You are gracious enough to host me for a few days while I attend a conference. After I leave, you're surprised to see a gift on the kitchen table. It's a box with a category inside! The objects aren't labeled so it's a little hard to tell what's going on with it, but you can see, for example, that there's a terminal object. There's also a note:
I've always been very fond of this category, and I thought you might like it too. It's a category of finitely generated algebras over some algebraically closed field - unfortunately I've forgotten which one! But I'm sure you'll figure it out.
Can you determine what the field in question is?
More generally, given just the dots and arrows of a category (no taking sections!) and the information that for some unknown ring $R$ the category is the category of
- Classical varieties over $R$
- Affine schemes over $R$
- Schemes over $R$
in what cases is there a unique ring $R$ (up to isomorphism) which produces my category (up to equivalence)?
(I posted this on stack exchange, but it seems like a reasonable overflow question so I am crossposting it: https://math.stackexchange.com/questions/3002833/can-one-recover-an-algebraically-closed-field-k-from-its-category-of-finitely)