We construct a Goodwillie tower of categories which interpolates
between the category of pointed spaces and the category of spectra.
This tower of categories refines the Goodwillie tower of the
identity functor in a precise sense. More gen-erally, we construct
such a tower for a large class of ?-categories C and classify such
Goodwillie towers in terms of the derivatives of the identity
functor of C.Asa particular application we show how this provides a
model for the homotopy theory of simply-connected spaces in terms
of coalgebras in spectra with Tate diagonals. Our classification of
Goodwillie towers simplifies considerably in settings where the
Tate cohomology of the symmetric groups vanishes. As an example we
apply our methods to rational homotopy theory. Another application
identifies the homotopy theory of p-local spaces with homotopy
groups in a certain finite range with the homotopy theory of
certain algebras over Ching's spectral version of the Lie operad.
This is a close analogue of Quillen's results on rational homotopy.
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