Shape Theory Categorical Methods of Approximation

by Cordier, J. M.; Porter, T.

• ISBN13: 9780486466231
• ISBN10: 048646623X
• Format: Paperback
• Copyright: 2008-05-19
• Publisher: Dover Publications

Summary

This in-depth treatment uses shape theory as a "case study" to illustrate situations common to many areas of mathematics, including the use of archetypal models as a basis for systems of approximations. It offers students a unified and consolidated presentation of extensive research from category theory, shape theory, and the study of topological algebras.

Author Biography

J.-M. Cordier Universite de Picardie-Jules Verne Faculte de Mathematiques et d'Informatique Amiens, France T. Porter Emeritus Professor, School of Computer Science University of Bangor Bangor, Wales, UK

Table of Contents

Introduction	p. 7
Advice to the reader	p. 13
Borsuk's shape theory for compact metric spaces	p. 15
Absolute retracts and absolute neighbourhood retracts	p. 17
Borsuk's shape theory	p. 19
Fundamental sequences and inverse systems of ANRs	p. 22
Historical Note	p. 27
Categorical shape theory	p. 28
Shape theory	p. 28
Examples of shape theories	p. 32
Pro-objects, procategories and proadjoints	p. 36
Shape theory and procategories	p. 53
Kan extensions and Holsztynski's shape functor	p. 58
Codensity monads	p. 64
Historical Note	p. 69
Shape theory for topological spaces	p. 70
Simplicial complexes and numerable covers	p. 70
Shape for topological spaces	p. 74
Comparison of the shape theories of Borsuk, Fox, Mardesic-Segal and Mardesic	p. 77
Historical Note	p. 85
Distributors and shape theory	p. 86
The bicategory, Dist	p. 86
Codensity monads and Kleisli categories in Dist	p. 98
Shape theories and distributors	p. 102
Notes on sources	p. 111
Functors between shape theories	p. 112
Change of models	p. 112
Absolute Kan extensions	p. 114
Exact squares	p. 127
Examples and applications of exact squares	p. 133
Some topological examples of exact squares with applications to suspensions of (pointed) spaces	p. 140
Notes on sources	p. 153
Stability and movability	p. 154
The interpretation of categorical notions in a shape category	p. 154
Stability	p. 157
Reindexing lemmas	p. 162
Stability in pro-Abelian categories	p. 166
The derived functors of Lim	p. 175
Movability and Mittag-Leffler	p. 178
Strong movability	p. 189
Notes on sources	p. 196
Categorical shape theory and Pattern Recognition, a possible link	p. 198
References	p. 201
Index	p. 205
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