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A. N. Dranishnikov introduced the notion of generalized cohomological dimension of compact metric spaces with respect to CW spectra. In this paper, taking an inverse system approach, we generalize this definition and obtain two types of generalized comological dimension with respect to general topological spaces, which are objects in the stable shape category. We characterize those two types of generalized cohomological dimension in terms of maps and obtain their fundamental properties. In particular, we obtain their relations to the integral cohomological dimension and the covering dimension. Moreover, we study the generalized cohomological dimensions of compact Hausdorff spaces with respect to the Kahn continuum and the Hawaiian earing.

A shape theoretic approach to generalized cohomological dimension with respect to topological spaces