Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F , often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor V Þ Ñ F pS V q , where S V is the one-point compactification of V . In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO . We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.
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.Comment: Reformatted for publication (this is the mockup), 23 pages, small
typos and formatting corrected as wel
Let M f n be the localization of the ∞-category of spaces at the vnperiodic equivalences, the case n = 0 being rational homotopy theory. We prove that M f n is for n ≥ 1 equivalent to algebras over a certain monad on the ∞-category of T (n)-local spectra. This monad is built from the Bousfield-Kuhn functor.
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