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

TL;DR

This paper develops a topos-theoretic framework for finitary, locally finite quantum gravity models based on Abstract Differential Geometry, providing a background-independent, singularity-free approach to Einstein's vacuum equations.

Contribution

It introduces a topos structure for finitary quantum causal sets within ADG, extending categorical methods to discrete quantum gravity models and generalizing Sorkin's finitary topologies.

Findings

Finitary sheaves of quantum causal sets form a topos structure.

The topos provides a background-independent, singularity-free formulation of Einstein's equations.

The framework generalizes Sorkin's finitary topologies within a differential geometric setting.

Abstract

Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity 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. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. 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.