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International Journal of Theoretical Physics

, Volume 46, Issue 3, pp 688–739 | Cite as

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

  • Ioannis Raptis
Article

Abstract

The pentalogy (Mallios, A. and Raptis, I. (2001). International Journal of Theoretical Physics 40, 1885; Mallios, A. and Raptis, I. (2002). International Journal of Theoretical Physics 41, 1857; Mallios, A. and Raptis, I. (2003).International Journal of Theoretical Physics 42, 1479; Mallios, A. and Raptis, I. (2004). ‘paper-book’/research monograph); I. Raptis (2005). International Journal of Theoretical Physics (to appear)is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantumcausal 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 Open image in new window . We show that the category of finitary differential triads Open image in new window is a finitary instance of an elementary topos proper in the original sense dueto 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 manifoldtopologies, the latter’s topological refinement inverse systems of locally finite coverings and their associated coarse graining sieves, the upshot being that Open image in new window is also a finitary example of a Grothendieck topos. In the process, we discover that the subobject classifier Ω fcq of Open image in new window 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 GeneralRelativity 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.

Keywords

quantum gravity causal sets quantum logic differential incidence algebras of locally finite partially ordered sets abstract differential geometry sheaf theory category theory topos theory 

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Authors and Affiliations

  1. 1.Algebra and Geometry Section, Department of MathematicsUniversity of AthensAthensGreece
  2. 2.Theoretical Physics Group, Blackett Laboratory, Imperial College of ScienceTechnology and MedicineLondonUK

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