This in-depth treatment uses shape theory as a qcase studyq 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 BAcnabou'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 theoryathe concept that leads to movability and strong movability of systemsaand illustrations of the equivalence of strong movability and stability in many contexts.Proposition 2 If [F = (K, L, Id, F, Id) is exact, then IFK: S, -+ SL is an isomorphism of categories with inverse K*. Proof Let u: C( Y, Fa) aagt; C(X, Fa) represent a morphism in SF; then by definition K*(F, , (u))(A) = lF, .(a)(KA), but the defining diagramanbsp;...
|Author||:||J. M. Cordier, Timothy Porter|
|Publisher||:||Courier Corporation - 2008|