  A quantum causal topology is presented. This is modeled after a
non-commutative scheme type of theory for the curved finitary spacetime sheaves
of the non-abelian incidence Rota algebras that represent `gravitational
quantum causal sets'. The finitary spacetime primitive algebra scheme
structures for quantum causal sets proposed here are interpreted as the
kinematics of a curved and reticular local quantum causality. Dynamics for
quantum causal sets is then represented by appropriate scheme morphisms, thus
it has a purely categorical description that is manifestly `gauge-independent'.
Hence, a schematic version of the Principle of General Covariance of General
Relativity is formulated for the dynamically variable quantum causal sets. We
compare our non-commutative scheme-theoretic curved quantum causal topology
with some recent $C^{*}$-quantale models for non-abelian generalizations of
classical commutative topological spaces or locales, as well as with some
relevant recent results obtained from applying sheaf and topos-theoretic ideas
to quantum logic proper. Motivated by the latter, we organize our finitary
spacetime primitive algebra schemes of curved quantum causal sets into a
topos-like structure, coined `quantum topos', and argue that it is a sound
model of a structure that Selesnick has anticipated to underlie Finkelstein's
reticular and curved quantum causal net. At the end we conjecture that the
fundamental quantum time-asymmetry that Penrose has expected to be the main
characteristic of the elusive `true quantum gravity' is possibly of a
kinematical or structural rather than of a dynamical character, and we also
discuss the possibility of a unified description of quantum logic and quantum
gravity in quantum topos-theoretic terms.
