# Lecture Notes on Topoi and Quasitopoi

World Scientific, 1991 - Mathematics - 290 pages
Quasitopoi generalize topoi, a concept of major importance in the theory of Categoreis, and its applications to Logic and Computer Science. In recent years, quasitopoi have become increasingly important in the diverse areas of Mathematics such as General Topology and Fuzzy Set Theory. These Lecture Notes are the first comprehensive introduction to quasitopoi, and they can serve as a first introduction to topoi as well.

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### Contents

 Categorical Toolchest 1 Functors 3 Natural Transformations 5 Universal Morphisms and Adjunctions 8 Special Adjunctions 12 Limits and Colimits 16 Properties of Limits and Colimits 18 Monads and Comonads 22
 Separated Objects and Sheaves 139 Associated Separated Objects and Sheaves 143 Reflectors for Sheaves and Separated Objects 146 Strong Topologies and Coarse Sheaves 149 Solid Quasitopoi 152 Grothendieck Topologies 157 Canonical Topologies 160 Geometric Morphisms 163

 Cartesian Closed Categories 26 Diagonal Polarity 30 Concrete and Topological Categories 32 Basic Properties 37 Subobjects 38 Relations and Powerset Objects 40 Subobject Classifiers 44 Topoi are Cartesian Closed 46 Partial Morphisms 49 Slice Categories 52 Locally Cartesian Closed Categories 56 Two Definitions of Quasitopoi 58 Universal Quantifiers 61 Coarse Objects of a Quasitopos 63 The Dual Category of a Topos is Algebraic 66 Exactness Properties of Quasitopoi 68 Examples of Topoi and Quasitopoi 73 Spectral Theory 78 Setvalued Presheaves 80 Examples and Complements 84 Sheaves for a Complete Heyting Algebra 87 Separated Presheaves 90 Sheaves on Topological Spaces 93 Examples of Topological Quasitopoi 96 Logic in a Quasitopos 101 Propositional Connectives 102 Quantifiers 107 The Language of a Quasitopos 111 Interpretations of Formulas 115 Internal Validity 118 Rules of Internal Logic 121 Some Constructions in a Quasitopos 125 Internal Unions and Intersections 128 Composition of Relations 130 Topologies and Sheaves 135 Closed and Dense Monomorphisms 136
 Coalgebras Define a Quasitopos 167 Geometric Morphisms 170 Topologies from Sheaves 173 Factorization of Geometric Morphisms 176 Internal Categories and Diagrams 179 Internal Diagrams 183 Internal Functors 186 Internal Limits and Colimits 189 Internal Diagrams over a Quasitopos 193 Topological Quasitopoi 197 Categories of pSieves 198 p Sieves Define a Quasitopos 200 Dense p Sieves 203 Coreflections for Dense Sieves 206 Quotient Sinks 209 Quasitopological Categories 212 Left Exact Concrete Categories 217 pTopologies 220 Properties of Dense Sieves 224 Dense Completions 227 Quasitopos Completions and Quasitopos Hulls 230 Examples 232 Quasitopoi of Fuzzy Sets 237 Hvalued Sets and Relations 242 Categories Set if and Mod H of H valued sets 245 Hvalued Subsets 249 Constructions for valued Sets 252 Sheaves and Presheaves in Set H 257 Set if is a Topos and Mod H a Quasitopos 261 The Topos Structure of ifSets 264 Fuz if and Related Categories 269 First Order Fuzzy Logic 272 Bibliography 277 Index 285 Copyright