Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

The pentalogy [60, 61, 62, 76, 63] is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure DTfcq. We show that the category of finitary differential triads DTfcq is a finitary instance of an elementary topos proper in the original sense due to Lawvere and Tierney. We present in the light of Abstract Differential Geometry (ADG) a Grothendieck-type of generalization of Sorkin’s finitary substitutes of continuous spacetime manifold topologies, the latter’s topological refinement inverse systems of locally finite coverings and their associated coarse graining sieves, the upshot being that DTfcq is also a finitary example of a Grothendieck topos. In the process, we discover that the subobject classifier Ωfcq of DTfcq is a Heyting algebra type of object, thus we infer that the internal logic of our finitary topos is intuitionistic, as expected. We also introduce the new notion of ‘finitary differential geometric morphism’ which, as befits ADG, gives a differential geometric slant to Sorkin’s purely topological acts of refinement (:coarse graining). Based on finitary differential geometric morphisms regarded as natural transformations of the relevant sheaf categories, we observe that the functorial ADG-theoretic version of the principle of general covariance of General Relativity is preserved under topological refinement. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research. PACS numbers: 04.60.-m, 04.20.Gz, 04.20.-q