Elementary Categories, Elementary Toposes

Author: Colin McLarty

Publisher: Clarendon Press

ISBN: 9780191589492

Category:

Page: 278

View: 5919

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. -
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Sketches of an Elephant: A Topos Theory Compendium

Author: P. T. Johnstone

Publisher: Oxford University Press

ISBN: 9780198515982

Category: Mathematics

Page: 716

View: 7861

Topos Theory is an important branch of mathematical logic of interest to theoretical computer scientists, logicians and philosophers who study the foundations of mathematics, and to those working in differential geometry and continuum physics. This compendium contains material that was previously available only in specialist journals. This is likely to become the standard reference work for all those interested in the subject.
Posted in Mathematics

Mathematisch-strukturelle Grundlagen der Informatik

Author: Hartmut Ehrig,Bernd Mahr,F. Cornelius,Martin Große-Rhode,P. Zeitz

Publisher: Springer-Verlag

ISBN: 3642567924

Category: Computers

Page: 622

View: 1641

In fünf sorgfältig aufeinander abgestimmten Teilen behandelt das Buch die wesentlichen mathematischen Elemente der formalen Spezifikation von Systemen und der Aussagen- und Prädikatenlogik, die für das Verständnis des formalisierten Problemlösens entscheidend und damit für Informatiker unerläßlich sind. Eine Einführung in die intuitive Mengentheorie vermittelt zunächst notwendige mathematische Grundlagen. Motiviert durch das Konzept von Datenstrukturen und abstrakten Datentypen werden dann algebraische Strukturen in der Informatik behandelt. Danach werden Aussagen- und Prädikatenlogik aus der Sicht der Mathematik und Informatik dargestellt. Schließlich führt die Kategorientheorie für Informatiker in die Welt der abstrakten Behandlung mathematischer Strukturen ein. Die Neuauflage wurde erweitert um Darstellungen zur Modellalgebra und zur Implementierung. Übungsaufgaben wurden ergänzt.
Posted in Computers

Synthetic Differential Topology

Author: Marta Bunge,Felipe Gago,Ana María San Luis

Publisher: Cambridge University Press

ISBN: 1108692206

Category: Mathematics

Page: N.A

View: 9120

This book formally introduces synthetic differential topology, a natural extension of the theory of synthetic differential geometry which captures classical concepts of differential geometry and topology by means of the rich categorical structure of a necessarily non-Boolean topos and of the systematic use of logical infinitesimal objects in it. Beginning with an introduction to those parts of topos theory and synthetic differential geometry necessary for the remainder, this clear and comprehensive text covers the general theory of synthetic differential topology and several applications of it to classical mathematics, including the calculus of variations, Mather's theorem, and Morse theory on the classification of singularities. The book represents the state of the art in synthetic differential topology and will be of interest to researchers in topos theory and to mathematicians interested in the categorical foundations of differential geometry and topology.
Posted in Mathematics

Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 2662

Posted in Mathematics

Actes

Proceedings

Author: International Association for Cybernetics

Publisher: N.A

ISBN: N.A

Category: Computers

Page: N.A

View: 7489

Posted in Computers

Categorical Logic

Author: Andrew M. Pitts

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: 94

View: 1752

Abstract: "This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists."
Posted in Logic, Symbolic and mathematical

Merging HOL with Set Theory

Preliminary Experiments

Author: Michael J. C. Gordon

Publisher: N.A

ISBN: N.A

Category: Mathematical statistics

Page: 40

View: 4942

Abstract: "Set theory is the standard foundation for mathematics, but the majority of general purpose mechanized proof assistants support versions of type theory (higher order logic). Examples include Alf, Automath, Coq, Ehdm, HOL, IMPS, Lambda, LEGO, Nuprl, PVS and Veritas. For many applications type theory works well and provides, for specification, the benefits of type-checking that are well-known in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientific or engineering background already know set theory, whereas type theory may appear inaccessable [sic] and so be an obstacle to the uptake of proof assistants based on it. This paper describes some experiments (using HOL) in combining set theory and type theory; the aim is to get the best of both worlds in a single system. Three approaches have been tried, all based on an axiomatically specified type V of ZF-like sets: (i) HOL is used without any additions besides V; (ii) an embedding of the HOL logic into V is provided; (iii) HOL axiomatic theories are not automatically translated into set-theoretic definitional theories. These approaches are illustrated with two examples: the construction of lists and a simple lemma in group theory."
Posted in Mathematical statistics

