as some sort of epistemic modality, then it’s quite a useful one. Firstly, if we know/believe/imagine Zero, that must mean the world really is no good, and there really is an element of Zero. Even better, if we know/believe/imagine S or T, that must mean we actually know/believe/imagine S or know/believe/imagine T. ‘Or’ behaves well, and always gives us a decision.
I wonder what we expect to hold. Well, naturally,
but surely we’d also like
but we’d also hope for
I wonder what else.
There’s an ‘idiomatic’ interface to these things.
Now, guess what. Every comonad is decisive. As it were,
That is, orwell revises the story so that, throughout, it’s consistently on whichever side we happen to be on at the (counit) moment.