zbMATH — the first resource for mathematics

Calculus. II: Analytic functors. (English) Zbl 0776.55008
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 over the literature which are needed for his theory.
For the convenience of the reader many of the results are reproved. Moreover, they are extended where extensions are necessary for the development of the theory. In the final two sections the author introduces $$\rho$$-analytic functors and proves the implications of analyticity mentioned above. For applications to $$A$$-theory, topological cyclic homology, and the free loop space functor $$\Lambda$$ he shows in an appendix that the homotopy equivalence from the derivative $$\partial_ X \Omega^ \infty \Sigma^ \infty(\Lambda X_ +)$$ to $$\Lambda\Sigma^ \infty(\Omega X_ +)$$ established in “Calculus I” respects the $$S^ 1$$-action on both functors.

MSC:
 55P65 Homotopy functors in algebraic topology 19D10 Algebraic $$K$$-theory of spaces
Full Text:
References:
 [1] Barratt, M. and Whitehead, J. H. C.: The first non-vanishing group of an (n+1)-ad, Proc. London Math. Soc (3) 6 (1956), 417-439. · Zbl 0072.18002 · doi:10.1112/plms/s3-6.3.417 [2] Bökstedt, M., Carlsson, G., Cohen, R., Goodwillie, T., Hsiang, W.-c. and Madsen, I.: On the algebraic K-theory of simply-connected spaces, preprint. · Zbl 0867.19003 [3] Bousfield, D. and Kan, D.: Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics 304, Springer-Verlag, New York (1972). · Zbl 0259.55004 [4] Ellis, G. and Steiner, R.: Higher-dimensional crossed modules and the homotopy groups of (n+1)-ads, J. Pure Appl. Algebra 46 (1987), 117-136. · Zbl 0622.55010 · doi:10.1016/0022-4049(87)90089-2 [5] Goodwillie, T.: Calculus I, The first derivative of pseudoisotopy theory, K-Theory 4 (1990), 1-27. · Zbl 0741.57021 · doi:10.1007/BF00534191 [6] Goodwillie, T.: Calculus III, The Taylor series of a homotopy functor, in preparation. · Zbl 1067.55006 [7] Goodwillie, T.: A multiple disjunction lemma for smooth concordance embeddings, Mem. Amer. Math. Soc. 431 (1990), 1-317. · Zbl 0717.57011 [8] Waldhausen, F.: Algebraic K-theory of topological spaces. II, in Lecture Notes in Mathematics 763, Springer-Verlag, New York (1979), pp. 356-394. · Zbl 0431.57004 [9] Waldhausen, F.: Algebraic K-theory of spaces, in Lecture Notes in Mathematics 1126, Springer-Verlag, New York (1985), pp. 318-419. · Zbl 0579.18006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.