UID:
almafu_9959240253902883
Format:
1 online resource (xvii, 522 pages) :
,
digital, PDF file(s).
ISBN:
1-139-88198-1
,
0-511-95795-5
,
1-107-10296-0
,
1-107-09468-2
,
1-107-08850-X
,
0-511-52587-7
Series Statement:
Encyclopedia of mathematics and its applications ; 52
Content:
The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of research they have chosen. The book is planned also to serve as a reference book for both specialists in the field and all those using category theory as a tool. Volume 3 begins with the essential aspects of the theory of locales, proceeding to a study in chapter 2 of the sheaves on a locale and on a topological space, in their various equivalent presentations: functors, etale maps or W-sets. Next, this situation is generalized to the case of sheaves on a site and the corresponding notion of Grothendieck topos is introduced. Chapter 4 relates the theory of Grothendieck toposes with that of accessible categories and sketches, by proving the existence of a classifying topos for all coherent theories.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
,
Cover; Half-title; Title; Copyright; Dedication; Contents; Preface to volume; Introduction to this handbook; 1 Locales; 1.1 The intuitionistic propositional calculus; 1.2 Heyting algebras; 1.3 Locales; 1.4 Limits and colimits of locales; 1.5 Nuclei; 1.6 Open morphisms of locales; 1.7 Etale morphisms of locales; 1.8 The points of a locale; 1.9 Sober spaces; 1.10 Compactness conditions; 1.11 Regularity conditions; 1.12 Exercises; 2 Sheaves; 2.1 Sheaves on a locale; 2.2 Closed subobjects; 2.3 Some categorical properties of sheaves; 2.4 Etale spaces; 2.5 The stalks of a topological sheaf
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2.6 Associated sheaves and etale morphisms2.7 Systems of generators for a sheaf; 2.8 The theory of Ω-sets; 2.9 Complete Ω-sets; 2.10 Some basic facts in ring theory; 2.11 Sheaf representation of a ring; 2.12 Change of base; 2.13 Exercises; 3 Grothendieck toposes; 3.1 A categorical glance at sheaves; 3.2 Grothendieck topologies; 3.3 The associated sheaf functor theorem; 3.4 Categorical properties of Grothendieck toposes; 3.5 Localizations of Grothendieck toposes; 3.6 Characterization of Grothendieck toposes; 3.7 Exercises; 4 The classifying topos; 4.1 The points of a topos
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4.2 The classifying topos of a finite limit theory4.3 The classifying topos of a geometric sketch; 4.4 The classifying topos of a coherent theory; 4.5 Diaconescu's theorem; 4.6 Exercises; 5 Elementary toposes; 5.1 The notion of a topos; 5.2 Examples of toposes; 5.3 Monomorphisms in a topos; 5.4 Some set theoretical notions in a topos; 5.5 Partial morphisms; 5.6 Injective objects; 5.7 Finite colimits; 5.8 The slice toposes; 5.9 Exactness properties of toposes; 5.10 Union of subobjects; 5.11 Morphisms of toposes; 5.12 Exercises; 6 Internal logic of a topos; 6.1 The language of a topos
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6.2 Categorical foundations of the logic of toposes6.3 The calculus of truth tables; 6.4 The point about ""ghost"" variables; 6.5 Coherent theories; 6.6 The Kripke-Joyal semantics; 6.7 The intuitionistic propositional calculus in a topos; 6.8 The intuitionistic predicate calculus in a topos; 6.9 Intuitionistic set theory in a topos; 6.10 The structure of a topos in its internal language; 6.11 Locales in a topos; 6.12 Exercises; 7 The law of excluded middle; 7.1 The regular elements of Ω; 7.2 Boolean toposes; 7.3 De Morgan toposes; 7.4 Decidable objects; 7.5 The axiom of choice; 7.6 Exercises
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8 The axiom of infinity8.1 The natural number object; 8.2 Infinite objects in a topos; 8.3 Arithmetic in a topos; 8.4 The trichotomy; 8.5 Finite objects in a topos; 8.6 Exercises; 9 Sheaves in a topos; 9.1 Topologies in a topos; 9.2 Sheaves for a topology; 9.3 The localizations of a topos; 9.4 The double negation sheaves; 9.5 Exercises; Bibliography; Index
,
English
Additional Edition:
ISBN 0-521-06124-5
Additional Edition:
ISBN 0-521-44180-3
Language:
English
URL:
https://doi.org/10.1017/CBO9780511525872
