Elementary Categories Elementary Toposes

Author : Colin McLarty
Publisher : Clarendon Press
Page : 282 pages
File Size : 53,7 Mb
Release : 1992-06-04
Category : Electronic
ISBN : 9780191589492

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Elementary Categories, Elementary Toposes by Colin McLarty Pdf

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. -

Author : P.T. Johnstone
Publisher : Courier Corporation
Page : 401 pages
File Size : 49,5 Mb
Release : 2014-01-15
Category : Mathematics
ISBN : 9780486493367

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Topos Theory by P.T. Johnstone Pdf

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.

Author : F.W. Lawvere,C. Maurer,G.C. Wraith
Publisher : Springer
Page : 352 pages
File Size : 50,5 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540374954

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Model Theory and Topoi by F.W. Lawvere,C. Maurer,G.C. Wraith Pdf

A Collection of Lectures by Variuos Authors

Author : M. Barr,C. Wells
Publisher : Springer
Page : 347 pages
File Size : 48,5 Mb
Release : 2013-06-09
Category : Mathematics
ISBN : 1489900233

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Toposes, Triples and Theories by M. Barr,C. Wells Pdf

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.

Author : P. T. Johnstone
Publisher : Oxford University Press
Page : 836 pages
File Size : 55,9 Mb
Release : 2002-09-12
Category : Computers
ISBN : 0198515987

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Sketches of an Elephant: A Topos Theory Compendium by P. T. Johnstone Pdf

Topos 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 to thereby 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.

Author : Olivia Caramello
Publisher : Oxford University Press
Page : 336 pages
File Size : 52,8 Mb
Release : 2018-01-19
Category : Philosophy
ISBN : 9780191076756

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Theories, Sites, Toposes by Olivia Caramello Pdf

According to Grothendieck, the notion of topos is "the bed or deep river where come to be married geometry and algebra, topology and arithmetic, mathematical logic and category theory, the world of the continuous and that of discontinuous or discrete structures". It is what he had "conceived of most broad to perceive with finesse, by the same language rich of geometric resonances, an "essence" which is common to situations most distant from each other, coming from one region or another of the vast universe of mathematical things". The aim of this book is to present a theory and a number of techniques which allow to give substance to Grothendieck's vision by building on the notion of classifying topos educed by categorical logicians. Mathematical theories (formalized within first-order logic) give rise to geometric objects called sites; the passage from sites to their associated toposes embodies the passage from the logical presentation of theories to their mathematical content, i.e. from syntax to semantics. The essential ambiguity given by the fact that any topos is associated in general with an infinite number of theories or different sites allows to study the relations between different theories, and hence the theories themselves, by using toposes as 'bridges' between these different presentations. The expression or calculation of invariants of toposes in terms of the theories associated with them or their sites of definition generates a great number of results and notions varying according to the different types of presentation, giving rise to a veritable mathematical morphogenesis.

Author : B. Jacobs
Publisher : Gulf Professional Publishing
Page : 784 pages
File Size : 54,6 Mb
Release : 2001-05-10
Category : Computers
ISBN : 0444508538

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Categorical Logic and Type Theory by B. Jacobs Pdf

This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.

Author : Marie La Palme Reyes,Gonzalo E. Reyes,Houman Zolfaghari
Publisher : Polimetrica s.a.s.
Page : 286 pages
File Size : 40,8 Mb
Release : 2004
Category : Mathematics
ISBN : 9788876990045

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Generic Figures and Their Glueings by Marie La Palme Reyes,Gonzalo E. Reyes,Houman Zolfaghari Pdf

Author : Saunders Mac Lane,Ieke Moerdijk
Publisher : Unknown
Page : 627 pages
File Size : 54,5 Mb
Release : 1992
Category : Algebraische Geometrie - Garbentheorie
ISBN : 3540977104

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Sheaves in Geometry and Logic by Saunders Mac Lane,Ieke Moerdijk Pdf

An introduction to the theory of toposes which begins with illustrative examples and goes on to explain the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.

Author : Jacob Lurie
Publisher : Princeton University Press
Page : 944 pages
File Size : 41,8 Mb
Release : 2009-07-06
Category : Mathematics
ISBN : 9781400830558

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Higher Topos Theory (AM-170) by Jacob Lurie Pdf

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Author : R. Goldblatt
Publisher : Elsevier
Page : 565 pages
File Size : 48,9 Mb
Release : 2014-06-28
Category : Mathematics
ISBN : 9781483299211

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Topoi by R. Goldblatt Pdf

The first of its kind, this book presents a widely accessible exposition of topos theory, aimed at the philosopher-logician as well as the mathematician. It is suitable for individual study or use in class at the graduate level (it includes 500 exercises). It begins with a fully motivated introduction to category theory itself, moving always from the particular example to the abstract concept. It then introduces the notion of elementary topos, with a wide range of examples and goes on to develop its theory in depth, and to elicit in detail its relationship to Kripke's intuitionistic semantics, models of classical set theory and the conceptual framework of sheaf theory (``localization'' of truth). Of particular interest is a Dedekind-cuts style construction of number systems in topoi, leading to a model of the intuitionistic continuum in which a ``Dedekind-real'' becomes represented as a ``continuously-variable classical real number''. The second edition contains a new chapter, entitled Logical Geometry, which introduces the reader to the theory of geometric morphisms of Grothendieck topoi, and its model-theoretic rendering by Makkai and Reyes. The aim of this chapter is to explain why Deligne's theorem about the existence of points of coherent topoi is equivalent to the classical Completeness theorem for ``geometric'' first-order formulae.

Author : M. Makkai,G.E. Reyes
Publisher : Springer
Page : 317 pages
File Size : 41,8 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540371007

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First Order Categorical Logic by M. Makkai,G.E. Reyes Pdf

Author : F. William Lawvere,Stephen H. Schanuel
Publisher : Cambridge University Press
Page : 409 pages
File Size : 48,9 Mb
Release : 2009-07-30
Category : Mathematics
ISBN : 9780521894852

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Conceptual Mathematics by F. William Lawvere,Stephen H. Schanuel Pdf

This truly elementary book on categories introduces retracts, graphs, and adjoints to students and scientists.

Author : Emily Riehl
Publisher : Courier Dover Publications
Page : 272 pages
File Size : 48,5 Mb
Release : 2017-03-09
Category : Mathematics
ISBN : 9780486820804

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Category Theory in Context by Emily Riehl Pdf

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.

Author : Benjamin C. Pierce
Publisher : MIT Press
Page : 117 pages
File Size : 48,7 Mb
Release : 1991-08-07
Category : Computers
ISBN : 9780262326452

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Basic Category Theory for Computer Scientists by Benjamin C. Pierce Pdf

Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial • Applications • Further Reading