## Lecture Notes on Topoi and QuasitopoiQuasitopoi 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 |

277 | |

285 | |

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### Common terms and phrases

adjunction bidense bijection cartesian closed coalgebras coarse objects codomain comonad complete Heyting algebra composition concrete category constant maps construct Conversely counit define denote diagram domain dual easily seen element epimorphic equalizer equivalent exponentially adjoint factors final lift finite limits following result follows immediately full subcategory functor F fuzzy sets geometric morphism Grothendieck topology H-set H-subset inclusion induced initial lift initial object injective internal category inverse image isomorphism lattice left adjoint left exact limit cone Mod H monad monomorphic and epimorphic morphism f natural transformations obtained p-dense phism powerset preserves finite presheaves Proof Proposition pullback functor pullback square quotient cone quotient sink relation represents partial morphisms restriction right adjoint satisfies Section Set H sheaf sheaves sieve singleton slice category spaces strong epimorphism strong monomorphism subfunctor subobject classifier subset surjective terminal object Theorem topoi topological category topos unique morphism valid variables