EMPIRICAL ALGEBRAIC-GEOMETRY

In a previous paper we introduced the notion of an orthogonal category and generalized the notion of a sheaf of sets on a complete Boolean algebra B to that of a sheaf on the complete Boolean algebra B with values in an orthogonal category K. By properly replacing the complete Boolean algebra B by a manual M of Boolean locales, we get a notion of a sheaf on M with values in K, which can be regarded as a quantum generalization of a sheaf on B. Taking K to be the category of sheaves of Abelian groups or that of schemes a la Grothendieck, we will discuss some fundamental aspects of the quantum generalizations of sheaves and schemes.