An approximate inverse system approach to shape fibrations

Abstract

The notion of shape fibration between compact met- ric spaces was introduced by S. Mardeysic and T. B. Rushing. Mardeysic extended the notion to arbitrary topological spaces. A shape fibration f : X ! Y between topological spaces is defined by using the notion of resolution (p; q; f) of the map f, where p : X ! X and q : Y ! Y are polyhedral resolutions of X and Y , respectively, and the approximate homotopy lifting property for the system map f : X ! Y. Although any map f : X ! Y between topological spaces admits a resolution (p; q; f), if polyhedral resolutions p : X ! X and q : Y ! Y are chosen in ad- vance, there may not exist a system map f : X ! Y so that (p; q; f) is a resolution of f. To overcome this deficiency, T. Watanabe introduced the notion of approximate resolution. An approximate resolution of a map f : X ! Y consists of approximate polyhedral resolutions p : X ! X and q : Y ! Y of X and Y , respectively, and an approximate map f : X ! Y. In this paper we obtain the approximate homotopy lifting property for ap- proximate maps and investigate its properties. Moreover, it is shown that the approximate homotopy lifting property is extended to the approximate pro-category and the approximate shape category in the sense of Watan- abe. It is also shown that the approximate pro-category together with fibrations defined as morphisms having the approximate homotopy lifting property with respect to arbitrary spaces and weak equivalences defined as morphisms inducing isomorphisms in the pro-homotopy category satisfies the composition axiom for a fibration category in the sense of H. J. Baues. As an application it is shown that shape fibrations can be defined in terms of our approximate homotopy lifting property for approximate maps and that every homeomorphism is a shape fibration.