## Topoi: The Categorial Analysis of LogicThe first of its kind, this book presents a widely accessible exposition of topos theory, aimed at the philosopher-logician as well as the mathematician. It is suitable for individual study or use in class at the graduate level (it includes 500 exercises). It begins with a fully motivated introduction to category theory itself, moving always from the particular example to the abstract concept. It then introduces the notion of elementary topos, with a wide range of examples and goes on to develop its theory in depth, and to elicit in detail its relationship to Kripke's intuitionistic semantics, models of classical set theory and the conceptual framework of sheaf theory (``localization'' of truth). Of particular interest is a Dedekind-cuts style construction of number systems in topoi, leading to a model of the intuitionistic continuum in which a ``Dedekind-real'' becomes represented as a ``continuously-variable classical real number''.The second edition contains a new chapter, entitled Logical Geometry, which introduces the reader to the theory of geometric morphisms of Grothendieck topoi, and its model-theoretic rendering by Makkai and Reyes. The aim of this chapter is to explain why Deligne's theorem about the existence of points of coherent topoi is equivalent to the classical Completeness theorem for ``geometric'' first-order formulae. |

### Contents

PROSPECTUS | 1 |

CHAPTER 1 MATHEMATICS SET THEORY? | 6 |

CHAPTER 2 WHAT CATEGORIES ARE | 17 |

CHAPTER 3 ARROWS INSTEAD OF EPSILON | 37 |

CHAPTER 4 INTRODUCING TOPOI | 75 |

FIRST STEPS | 109 |

CHAPTER 6 LOGIC CLASSICALLY CONCEIVED | 125 |

CHAPTER 7 ALGEBRA OF SUBOBJECTS | 146 |

CHAPTER 11 ELEMENTARY TRUTH | 230 |

CHAPTER 12 CATEGORIAL SET THEORY | 289 |

CHAPTER 13 ARITHMETIC | 332 |

CHAPTER 14 LOCAL TRUTH | 359 |

CHAPTER 15 ADJOINTNESS AND QUANTIFIERS | 438 |

CHAPTER 16 LOGICAL GEOMETRY | 458 |

REFERENCES | 521 |

CATALOGUE OF NOTATION | 531 |

CHAPTER 8 INTUITIONISM AND ITS LOGIC | 173 |

CHAPTER 9 FUNCTORS | 194 |

CHAPTER 10 SET CONCEPTS AND VALIDITY | 211 |

INDEX OF DEFINITIONS | 541 |

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### Common terms and phrases

6-arrow 6-object arrow f assigns axioms bijection Bn(I Boolean category theory characterisation classical co-product codomain colimits collection component concept construction corresponding defined definition denoted diagram commute disjoint domain epic epimorphic equaliser equivalent ExAMPLE ExERCISE exists extensional f and g finite formula function f functor F geometric morphism germs given gives Grothendieck topos hence Heyting algebra homeomorphism identity arrow implies inclusion interpretation intuitionistic inverse language lattice left adjoint Lemma Let f logic mathematical monic monoid natural numbers object natural transformation notion open sets pair partial elements poset pre-order precisely preserves PROOF pseudo-complement pullback Q-Set relation satisfies semantics sentence set theory set-theoretic Sh(C sheaf sheaves Show singleton stalk structure Sub(d subobject classifier subset symbol terminal object Theorem Top(I topoi topological space topology true truth truth-values unique arrow valid variables well-pointed