Cover Schemes, Frame-Valued Sets and Their Potential Uses in Spacetime Physics

TL;DR

This paper explores the concept of cover schemes in preordered sets, their relationship with frames and Kripke models, and applies these ideas to model discrete spacetime in quantum gravity.

Contribution

It introduces and develops cover schemes in the context of preordered sets and connects them with frame-valued set theory for potential applications in spacetime physics.

Findings

Cover schemes generalize Grothendieck topologies to preordered sets.

Frame-valued set theory can model evolving discrete spacetime.

Application to causal sets in quantum gravity.

Abstract

The immensely fruitful concept of Grothendieck topology or covering issued from the efforts of algebraic geometers to study "sheaf-like" objects defined on categories more general than the lattice of open sets on a topological space. In the present paper the covering concept - here called a cover scheme -is presented and developed in the simple case when the underlying category is a preordered set. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and frame-valued set theory is discussed. Finally cover schemes and frame-valued set theory are applied in the context of Markopoulou's 1999 account of discrete spacetime as sets "evolving" over a causal set.