Relations between different
kinds of homology theories on the category Com (of compacta) resp. the related
strong shape category Com are studied. In particular homology theories satisfying a
clusteraxiom (as for example the strong shape homology E∗ with coefficients in a
spectrum E, for a restricted class of spectra being defined on the category of finite
dimensional compacts) allow interesting characterizations. As an application
this provides new proofs of classical theorems concerning Steenrod-Sitnikov
homology theories, including a result on the Brown-Douglas-Fillmore homology
𝜖∗.
