Sketches of an Elephant: A Topos Theory Compendium: Volume 2Topos Theory is a subject that stands at the junction of geometry, mathematical logic and theoretical computer science, and it derives much of its power from the interplay of ideas drawn from these different areas. Because of this, an account of topos theory which approaches the subject from one particular direction can only hope to give a partial picture; the aim of this compendium is to present as comprehensive an account as possible of all the main approaches and thereby to demonstrate the overall unity of the subject. The material is organized in such a way that readers interested in following a particular line of approach may do so by starting at an appropriate point in the text. |
Contents
C1 Sheaves on a locale | 471 |
B 2CATEGORICAL ASPECTS OF TOPOS THEORY 233 | 474 |
| 477 | |
C2 Sheaves on a site | 536 |
C3 Classes of geometric morphisms | 606 |
F TOPOSES AS MATHEMATICAL UNIVERSES | 702 |
C4 Local compactness and exponentiability | 711 |
C5 Toposes as groupoids | 755 |
D4 Higherorder logic | 940 |
XCalculus and cartesian closed categories | 951 |
Toposes as type theories | 963 |
F3 The free topos | 970 |
Predicative type theories | 976 |
Axioms of choice and booleanness | 987 |
De Morgans law and the Gleason cover | 998 |
Real numbers in a topos | 1012 |
TOPOSES AS THEORIES | 805 |
D1 Firstorder categorical logic | 807 |
Categorical semantics | 817 |
Firstorder logic | 828 |
Syntactic categories | 841 |
Classical completeness | 852 |
D2 Sketches | 861 |
Sketches and theories | 868 |
Sketchable and accessible categories | 874 |
Properties of model categories | 882 |
D3 Classifying toposes | 890 |
The object classifier | 901 |
Coherent toposes | 910 |
Boolean classifying toposes | 918 |
Conceptual completeness | 931 |
D5 Aspects of finiteness | 1033 |
Finite cardinals | 1041 |
Finitary algebraic theories | 1049 |
Kuratowskifiniteness | 1058 |
Orbitals and numerals | 1075 |
F4 Topos theory and set theory | 1078 |
F TOPOSES AS MATHEMATICAL UNIVERSES | 1080 |
Bibliography follows p | 1089 |
A TOPOSES AS CATEGORIES | 1 |
| 20 | |
Index of notation 55 | 55 |
General Index 61 | 61 |
A2 Toposes basic theory | 68 |
| 67 | |
Other editions - View all
Sketches of an Elephant: A Topos Theory Compendium: Volume 2 P. T. Johnstone No preview available - 2002 |
Common terms and phrases
algebra arbitrary atomic axiom Beck-Chevalley condition Boolean BTop/S canonical cartesian closed codomain coequalizer coherent commutative compact composite construction Cont(G context coproduct corresponding covering deduce defined definition denote dense diagram element epimorphism equivalent fact factorization fibration fibrewise filtered colimits finite follows forgetful functor frame homomorphism full subcategory functor F geometric morphism Grothendieck topos hence Heyting hyperconnected inclusion Ind-C induced injective internal locale inverse image functor isomorphism J-covering K-finite lattice left adjoint Lemma local homeomorphism locally connected logic Math monic monoid monomorphism Morgan's law morphism f natural number object natural transformation obtain open sublocales poset preserves Proof Proposition provable pullback pullback square regular resp right adjoint S-indexed S-topos satisfies sequent Sh(C Sh(X Sh(Y sheaf sheaves sieve simply space subobject subsets subterminal objects surjective T-model terminal object theorem theory topology toposes unique verify