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:


Dear X,


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.


Regards,


Sarah


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 $R$
* etc.


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>)



Can one recover an algebraically closed field $k$ from the dots and arrows of its category of finitely generated $k$-algebras?
I think the following works for any commutative ring: Consider the category of abelian group objects in the overcategory above the initial element. This is equivalent to the category of finitely generated $R$-Modules (see for example <https://ncatlab.org/nlab/show/module>). By the general theory of Morita equivalence, this determines $R$.


