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Table of Contents Chapter 1 A Taste of Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1. Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2. Grothendieck Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3. Separable Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 2 Noncommutative Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 2.1. Small Categories, Posets and Noncommutative Topologies . . . . . . . . . 29 2.2. The Topology of Virtual Opens and its Commutative Shadow . . . . . 40 2.3. Points and the Point Spectrum. Points in a Pointless world . . . . . . . 54 2.4. Presheaves and Sheaves over Noncommutative Topologies. . . . . . . . . .63 2.5. Noncommutative Grothendieck Topologies . . . . . . . . . . . . . . . . . . . . . . . . .68 2.6. The Fundamental Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 I. Torsion Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 II. L(H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 III. Ore sets in Schematic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Chapter 3 Grothendieck Categorical Representations . . . . . . . . . 121 3.1. Spectral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.2. Affine Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.3. Quotient representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.4. Noncommutative Projective Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154 Chapter 4 Sheaves and Dynamical Topology . . . . . . . . . . . . . . . . . . . 163 4.1. Introducing Structure Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163 4.2. Dynamical Presheaves and Temporal Points . . . . . . . . . . . . . . . . . . . . . . 176 4.3. The ¿Spaced-Time¿ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5
Library of Congress Subject Headings for this publication:
Categories (Mathematics).
Grothendieck categories.
Representations of congruence lattices.
Sheaf theory.
Dynamics.
Noncommutative function spaces.