Topos Theory

Topos Theory
Author: P.T. Johnstone
Publisher: Courier Corporation
Total Pages: 401
Release: 2014-01-15
Genre: Mathematics
ISBN: 0486493369

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.

Higher Topos Theory

Higher Topos Theory
Author: Jacob Lurie
Publisher: Princeton University Press
Total Pages: 944
Release: 2009-07-26
Genre: Mathematics
ISBN: 0691140480

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.

Toposes and Local Set Theories

Toposes and Local Set Theories
Author: John L. Bell
Publisher: Courier Corporation
Total Pages: 290
Release: 2008-01-01
Genre: Mathematics
ISBN: 0486462862

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.

Sheaves in Geometry and Logic

Sheaves in Geometry and Logic
Author: Saunders Mac Lane
Publisher:
Total Pages: 627
Release: 1992
Genre: Algebraische Geometrie - Garbentheorie
ISBN: 9783540977100

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.

Topoi

Topoi
Author: R. Goldblatt
Publisher: Elsevier
Total Pages: 569
Release: 2014-06-28
Genre: Mathematics
ISBN: 148329921X

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.

The Topos of Music

The Topos of Music
Author: Guerino Mazzola
Publisher: Birkhäuser
Total Pages: 1310
Release: 2012-12-06
Genre: Mathematics
ISBN: 303488141X

With contributions by numerous experts

Model Theory and Topoi

Model Theory and Topoi
Author: F.W. Lawvere
Publisher: Springer
Total Pages: 352
Release: 2006-11-15
Genre: Mathematics
ISBN: 3540374957

A Collection of Lectures by Variuos Authors

Sketches of an Elephant: A Topos Theory Compendium

Sketches of an Elephant: A Topos Theory Compendium
Author: P. T. Johnstone
Publisher: Oxford University Press
Total Pages: 836
Release: 2002-09-12
Genre: Computers
ISBN: 9780198515982

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.

Basic Category Theory

Basic Category Theory
Author: Tom Leinster
Publisher: Cambridge University Press
Total Pages: 193
Release: 2014-07-24
Genre: Mathematics
ISBN: 1107044243

A short introduction ideal for students learning category theory for the first time.

Theories, Sites, Toposes

Theories, Sites, Toposes
Author: Olivia Caramello
Publisher: Oxford University Press
Total Pages: 381
Release: 2018
Genre: Mathematics
ISBN: 019875891X

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.