A -locally affine space is a sheaf of sets on the category (affine schemes) opposite to the category of commutative unital rings in some subcanonical Grothendieck topology . If is the Zariski topology, then -locally affine spaces are schemes. Other examples are sheaves in fppf, fpqc, étale or lisse (=smooth) topology. The relative version uses the category of relative affine schemes which is, in the case of where is a commutative ring, simply the opposite of the category of -algebras.
In noncommutative algebraic geometry, one can consider , that is the opposite to the category of all unital associative rings instead of . There is a noncommutative smooth topology on (M. Konstevich, A. L. Rosenberg, Noncommutative smooth spaces, arXiv:math/9812158), which is a genuine Grothendieck topology. On the other hand, one can not obtain a Grothendieck topology using exact affine lcoalizations: the stability axiom of Grothendieck topologies fails. However some other generalizations of topologies on are available for the noncommutative case (e.g. Q-categories and “quasi-topologies” of Rosenberg) or one can instead of working with sheaves on the big site work with noncommutative generalization of small cotopologies on rings (cf. work of van Oystaeyen on “noncommutative topologies”).