A generator (or a cogenrerator) for the category of schemes Does the category of Schemes admit a (single) generator (or a cogenerator)?
What if we restrict to the category of schemes of finite type over a field $k$?

The category of schemes does not admit a generating family, see MO/2015.
But the category of schemes which are (locally) of finite type over some ring $k$ has a generating family: First of all, the category of affine schemes of finite type is a large generating family. Next, one observes that this subcategory is essentially small: Up to isomorphism there is only a set of $k$-algebras of finite type. In fact, they are all isomorphic to $k[x_1,x_2,\dotsc]/I$ for some ideal $I$ such that the quotient is of finite type. Taking the disjoint union of all these affine schemes, we obtain a generating object if we work with schemes locally of finite type.
I doubt that the category of schemes of finite type has a generating object.
