From Objects to Diagrams for Ranges of Functors
This work introduces tools from the field of category theory that make it possible to tackle a number of representation problems that have remained unsolvable to date (e.g. the determination of the range of a given functor). The basic idea is: if a functo
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For further volumes: http://www.springer.com/series/304
2029
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Pierre Gillibert
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Friedrich Wehrung
From Objects to Diagrams for Ranges of Functors
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Pierre Gillibert Charles University in Prague Department of Mathematics Sokolovsk´a 83 18600 Prague Praha Czech Republic [email protected]
Friedrich Wehrung University of Caen, LMNO, CNRS UMR 6139, Department of Mathematics 14032 Caen, Cedex France [email protected]
ISBN 978-3-642-21773-9 e-ISBN 978-3-642-21774-6 DOI 10.1007/978-3-642-21774-6 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011933135 Mathematics Subject Classification (2010): Primary 18A30, 18A25, 18A20, 18A35; Secondary 03E05, 05D10, 06A07, 06A12 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Foreword
The aim of the present work is to introduce a general method, applicable to various fields of mathematics, that enables us to gather information on the range of a functor Φ, thus making it possible to solve previously intractable representation problems with respect to Φ. This method is especially effective in case the problems in question are “cardinality-sensitive”, that is, an analogue of the cardinality function turns out to play a crucial role in the description of the members of the range of Φ. Let us first give a few examples of such problems. The first three belong to the field of universal algebra, the fourth to the field of ring theory (nonstable K-theory of rings). Context 1. The classical Gr¨ atzer–Schmidt Theorem, in universal algebra, states that every (∨, 0)-semilattice is isomorphic to the compact congruence lattice of some algebra. Can this result be extended to diagrams of (∨, 0)semilattices? Context 2. For a member A of a quasivariety A of algebraic systems, we denote by ConA c A the (∨, 0)-semilattice of all compact elements of the lattice of all congruences of A with quotient in A; further, we denote by Conc,r A the class of all isomorphic copies of ConA c A where A ∈ A. For quasiv
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