Shape Theory

Categorical Methods of Approximation

J. M. Cordier, T. Porter, T. Porter

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486466231
• eBook ISBN: 9780486783475
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

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.
A short introduction to geometric shape explains specifics of the construction of the shape category and relates it to an abstract definition of shape theory. Upon returning to the geometric base, the text considers simplical complexes and numerable covers, in addition to Morita's form of shape theory. Subsequent chapters explore Bénabou's theory of distributors, the theory of exact squares, Kan extensions, the notion of a stable object, and stability in an Abelian context. The text concludes with a brief description of derived functors of the limit functor theory—the concept that leads to movability and strong movability of systems—and illustrations of the equivalence of strong movability and stability in many contexts.

1

Borsuk’s Shape Theory for Compact Metric Spaces

Before we start on a detailed introduction to Borsuk’s shape theory, it will pay to give a brief description of what it aimed to do, why, and how, thus indicating the main ideas to be developed later on in this chapter. We cannot hope to give here more than an idea of the overall theory, but this should not matter as the main point of this chapter is to lay down some of the foundations for our use of the geometric side of shape theory as a ‘case-study’ in the general ideas behind the use of approximations, and in the general philosophy of shape theory. It should also be pointed out that there are several good introductory articles on geometric shape theory plus at least three books giving fully detailed accounts (namely Borsuk [13], Dydak and Segal [32] and Mardešić and Segal [78]).

In 1978, Borsuk introduced the theory of shape in an article [10]. This aimed to give a classification of compact metric spaces that was coarser than homotopy but which would coincide with homotopy theory on spaces that had reasonable local properties, namely the ANRs (absolute neighbourhood retracts—see section 1.1). These latter spaces have some of the same homotopy theoretic properties as polyhedra, and one can usually replace general ANRs with polyhedra in their application within shape theory.

Shape is often stated to be a sort of Čech homotopy theory. Using the results of Alexandroff, Čech had managed to extend homological and cohomological inv...

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• Introduction
• Advice to the reader
• 1. Borsuk’s shape theory for compact metric spaces
• 2. Categorical shape theory
• 3. Shape theory for topological spaces
• 4. Distributors and shape theory
• 5. Functors between shape theories
• 6. Stability and movability
• Appendix: Categorical shape theory and Pattern Recognition, a possible link
• References
• Index