Are there any results that allow us to characterize affine schemes via morphisms to/from other schemes? Say we have a scheme $X$. We want to know if it is affine, but:

• we don't know anything about the structure of the scheme itself (particularly its structure sheaf)
• all we can do is ask how it maps to/from other schemes, whether these morphisms satisfy certain properties, form certain categorical constructions, etc.

Is this possible?

Are there any results that allow us to characterize affine schemes via morphisms to/from other schemes?

Suppose we have a category $\mathcal{C}$ which is equivalent to $\mathbf{Sch}$, the category of schemes. We do not know anything about the equivalence. We want to find the objects $X \in \mathcal{C}$ such that these objects are equivalent exactly to the affine schemes in $\mathcal{C}$.

Is this possible? If not, can we modify the problem slightly (e.g. replace $\mathbf{Sch}$ with a reasonable subcategory, or assume the existence of some reasonable functors) to get an answer? Or is the impossibility fundamental?