Categories of Boolean Sheaves of Simple Algebras
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1187
Yves Diers
Categories of Boolean Sheaves of Simple Algebras
Springer-Verlag Berlin Heidelberg New York Tokyo
Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann
1187
Yves Diers
Categories of Boolean Sheaves of Simple Algebras
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Yves Diers Departernent de Mathematiques Universite de Valenciennes et du Hainaut Cambresis 59326 Valenciennes - Cedex, France
Mathematics Subject Classification (1980): 18B99, 18F20, 18C 10, 18C20, 18020, 16A90, 16A30, 16A32, 16A74, 20E18, 20F29, 03C90, 03C05 ISBN 3-540-16459-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16459-6 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
This book studies and classifies categories of commutative regular algebras, with or without unit, equipped with structure such as order, lattice-order, differential structure, continuous group representation, with integral, algebraic, or separable elements, etc. It is the continuation and development of the study of Locally Finitely Presentable Categories done by P. Gabriel and F. Ulmer, directed mainly towards the study of categories of regular algebras. It uses the axiomatic method that deals with categories equipped with structure instead of categories given concretely by their objects and morphisms. The axiomatic method allows one to give a unified treatment of these categories, to highlight their specific features and to get a fine classification based on universal properties. It gives a construction of the localisation and globalisation processes which leads to a single proof of numerous representation theorems of regular algebras by sheaves. It is shown how well known categorical constructions can be used to get new sectional representations of algebras, algebraic closure constructions, and Galois theories. Categories of continuous representations of finite, or profinite groups, in algebraic categories, and in the category of Boolean algebras in particular, are characterized by universal properties. There is a proof that the notion of commutative regular algebra without unit is 2-equivalent to the notion of commutative unitary regular algebra. All the Sheaves of algebras which appear here being based on Boolean OY locally Boolean spaces, the passage from stalks to global sections preserves the yalidity of numerous formUlas, and leads to transfer theorems as used by logicians. In order to achieve this work, a lot of new notions have been int
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