Colocalizations and their realizations as spectra (Q2783424)

The notion of homology of spaces can be generalized via the notion of a chain functor; i.e. a pair of functors \(\{\text{pairs of spaces}\}\) to \(\{\text{chain complexes}\}\) with certain properties. Every (generalized) homology theory can be got in this way. NEWLINENEWLINENEWLINEGiven a full subcategory \(\mathcal{L}\) of the category of pairs of spaces, every chain functor \(A_*\) has a localization sequence \(_{\mathcal{L}}A_* \to A_* \to {A_{\mathcal{L}}}_*\). This is analogous to Bousfield's theorem on the existence of localization in the category of spectra with respect to another spectrum. NEWLINENEWLINENEWLINEThe main result of the present paper gives, in full generality, the existence of a colocalization sequence \(A^{\mathcal{L}}_{*} \to A_* \to ^{\mathcal{L}}A_*\) of chain functors. This is remarkable because it is known that an analogous result in the category of spectra does not hold.