Toposes and Local Set Theories

An Introduction
Author: John L. Bell
Publisher: Courier Corporation
ISBN: 0486462862
Category: Mathematics
Page: 267
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This text introduces topos theory, a development in category theory that unites important but seemingly diverse notions from algebraic geometry, set theory, and intuitionistic logic. Topics include local set theories, fundamental properties of toposes, sheaves, local-valued sets, and natural and real numbers in local set theories. 1988 edition.

Topos Theory


Author: P.T. Johnstone
Publisher: Courier Corporation
ISBN: 0486493369
Category: Mathematics
Page: 400
View: 4694
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Focusing on topos theory's integration of geometric and logical ideas into the foundations of mathematics and theoretical computer science, this volume explores internal category theory, topologies and sheaves, geometric morphisms, and other subjects. 1977 edition.

Elementary Categories, Elementary Toposes


Author: Colin McLarty
Publisher: Clarendon Press
ISBN: 9780191589492
Category:
Page: 278
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The book covers elementary aspects of category theory and topos theory. It has few mathematical prerequisites, and uses categorical methods throughout rather than beginning with set theoretic foundations. It works with key notions such as cartesian closedness, adjunctions, regular categories, and the internal logic of a topos. Full statements and elementary proofs are given for the central theorems, including the fundamental theorem of toposes, the sheafification theorem, and the construction of Grothendieck toposes over any topos as base. Three chapters discuss applications of toposes in detail, namely to sets, to basic differential geometry, and to recursive analysis. - ;Introduction; PART I: CATEGORIES: Rudimentary structures in a category; Products, equalizers, and their duals; Groups; Sub-objects, pullbacks, and limits; Relations; Cartesian closed categories; Product operators and others; PART II: THE CATEGORY OF CATEGORIES: Functors and categories; Natural transformations; Adjunctions; Slice categories; Mathematical foundations; PART III: TOPOSES: Basics; The internal language; A soundness proof for topos logic; From the internal language to the topos; The fundamental theorem; External semantics; Natural number objects; Categories in a topos; Topologies; PART IV: SOME TOPOSES: Sets; Synthetic differential geometry; The effective topos; Relations in regular categories; Further reading; Bibliography; Index. -

Models for Smooth Infinitesimal Analysis


Author: Ieke Moerdijk,Gonzalo E. Reyes
Publisher: Springer Science & Business Media
ISBN: 147574143X
Category: Mathematics
Page: 400
View: 6691
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The aim of this book is to construct categories of spaces which contain all the C?-manifolds, but in addition infinitesimal spaces and arbitrary function spaces. To this end, the techniques of Grothendieck toposes (and the logic inherent to them) are explained at a leisurely pace and applied. By discussing topics such as integration, cohomology and vector bundles in the new context, the adequacy of these new spaces for analysis and geometry will be illustrated and the connection to the classical approach to C?-manifolds will be explained.

Models and Ultraproducts

An Introduction
Author: John Lane Bell,A. B. Slomson
Publisher: Courier Corporation
ISBN: 0486449793
Category: Mathematics
Page: 322
View: 7602
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In this text for first-year graduate students, the authors provide an elementary exposition of some of the basic concepts of model theory--focusing particularly on the ultraproduct construction and the areas in which it is most useful. The book, which assumes only that its readers are acquainted with the rudiments of set theory, starts by developing the notions of Boolean algebra, propositional calculus, and predicate calculus. Model theory proper begins in the fourth chapter, followed by an introduction to ultraproduct construction, which includes a detailed look at its theoretic properties. An overview of elementary equivalence provides algebraic descriptions of the elementary classes. Discussions of completeness follow, along with surveys of the work of Jónsson and of Morley and Vaught on homogeneous universal models, and the results of Keisler in connection with the notion of a saturated structure. Additional topics include classical results of Gödel and Skolem, and extensions of classical first-order logic in terms of generalized quantifiers and infinitary languages. Numerous exercises appear throughout the text.

Topology Via Logic


Author: Steven Vickers
Publisher: Cambridge University Press
ISBN: 9780521576512
Category: Computers
Page: 200
View: 434
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This is an advanced textbook on topology for computer scientists. It is based on a course given by the author to postgraduate students of computer science at Imperial College.

Toposes, Triples and Theories


Author: M. Barr,C. Wells
Publisher: Springer Science & Business Media
ISBN: 1489900217
Category: Mathematics
Page: 347
View: 7866
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As its title suggests, this book is an introduction to three ideas and the connections between them. Before describing the content of the book in detail, we describe each concept briefly. More extensive introductory descriptions of each concept are in the introductions and notes to Chapters 2, 3 and 4. A topos is a special kind of category defined by axioms saying roughly that certain constructions one can make with sets can be done in the category. In that sense, a topos is a generalized set theory. However, it originated with Grothendieck and Giraud as an abstraction of the of the category of sheaves of sets on a topological space. Later, properties Lawvere and Tierney introduced a more general id~a which they called "elementary topos" (because their axioms did not quantify over sets), and they and other mathematicians developed the idea that a theory in the sense of mathematical logic can be regarded as a topos, perhaps after a process of completion. The concept of triple originated (under the name "standard construc in Godement's book on sheaf theory for the purpose of computing tions") sheaf cohomology. Then Peter Huber discovered that triples capture much of the information of adjoint pairs. Later Linton discovered that triples gave an equivalent approach to Lawverc's theory of equational theories (or rather the infinite generalizations of that theory). Finally, triples have turned out to be a very important tool for deriving various properties of toposes.

