  This note aims to give a self-contain and detail explanation about U(k')times
U(k) Pati-Salam model in curved space-time which derived from (1+n)-dimensional
square root Lorentz manifold by self-parallel transportation principle. The
concepts foundation of manifold from view point of category theory, fiber
bundle and sheaf theories are reviewed. There are extra U(k')times
U(k)-principal bundle and U(k)-associated bundle than (1+n)-dimensional Lorentz
manifold. The conservative currents on square root Lorentz manifold are
discussed preliminary. A detail proof of relation from sheaf quantization to
path integral quantization is given.
