caa a22 4500 463196481 CHVBK 20180406164924.0 cr unu---uuuuu 170326e20070301xx s 000 0 eng 10.1007/s10773-006-9240-y doi (NATIONALLICENCE)springer-10.1007/s10773-006-9240-y Raptis Ioannis Algebra and Geometry Section, Department of Mathematics, University of Athens, Panepistimioupolis, 157 84, Athens, Greece aut Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity [Elektronische Daten] [Ioannis Raptis] 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 . We show that the category of finitary differential triads 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 is also a finitary example of a Grothendieck topos. In the process, we discover that the subobject classifier Ω fcq of 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. Springer Science+Business Media, LLC, 2007 quantum gravity nationallicence causal sets nationallicence quantum logic nationallicence differential incidence algebras of locally finite partially ordered sets nationallicence abstract differential geometry nationallicence sheaf theory nationallicence category theory nationallicence topos theory nationallicence International Journal of Theoretical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 46/3(2007-03-01), 688-739 0020-7748 46:3<688 2007 46 10773 https://doi.org/10.1007/s10773-006-9240-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10773-006-9240-y text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Raptis Ioannis Algebra and Geometry Section, Department of Mathematics, University of Athens, Panepistimioupolis, 157 84, Athens, Greece aut NATIONALLICENCE 773 0- International Journal of Theoretical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 46/3(2007-03-01), 688-739 0020-7748 46:3<688 2007 46 10773 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer