nLab locally affine space

A $\tau$-locally affine space is a sheaf of sets on the category $Aff$ (affine schemes) opposite to the category of commutative unital rings in some subcanonical Grothendieck topology $\tau$. If $\tau$ is the Zariski topology, then $\tau$-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 $Aff/X$ which is, in the case of $X=Spec(A)$ where $A$ is a commutative ring, simply the opposite of the category of $A$-algebras.

• Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, 203, Springer (1971) [doi:10.1007/BFb0059750]

In noncommutative algebraic geometry, one can consider $NAff$, that is the opposite to the category of all unital associative rings instead of $Aff$. There is a noncommutative smooth topology on $NAff$ (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 $NAff$ 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 $NAff$ work with noncommutative generalization of small cotopologies on rings (cf. work of van Oystaeyen on “noncommutative topologies”).