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Shape theory: categorical methods of approximation. (English) Zbl 0663.18001

Mathematics and Its Applications. Chichester: Ellis Horwood Limited; New York etc.: Halsted Press. 207 p. £29.95 (1989).
Geometric shape theory as developed from 1968 onwards by Borsuk, Mardešić, Segal and others exemplifies a process that recurs many times in mathematics. One has a class (or category), \(A\), of nice objects (models) and a larger class (or category), \(B\), of objects that one wishes to study together with some means (usually a functor, \(K\), from \(A\) to \(B)\) of picking out the models within the larger category. The process consists of approximating each object \(X\) of \(B\) by a system of models. In the Borsuk theory, \(B\) is the homotopy category of compact metric spaces whilst \(A\) is the full subcategory of those spaces having the homotopy type of a polyhedron. The algebraic topology of polyhedra is relatively well known and one tries in shape theory to extend results from \(A\) to \(B\). Similar ideas are used in the theory of topological groups and, in a dual form, in Galois theory.
This monograph explores the categorical theory of these approximation situations. Starting with a functor \(K: A\to B\) and an object \(X\) of \(B\) one forms the comma category \(X\downarrow K\) and one uses this as a categorical model of \(X\). Using functors between such comma categories that are compatible with the natural codomain functors to \(A\) yields the shape category, \(S_K\). The study of \(S_K\) and of the interaction between this and other methods (Kan extensions, procategories, distributors, exact squares, etc.) form the bulk of the material considered. The geometric form of shape is introduced and is used as a case study for the general theory, illustrating the ideas and providing a ready ground for application of the abstract theory. There is some potential for application of these ideas in pattern recognition theory and an appendix is included that aims to interpret the categorical terms in that context.
The chapter headings are as follows: I. Borsuk’s shape theory for compact metric spaces, II. Categorical shape theory, III. Shape theory for topological spaces, IV. Distributors ad shape theory, V. Functors between shape theories, VI. Stability and movability, Appendix.

MSC:

18-02 Research exposition (monographs, survey articles) pertaining to category theory
18A25 Functor categories, comma categories
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
68T10 Pattern recognition, speech recognition