## Sheaves in Geometry and Logic: A First Introduction to Topos TheoryWe dedicate this book to the memory of J. Frank Adams. His clear insights have inspired many mathematicians, including both of us. In January 1989, when the first draft of our book had been completed, we heard the sad news of his untimely death. This has cast a shadow on our subsequent work. Our views of topos theory, as presented here, have been shaped by continued study, by conferences, and by many personal contacts with friends and colleagues-including especially O. Bruno, P. Freyd, J.M.E. Hyland, P.T. Johnstone, A. Joyal, A. Kock, F.W. Lawvere, G.E. Reyes, R Solovay, R Swan, RW. Thomason, M. Tierney, and G.C. Wraith. Our presentation combines ideas and results from these people and from many others, but we have not endeavored to specify the various original sources. Moreover, a number of people have assisted in our work by pro viding helpful comments on portions of the manuscript. In this respect, we extend our hearty thanks in particular to P. Corazza, K. Edwards, J. Greenlees, G. Janelidze, G. Lewis, and S. Schanuel. |

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### Contents

II | 1 |

III | 10 |

IV | 24 |

V | 29 |

VI | 31 |

VII | 35 |

VIII | 39 |

IX | 44 |

LXII | 296 |

LXIII | 302 |

LXIV | 315 |

LXV | 318 |

LXVI | 324 |

LXVII | 331 |

LXVIII | 343 |

LXIX | 347 |

X | 48 |

XI | 50 |

XII | 57 |

XIII | 62 |

XIV | 64 |

XV | 65 |

XVI | 69 |

XVII | 73 |

XVIII | 79 |

XIX | 83 |

XX | 88 |

XXI | 95 |

XXII | 97 |

XXIII | 99 |

XXIV | 103 |

XXV | 106 |

XXVI | 109 |

XXVII | 116 |

XXVIII | 121 |

XXIX | 128 |

XXX | 134 |

XXXI | 140 |

XXXII | 145 |

XXXIII | 150 |

XXXIV | 155 |

XXXV | 161 |

XXXVI | 167 |

XXXVII | 171 |

XXXVIII | 176 |

XXXIX | 180 |

XL | 184 |

XLI | 190 |

XLII | 198 |

XLIII | 204 |

XLIV | 210 |

XLV | 213 |

XLVI | 218 |

XLVII | 219 |

XLVIII | 223 |

XLIX | 227 |

L | 233 |

LI | 235 |

LII | 237 |

LIII | 240 |

LIV | 247 |

LV | 256 |

LVI | 263 |

LVII | 267 |

LVIII | 268 |

LIX | 277 |

LX | 284 |

LXI | 291 |

LXX | 348 |

LXXI | 353 |

LXXII | 361 |

LXXIII | 366 |

LXXIV | 378 |

LXXV | 384 |

LXXVI | 390 |

LXXVII | 394 |

LXXVIII | 399 |

LXXIX | 409 |

LXXX | 416 |

LXXXI | 421 |

LXXXII | 422 |

LXXXIII | 425 |

LXXXIV | 434 |

LXXXV | 436 |

LXXXVI | 439 |

LXXXVII | 447 |

LXXXVIII | 452 |

LXXXIX | 457 |

XC | 468 |

XCI | 472 |

XCII | 473 |

XCIII | 475 |

XCIV | 477 |

XCV | 482 |

XCVI | 489 |

XCVII | 493 |

XCVIII | 502 |

XCIX | 508 |

C | 513 |

CI | 516 |

CII | 521 |

CIII | 523 |

CIV | 528 |

CV | 529 |

CVI | 532 |

CVII | 535 |

CVIII | 541 |

CIX | 555 |

CX | 561 |

CXI | 568 |

CXII | 571 |

CXIII | 574 |

CXV | 577 |

CXVI | 580 |

CXVII | 589 |

CXVIII | 598 |

605 | |

615 | |

619 | |

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### Common terms and phrases

adjunction arrow f associated sheaf axiom bijection Boolean algebra bundle classifying topos cocomplete coequalizer colimits composite condition consider construction continuous map coproduct Corollary corresponding counit covering sieve Def(M defined definition element embedding epimorphic epimorphic family equalizer equivalence of categories etale example exists exponential factors filtering finite limits follows free variables function functor category functor f G-torsor geometric morphism given Grothendieck topology Grothendieck topos hence Heyting algebra homomorphism identity implies inclusion induced inverse image kernel pair lattice left adjoint left exact Lemma locales map f matching family monic mono monomorphism Moreover morphism f natural isomorphism natural numbers natural transformation open sets open subsets phism poset presheaf Proposition prove pullback right adjoint ring sends Setsc Setsc°P Sh(C Sh(X sheaf F sheaves small category square subobject classifier subsheaf surjection tensor product terminal object Theorem topoi topological space unique yields Yoneda