Calculus. II: Analytic functors
type: Item
instance of: scholarly article
title: Calculus. II: Analytic functors
zbMATH Open document ID: 0776.55008
DOI: 10.1007/BF00535644
author: T. G. Goodwillie
published in: \(K\)-Theory
publication date: 1992-08-03T00:00:00Z
review text: The paper is a sequel to ``Calculus I'' (Zbl 0741.57021), where derivatives of homotopy functors were introduced. Its contents is the development of the theory of \(\rho\)-analytic functors, i.e. homotopy functors \(F\) from spaces to (based) spaces or spectra satisfying the following property: Given an \(n\)-cube diagram \(\mathcal X\) of spaces, which is an iterated homotopy pushout, and suppose the \(n\) maps from the initial vertex are all \((\rho+1)\)-connected, then there is an integer \(q\) independent of \(n\) such that \(F({\mathcal X})\) behaves like an \(n\)- dimensional homotopy pullback diagram up to the dimension \((n-1)\rho-q\). A number of results are proved which imply the equivalence of \(\rho\)- analytic functors under additional connectivity assumptions once the equivalence of their differentials or derivatives is established. To give a flavour of the main results we quote the ``First Derivative Criterion'': Let \(F\to G\) be a natural transformation of \(\rho\)-analytic functors from spaces to spectra which induces homotopy equivalences of the differentials \(D_ X F(Y)\to D_ X G(Y)\) for all \(X\), \(Y\). Then for any \((\rho+1)\)-connected map \(Y\to X\) the following diagram is a homotopy pullback \[ \begin{matrix} F(Y) & \longrightarrow & G(Y)\\ \downarrow && \downarrow\\ F(X) & \longrightarrow & G(X).\end{matrix} \] As examples may serve Waldhausen's \(A\)-theory functor, which is 1-analytic, or the functor \(X\mapsto\) \(\Omega^ \infty \Sigma^ \infty({\mathcal T}\text{ op}(K,X)_ +)\) with a finite CW-complex \(K\), which is \(\dim K\)-analytic. A sketch of applications can be found in [the author, Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 621-630 (1991; Zbl 0759.55011)]. The theory of analytic functors relies heavily on the homotopy theory of cubical diagrams. In the first three sections of the paper the author collects those results about cubes, \(n\)-dimensional homotopy pushouts and pullbacks and their homotopy theory spread ove…
Mathematics Subject Classification ID: 55P65
Mathematics Subject Classification ID: 19D10
zbMATH DE Number: 39905
zbMATH Keywords: algebraic \(K\)-theory of spaces
zbMATH Keywords: analytic functors
zbMATH Keywords: \(n\)-cube diagram
zbMATH Keywords: topological cyclic homology
zbMATH Keywords: free loop space functor
reviewed by: Rainer Vogt
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00535644
OpenAlex ID: W2009989341
description: scientific article; zbMATH DE number 39905