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Full Description
This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to 'physical geometry'. In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory of adjoint functors in category theory and the elucidation of the concepts of sheaf theory and homological algebra in relation to the description and analysis of dynamically constituted physical geometric spectrums.
Contents
Prolegomena: Exordium; Basic Working Notions; Observables and States; Connections and Differential Analysis; The Functorial Imperative; Grothendieck Topos Interpretation of the Hom-Tensor Adjunction; The Grothendieck Topology of Epimorphic Families; Unit and Counit of the Hom-Tensor Adjunction; General Theory: General Introduction; Basic Assumptions of ADG (Abstract Differential Geometry); Basic Framework; Bohr's Correspondence; Functorial, Topos-Theoretic Mechanism of ADG; Kahler Construction; Elementary Particles in the Jargon of ADG; Relational Aspect of Space, Again; Dynamical Dressing, Extension: Kahler Construction (Contn'd); Adjunction, Least Action Principle; Transformation Law of Potentials, in Terms of ADG; Characteristics of a Physical Law; Complementary Remarks; Epilogue; Applications: Fundamental Adjunctions: On Utiyama's Theme/Principle Through "A-Invariance"; "Affine Geometry" and "Quantum"; Chasing Feynman; Stone-von Neumann Adjunction; Quantized Einstein's Equation; The Essence of ADG; Peroration;
