elementary categories elementary toposes oxford logic guides

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Elementary Categories Elementary Toposes

Author : Colin McLarty
ISBN : 0191589497
Genre :
File Size : 67. 82 MB
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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. -

Mathematisch Strukturelle Grundlagen Der Informatik

Author : Hartmut Ehrig
ISBN : 9783642567926
Genre : Computers
File Size : 75. 96 MB
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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.

Sketches Of An Elephant A Topos Theory Compendium

Author : P. T. Johnstone
ISBN : 0198515987
Genre : Mathematics
File Size : 75. 92 MB
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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.

Synthetic Differential Topology

Author : Marta Bunge
ISBN : 9781108692205
Genre : Mathematics
File Size : 56. 78 MB
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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.

Mathematical Reviews

Author :
ISBN : UOM:39015078588582
Genre : Mathematics
File Size : 66. 69 MB
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Category Theory For Computing Science

Author : Michael Barr
ISBN : UOM:39015034447873
Genre : Mathematics
File Size : 41. 79 MB
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Categorical Logic

Author : Andrew M. Pitts
ISBN : UCSC:32106013157604
Genre : Logic, Symbolic and mathematical
File Size : 64. 92 MB
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Abstract: "This document provides an introduction to the interaction between category theory and mathematical logic which is slanted towards computer scientists."

Merging Hol With Set Theory

Author : Michael J. C. Gordon
ISBN : UCSC:32106013459703
Genre : Mathematical statistics
File Size : 84. 24 MB
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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."

Actes

Author : International Association for Cybernetics
ISBN : UOM:39015047364727
Genre : Computers
File Size : 70. 86 MB
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Algebra Of Programming

Author : Richard Bird
ISBN : 013507245X
Genre : Computers
File Size : 63. 45 MB
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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.

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