  We call attention to the intermediate constructions $\T_n F$ in Goodwillie's
Calculus of homotopy functors, giving a new model which naturally gives rise to
a family of towers filtering the Taylor Tower of a functor. We also establish a
surprising equivalence between the homotopy inverse limits of these towers and
the homotopy inverse limits of certain cosimplicial resolutions. This
equivalence gives a greatly simplified construction for the homotopy inverse
limit of the Taylor tower of a functor $F$ under general assumptions.
