VIRTUAL TOPOLOGY AND FUNCTOR GEOMETRY
CONTENTS A TASTE OF CATEGORY THEORY BASIC NOTIONS GROTHENDIECK CATEGORIES SEPARABLE FUNCTORS NONCOMMUTATIVE SPACES SMALL CATEGORIES, POSETS, AND NONCOMMUTATIVE TOPOLOGIES DIRECTED SUBSETS AND THE LIMIT POSET POSET DYNAMICS THE TOPOLOGY OF VIRTUAL OPENS AND ITS COMMUTATIVE SHADOW PROJECTS MORE NONCOMMUTATIVE TOPOLOGY SOME DIMENSION THEORY POINTS AND THE POINT SPECTRUM: POINTS IN A POINTLESS WORLD THE RELATION BETWEEN QUANTUM POINTS AND STRONG IDEMPOTENTS FUNCTIONS ON SETS OF QUANTUM POINTS NONCOMMUTATIVE GROTHENDIECK TOPOLOGIES WARNING A NONCOMMUTATIVE TOPOS THEORY NONCOMMUTATIVE PROBABILITY (AND MEASURE) THEORY COVERS AND COHOMOLOGY THEORIES THE FUNDAMENTAL EXAMPLES I: TORSION THEORIES PROJECT: MICROLOCALIZATION IN A GROTHENDIECK CATEGORY THE FUNDAMENTAL EXAMPLES II: L(H) THE GENERALIZED STONE TOPOLOGY NOTE PROJECT: NONCOMMUTATIVE GELFAND DUALITY ORE SETS IN SCHEMATIC ALGEBRAS SPECTRAL REPRESENTATIONS AF?NE ELEMENTS OBSERVATION AND EXAMPLE PROJECT: GEOMETRICALLY GRADED RINGS NONCOMMUTATIVE PROJECTIVE SPACE PROJECT: EXTENDED THEORY FOR GABRIEL DIMENSION PROPERTIES OF GABRIEL DIMENSION PROJECT: GENERAL BIRATIONALITY SHEAVES AND DYNAMICAL TOPOLOGY INTRODUCING STRUCTURE SHEAVES CLASSICAL EXAMPLE AND MOTIVATION ABSTRACT NONCOMMUTATIVE SPACES AND SCHEMES PROJECT: REPLACING ESSENTIAL BY SEPARABLE FUNCTORS EXAMPLE: ORE SETS IN SCHEMATIC ALGEBRAS PROJECT: MONADS IN BICATEGORIES PROJECT: SPECTRAL FAMILIES ON THE SPECTRUM PROJECT: TEMPORAL ?CECH AND SHEAF COHOMOLOGY SUBPROJECT 1: TEMPORAL GROTHENDIECK REPRESENTATIONS PROJECT: DYNAMICAL GROTHENDIECK TOPOLOGIES THE SPACED-TIME MODEL NONCOMMUTATIVE MANIFOLDS TOWARD REAL NONCOMMUTATIVE MANIFOLDS FOOD FOR THOUGHT: FROM PHYSICS TO PHILOSOPHY CHAPTER EXAMPLES AND NOTATION I THE CATEGORY SET IS OBTAINED BY TAKING THE CLASS OF ALL SETS USING MAPS FOR THE MORPHISMS II THE CATEGORY TOP IS OBTAINED BY TAKING THE CLASS OF ALL TOPOLOGICAL SPACES USING CONTINUOUS FUNCTIONS FOR THE MORPHISMS III THE CATEGORY AB IS OBTAINED BY TAKING THE CLASS OF ALL ABELIAN GROUPS USING GROUPS HOMOMORPHISMS FOR THE MORPHISMS IV THE CATEGORY GR IS OBTAINED BY TAKING THE CLASS OF ALL GROUPS USING GROUP HOMOMORPHISMS FOR THE MORPHISMS