Topos Theory
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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.
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.
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.
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.
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.
Author | : Guerino Mazzola |
Publisher | : Birkhäuser |
Total Pages | : 1310 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 303488141X |
With contributions by numerous experts
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.
Author | : F.W. Lawvere |
Publisher | : Springer |
Total Pages | : 352 |
Release | : 2006-11-15 |
Genre | : Mathematics |
ISBN | : 3540374957 |
A Collection of Lectures by Variuos Authors
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.
Author | : Cecilia Flori |
Publisher | : Springer |
Total Pages | : 452 |
Release | : 2013-03-27 |
Genre | : Science |
ISBN | : 364235713X |
In the last five decades various attempts to formulate theories of quantum gravity have been made, but none has fully succeeded in becoming the quantum theory of gravity. One possible explanation for this failure might be the unresolved fundamental issues in quantum theory as it stands now. Indeed, most approaches to quantum gravity adopt standard quantum theory as their starting point, with the hope that the theory’s unresolved issues will get solved along the way. However, these fundamental issues may need to be solved before attempting to define a quantum theory of gravity. The present text adopts this point of view, addressing the following basic questions: What are the main conceptual issues in quantum theory? How can these issues be solved within a new theoretical framework of quantum theory? A possible way to overcome critical issues in present-day quantum physics – such as a priori assumptions about space and time that are not compatible with a theory of quantum gravity, and the impossibility of talking about systems without reference to an external observer – is through a reformulation of quantum theory in terms of a different mathematical framework called topos theory. This course-tested primer sets out to explain to graduate students and newcomers to the field alike, the reasons for choosing topos theory to resolve the above-mentioned issues and how it brings quantum physics back to looking more like a “neo-realist” classical physics theory again.