Elementary Categories Elementary Toposes PDF Download
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Author | : Colin McLarty |
Publisher | : Clarendon Press |
Total Pages | : 282 |
Release | : 1992-06-04 |
Genre | : |
ISBN | : 0191589497 |
Download Elementary Categories, Elementary Toposes Book in PDF, ePub and Kindle
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 |
Total Pages | : 401 |
Release | : 2014-01-15 |
Genre | : Mathematics |
ISBN | : 0486493369 |
Download Topos Theory Book in PDF, ePub and Kindle
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 |
Publisher | : Springer |
Total Pages | : 352 |
Release | : 2006-11-15 |
Genre | : Mathematics |
ISBN | : 3540374957 |
Download Model Theory and Topoi Book in PDF, ePub and Kindle
A Collection of Lectures by Variuos Authors
Author | : P. T. Johnstone |
Publisher | : Oxford University Press |
Total Pages | : 836 |
Release | : 2002-09-12 |
Genre | : Computers |
ISBN | : 9780198515982 |
Download Sketches of an Elephant: A Topos Theory Compendium Book in PDF, ePub and Kindle
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 | : R. Goldblatt |
Publisher | : Elsevier |
Total Pages | : 569 |
Release | : 2014-06-28 |
Genre | : Mathematics |
ISBN | : 148329921X |
Download Topoi Book in PDF, ePub and Kindle
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 | : F. William Lawvere |
Publisher | : Cambridge University Press |
Total Pages | : 409 |
Release | : 2009-07-30 |
Genre | : Mathematics |
ISBN | : 0521894859 |
Download Conceptual Mathematics Book in PDF, ePub and Kindle
This truly elementary book on categories introduces retracts, graphs, and adjoints to students and scientists.
Author | : Saunders Mac Lane |
Publisher | : |
Total Pages | : 627 |
Release | : 1992 |
Genre | : Algebraische Geometrie - Garbentheorie |
ISBN | : 9783540977100 |
Download Sheaves in Geometry and Logic Book in PDF, ePub and Kindle
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 | : M. Barr |
Publisher | : Springer |
Total Pages | : 347 |
Release | : 2013-06-09 |
Genre | : Mathematics |
ISBN | : 9781489900234 |
Download Toposes, Triples and Theories Book in PDF, ePub and Kindle
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 | : B. Jacobs |
Publisher | : Gulf Professional Publishing |
Total Pages | : 784 |
Release | : 2001-05-10 |
Genre | : Computers |
ISBN | : 9780444508539 |
Download Categorical Logic and Type Theory Book in PDF, ePub and Kindle
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 | : Jacob Lurie |
Publisher | : Princeton University Press |
Total Pages | : 944 |
Release | : 2009-07-26 |
Genre | : Mathematics |
ISBN | : 0691140480 |
Download Higher Topos Theory Book in PDF, ePub and Kindle
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.