Lecture Notes on Topoi and Quasitopoi

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World Scientific, 1991 - Mathematics - 290 pages
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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
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