Morphisms of inverse systems require meshes

Abstract

Resolutions of spaces can be viewed as special inverse systems, which behave very much like inverse systems behave in the compact case. T. Watanabe defined a category of polyhedral resolutions and showed that the limit functor defines a natural equivalence between this category and the category of topologically complete spaces. In order to develop his theory he had to consider gauged inverse systems, i.e., inverse systems whose terms are endowed with certain coverings, called meshes. This paper is devoted to the question if one can develop an analogous theory for usual (nongauged) inverse systems. An example is exhibited, which suggests a negative answer.