Algebra of Programming

Author: Richard Bird,Oege de Moor

Publisher: N.A

ISBN: 9780135072455

Category: Computers

Page: 295

View: 2981

Describes an algebraic approach to programming that permits the calculation of programs. Introduces the fundamentals of algebra for programming. Presents paradigms and strategies of program construction that form the core of Algorithm Design. Discusses functions and categories; applications; relations and allegories; datatypes; recursive programs, optimization issues, thinning algorithms, dynamic programming and greedy algorithms. Appropriate for all programmers.
Posted in Computers

Forthcoming Books

Author: Rose Arny

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 2220

Posted in American literature

Relative category theory and geometric morphisms

a logical approach

Author: Jonathan Chapman,Frederick Rowbottom

Publisher: Oxford University Press, USA

ISBN: N.A

Category: Mathematics

Page: 263

View: 2375

Topos theory provides an important setting and language for much of mathematical logic and set theory. It is well known that a typed language can be given for a topos to be regarded as a category of sets. This enables a fruitful interplay between category theory and set theory. However, one stumbling block to a logical approach to topos theory has been the treatment of geometric morphisms. This book presents a convenient and natural solution to this problem by developing the notion of a frame relative to an elementary topos. The authors show how this technique enables a logical approach to be taken to topics such as category theory relative to a topos and the relative Giraud theorem. The work is self-contained except that the authors presuppose a familiarity with basic category theory and topos theory. Logicians, set and category theorists, and computer scientist working in the field will find this work essential reading.
Posted in Mathematics

Books in Print

Author: N.A

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 488

Books in print is the major source of information on books currently published and in print in the United States. The database provides the record of forthcoming books, books in-print, and books out-of-print.
Posted in American literature

The Bulletin of Symbolic Logic

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Logic, Symbolic and mathematical

Page: N.A

View: 8128

Posted in Logic, Symbolic and mathematical

Kategorien und Funktoren

Author: Bodo Pareigis

Publisher: N.A

ISBN: N.A

Category: Categories (Mathematics)

Page: 192

View: 1048

Posted in Categories (Mathematics)

Einführung in die Kategorientheorie

Mit ausführlichen Erklärungen und zahlreichen Beispielen

Author: Martin Brandenburg

Publisher: Springer-Verlag

ISBN: 3662535211

Category: Mathematics

Page: 343

View: 7455

Die Kategorientheorie deckt die innere Architektur der Mathematik auf. Dabei werden die strukturellen Gemeinsamkeiten zwischen mathematischen Disziplinen und ihren spezifischen Konstruktionen herausgearbeitet. Dieses Buch gibt eine systematische Einführung in die Grundbegriffe der Kategorientheorie. Zahlreiche ausführliche Erklärungstexte sowie die große Menge an Beispielen helfen beim Einstieg in diese verhältnismäßig abstrakte Theorie. Es werden viele konkrete Anwendungen besprochen, welche die Nützlichkeit der Kategorientheorie im mathematischen Alltag belegen. Jedes Kapitel wird mit einem motivierenden Text eingeleitet und mit einer großen Aufgabensammlung abgeschlossen. An Vorwissen muss der Leser lediglich ein paar Grundbegriffe des Mathematik-Studiums mitbringen. Die vorliegende zweite vollständig durchgesehene Auflage ist um ausführliche Lösungen zu ausgewählten Aufgaben ergänzt.
Posted in Mathematics

The British National Bibliography

Author: Arthur James Wells

Publisher: N.A

ISBN: N.A

Category: English literature

Page: N.A

View: 6929

Posted in English literature