The self-contained theory of certain singular coverings of toposes called complete spreads, that is presented in this volume, is a field of interest to topologists working in knot theory, as well as to various categorists. It extends the complete spreads in topology due to R. H. Fox (1957) but, unlike the classical theory, it emphasizes an unexpected connection with topos distributions in the sense of F. W. Lawvere (1983). The constructions, though often motivated by classical theories, are sometimes quite different from them. Special classes of distributions and of complete spreads, inspired respectively by functional analysis and topology, are studied. Among the former are the probability distributions; the branched coverings are singled out amongst the latter.

This volume may also be used as a textbook for an advanced one-year graduate course introducing topos theory with an emphasis on geometric applications. Throughout the authors emphasize open problems. Several routine proofs are left as exercises, but also as 'exercises' the reader will find open questions for possible future work in a variety of topics in mathematics that can profit from a categorical approach.

Autorentext

Marta Bunge was born in Argentina where she studied philosophy and mathematics. She did her graduate work in mathematics at the University of Pennsylvania where she obtained her Ph. D. degree in 1966 under the supervision of Peter Freyd and F. William Lawvere. She has worked at McGill University since 1966, where she is currently Professor Emerita. She has been a visitor at mathematics institutes in Aarhus, Zurich, Mexico, Geneva, Mallorca, Genoa, and Sydney.

Jonathon Funk was born in Saskatchewan where he studied mathematics before coming to McGill University, where in 1991 he completed a Ph. D. under the supervision of Marta Bunge. Since then he has worked in Newfoundland, Cyprus, British Columbia, and Regina. Presently, he is Lecturer at the University of the West Indies in Barbados. He has been a visitor to McGill University on several occasions.

Klappentext

This volume presents a self-contained theory of certain singular coverings of toposes, including branched coverings. This book is distinguished from classical treatments of the subject by its unexpected connection with a topic from functional analysis, namely, distributions. Although primarily aimed at topos theorists, this book may also be used as a textbook for advanced graduate courses introducing topos theory with an emphasis on geometric applications.

Inhalt

Distributions and Complete Spreads.- Lawvere Distributions on Toposes.- Complete Spread Maps of Toposes.- The Spread and Completeness Conditions.- An Axiomatic Theory of Complete Spreads.- Completion KZ-Monads.- Complete Spreads as Discrete M-fibrations.- Closed and Linear KZ-Monads.- Aspects of Distributions and Complete Spreads.- Lattice-Theoretic Aspects.- Localic and Algebraic Aspects.- Topological Aspects.