## The Boardman category of spectra, chain complexes and (co)-localizations.(English)Zbl 0921.55007

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.

### MSC:

 55P42 Stable homotopy theory, spectra 55P60 Localization and completion in homotopy theory 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 55U15 Chain complexes in algebraic topology 55N07 Steenrod-Sitnikov homologies
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