Our object of study is the natural tower which, for any given map
f[ratio ]A→B and each space X,
the localization of X with respect to f and converges
itself. These towers can be used to produce
approximations to localization with respect to any generalized homology
E∗, yielding, for example,
an analogue of Quillen's plus-construction for E∗.
We discuss in detail the case of ordinary homology with
coefficients in ℤ/p or ℤ[1/p].
Our main tool is a comparison theorem for nullification functors (that
localizations with respect to maps of the form f[ratio ]A→pt),
which allows us, among other things, to generalize
Neisendorfer's observation that p-completion of
simply-connected spaces coincides with nullification with
respect to a Moore space M(ℤ[1/p],