We construct a sheaf-theoretic representation of quantum observables algebras
over a base category equipped with a Grothendieck topology, consisting of
epimorphic families of commutative observables algebras, playing the role of
local arithmetics in measurement situations. This construction makes possible
the adaptation of the methodology of Abstract Differential Geometry (ADG), a la
Mallios, in a topos-theoretic environment, and hence, the extension of the
"mechanism of differentials" in the quantum regime. The process of gluing
information, within diagrams of commutative algebraic localizations, generates
dynamics, involving the transition from the classical to the quantum regime,
formulated cohomologically in terms of a functorial quantum connection, and
subsequently, detected via the associated curvature of that connection.Comment: 81 pages, replaced by expanded published versio
