Theories, Sites, Toposes: Relating and Studying Mathematical Theories Through Topos-Theoretic 'Bridges'

ISBN : 9780198758914

Olivia Caramello
336 Pages
156 x 234 mm
Pub date
Feb 2017
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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.


1 Topos-theoretic background
2 Classifying toposes and the 'bridge' technique
3 A duality theorem
4 Lattices of theories
5 Flat functors and classifying toposes
6 Theories of presheaf type: general criteria
7 Expansions and faithful interpretations
8 Quotients of a theory of presheaf type
9 Examples of theories of presheaf type
10 Some applications

About the author: 

Dr Olivia Caramello is a research mathematician working as a Marie Curie Fellow at the Universite de Paris VII, also with an affiliation to IHES (Institut des Hautes Etudes Scientifiques). Her research focuses on investigating the prospective role of Grothendieck toposes as unifying spaces in Mathematics and Logic. Her main contribution has been the development of methods and techniques for transferring information between distinct mathematical theories by using toposes. She was previously a junior researcher at Centro di Recerca Matematica Ennio De Giorgi Scuola Normale Superiore (Pisa), Research Fellow at Jesus College, Cambridge and the DPMMS, Visiting Researcher at the Max Planck Institute for Mathematics (Bonn) and CARMIN Fellow at IHES.

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