The Boardman category of spectra, chain complexes and (co)-localizations
A chain functor associates a pair \((X,A)\) of topological spaces to chain complexes \(C(X,A)\) and \(C'(X,A)\). In addition, we require that the complexes \(C(A), C(X), C(X,A)\) and \(C'(X,A)\) share a certain relationship, which ensures that \((X,A)\mapsto H_*C(X,A)\) is a homology theory on the homotopy category of pairs of topological spaces. Singular homology is a special case (with \(C=C'\)). However, the more intricate structural data of ``chain functor'' are designed so that compactly supported homology theories on pairs of CW-spaces defined via chain functors correspond bijectively to such theories defined via spectra (in the sense of Boardman). It follows that, whenever a chain complex construction on chain functors yields again a chain functor, then there is a corresponding construction on spectra. The present article is a survey of the author's recent work in this direction. For example, the tensor product of chain functors is again a chain functor, and so there is a corresponding operation on spectra (which is not the smash product). Notably, this leads to a different view of (co-)localization in the homotopy category of spaces.
scientific article; zbMATH DE number 1295632