Topoi

The Categorial Analysis of Logic
Author: Robert Goldblatt
Publisher: Courier Corporation
ISBN: 048631796X
Category: Mathematics
Page: 576
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A classic exposition of a branch of mathematical logic that uses category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers.

An Introduction to Category Theory


Author: Harold Simmons
Publisher: Cambridge University Press
ISBN: 1139503324
Category: Mathematics
Page: N.A
View: 5156
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Category theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course.

Sets for Mathematics


Author: F. William Lawvere,Robert Rosebrugh
Publisher: Cambridge University Press
ISBN: 9780521010603
Category: Mathematics
Page: 261
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In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.

Intuitionistic Set Theory


Author: John L. Bell
Publisher: N.A
ISBN: 9781848901407
Category: Mathematics
Page: 132
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While intuitionistic (or constructive) set theory IST has received a certain attention from mathematical logicians, so far as I am aware no book providing a systematic introduction to the subject has yet been published. This may be the case in part because, as a form of higher-order intuitionistic logic - the internal logic of a topos - IST has been chiefly developed in a tops-theoretic context. In particular, proofs of relative consistency with IST for mathematical assertions have been (implicitly) formulated in topos- or sheaf-theoretic terms, rather than in the framework of Heyting-algebra-valued models, the natural extension to IST of the well-known Boolean-valued models for classical set theory. In this book I offer a brief but systematic introduction to IST which develops the subject up to and including the use of Heyting-algebra-valued models in relative consistency proofs. I believe that IST, presented as it is in the familiar language of set theory, will appeal particularly to those logicians, mathematicians and philosophers who are unacquainted with the methods of topos theory.

A Book of Set Theory


Author: Charles C Pinter
Publisher: Courier Corporation
ISBN: 0486497089
Category: Mathematics
Page: 256
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"This accessible approach to set theory for upper-level undergraduates poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. 1971 edition with new material by the author"--

Set Theory and the Continuum Hypothesis


Author: Paul J. Cohen,Martin Davis
Publisher: Courier Corporation
ISBN: 0486469212
Category: Mathematics
Page: 154
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This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

Axiomatic Method and Category Theory


Author: Andrei Rodin
Publisher: Springer Science & Business Media
ISBN: 3319004042
Category: Philosophy
Page: 285
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This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.

A Primer of Infinitesimal Analysis


Author: John L. Bell
Publisher: Cambridge University Press
ISBN: 0521887186
Category: Mathematics
Page: 124
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A rigorous, axiomatically formulated presentation of the 'zero-square', or 'nilpotent' infinitesimal.

Abstract and Concrete Categories

The Joy of Cats
Author: Jiri Adamek,Jiří Adámek (ing.),Horst Herrlich,George E. Strecker
Publisher: N.A
ISBN: 9780486469348
Category: Mathematics
Page: 517
View: 8970
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This up-to-date introductory treatment employs the language of category theory to explore the theory of structures. Its unique approach stresses concrete categories, and each categorical notion features several examples that clearly illustrate specific and general cases. A systematic view of factorization structures, this volume contains seven chapters. The first five focus on basic theory, and the final two explore more recent research results in the realm of concrete categories, cartesian closed categories, and quasitopoi. Suitable for advanced undergraduate and graduate students, it requires an elementary knowledge of set theory and can be used as a reference as well as a text. Updated by the authors in 2004, it offers a unifying perspective on earlier work and summarizes recent developments.

Logic and Structure


Author: Dirk van Dalen
Publisher: Springer Science & Business Media
ISBN: 3662029626
Category: Mathematics
Page: 220
View: 8848
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New corrected printing of a well-established text on logic at the introductory level.

Algebraic Logic


Author: Paul R. Halmos
Publisher: Courier Dover Publications
ISBN: 0486810410
Category: Mathematics
Page: 272
View: 5835
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Beginning with an introduction to the concepts of algebraic logic, this concise volume features ten articles by a prominent mathematician that originally appeared in journals from 1954 to 1959. Covering monadic and polyadic algebras, these articles are essentially self-contained and accessible to a general mathematical audience, requiring no specialized knowledge of algebra or logic. Part One addresses monadic algebras, with articles on general theory, representation, and freedom. Part Two explores polyadic algebras, progressing from general theory and terms to equality. Part Three offers three items on polyadic Boolean algebras, including a survey of predicates, terms, operations, and equality. The book concludes with an additional bibliography and index.

Model Theory

Third Edition
Author: C.C. Chang,H. Jerome Keisler
Publisher: Courier Corporation
ISBN: 0486310957
Category: Mathematics
Page: 672
View: 7336
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This bestselling textbook for higher-level courses was extensively revised in 1990 to accommodate developments in model theoretic methods. Topics include models constructed from constants, ultraproducts, and saturated and special models. 1990 edition.

Category Theory in Context


Author: Emily Riehl
Publisher: Courier Dover Publications
ISBN: 0486820807
Category: Mathematics
Page: 272
View: 4283
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Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.