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

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

2003-01-09

arxiv.org/abs/gr-qc/0301034v2

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

34 pages

Scientific paper

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.

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