Suppose I hand you a commutative ring . I stipulate that you are only allowed to work in the language of the category of commutative rings; you can only refer to objects and morphisms. (That means you can’t refer directly to elements of , and you also can’t refer directly to the multiplication or addition maps , since these aren’t morphisms.) Geometrically, I might equivalently say that you are only allowed to work in the language of the category of affine schemes, since the two are dual. Can you recover as a set, and can you recover the ring operations on ?
The answer turns out to be yes. Today we’ll discuss how this works, and along the way we’ll run into some interesting ideas.