We present a comprehensive reformulation of General Relativity using category theory, sheaf theory, and topos theory, providing an alternative to the traditional differential geometric framework. The fundamental construct is a category Loc of local spacetime regions equipped with a Grothendieck topology, forming a site (Loc, J). Physical observables are represented as sheaves on this site: the metric sheaf Met, matter sheaf Mat, and their derived structures. The EinsteinField Equations emerge not as differential equations but as natural transformations between functors, defining the solution sheaf Sol—the subsheaf of local configurations satisfying the equations of motion. We develop internal differential geometry within the topos Sh(Loc, J), constructing the Hilbert action as a natural transformation and deriving the field equations from an internal variational principle. A model of General Relativity corresponds to a global section of Sol. This formulation provides a robust mathematical foundation for quantum gravity research and offers natural pathways for unification with quantum theory within a common topos-theoretic framework, while maintaining complete consistency with established experimental results.