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A chain rule for Goodwillie derivatives of functors from spectra to spectra. (English) Zbl 1189.55003

Let Spec be the category of spectra. Let \(F, G: \text{Spec}\to\text{Spec}\) be homotopy functors. Let \(\partial_nF(X)\) be the \(n\)-th Goodwillie derivative of \(F\) at \(X\) [T. G. Goodwillie, Geom. Topol. 7, 645–711 (2003; Zbl 1067.55006)]. By construction, \(\partial_nF(X)\) is a spectrum with an action of the symmetric group \(\Sigma_n\). Thus the sequence of derivatives \(\partial_*F(X):=\{\partial_nF(X)\}_{n=1}^\infty\) can be viewed as a symmetric sequence of spectra.
This paper is concerned with the chain rule for functors from Spec to Spec. A chain rule is a formula that expresses \(\partial_*(FG)\) in terms of \(\partial_*F\) and \(\partial_*G\). The author’s main theorem states that (under a mild hypothesis on \(F\)) there is an equivalence of symmetric sequences
\[ \partial_*FG(X)\simeq \partial_*F(GX) \circ \partial_*G(X). \]
Here the symbol \(\circ\) on the right hand side denotes the composition product of symmetric sequences. More explicitly this means that for each \(n\) there is an equivalence of spectra with an action of \(\Sigma_n\)
\[ \partial_n FG(X)\simeq \bigvee_{\alpha\in \text{Sur}(n,i)/\Sigma_i} \partial_iF(GX)\wedge \partial_{n_1}G(X)\wedge \dots\wedge \partial_{n_i}G(X). \tag{1} \]
Here the wedge sum is taken over equivalence classes of surjective functions \(\alpha: n \twoheadrightarrow i\) from a set with \(n\) elements to another set. \(n_1, \dots, n_i\) denote the pre-images of elements of \(i\) under \(\alpha\).
The author’s result is in fact entirely analogous to the the classical Faà di Bruno formula for the higher derivatives of a composite function [W. P. Johnson, Am. Math. Mon. 109, No. 3, 217–234 (2002; Zbl 1024.01010)]. For example, when \(n=1\) one obtains the formula
\[ \partial_1FG(X) \simeq \partial_1F(GX) \wedge \partial_1 G(X) \]
which is clearly similar to the familiar chain rule for first derivatives. However, the proof in the case of functor calculus is not just an adaptation of the proof in the classical case. It is fairly easy to see that \(\partial_n FG(X)\) admits a filtration whose associated graded complex is equivalent to the wedge sum on right hand side of (1), but it is by no means obvious that the filtration splits. An important role in the author’s proof is played by a “co-cross-effects” construction due to R. McCarthy [Greenlees, J. P. C. (ed.) et al., Homotopy methods in algebraic topology. Proceedings of an AMS-IMS-SIAM joint summer research conference, University of Colorado, Boulder, CO, USA, June 20–24, 1999. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 271, 183–215 (2001; Zbl 0996.19005)].
The author also considers the problem of expressing the full Taylor tower of \(FG\) in terms of the Taylor towers of \(F\) and \(G\), and offers some results in this direction.
Reviewer’s remark: the paper is very well-written, but I did catch one typo: in diagram (*) near the top of page 407
\[ F(\text{hofib}(G(X\wedge Z)\to GX)\wedge GX) \]
should be
\[ F(\text{hofib}(G(X\vee Z)\to GX)\vee GX). \]

MSC:

55P42 Stable homotopy theory, spectra
55P65 Homotopy functors in algebraic topology
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References:

[1] Thomas G. Goodwillie, Calculus. II. Analytic functors, \?-Theory 5 (1991/92), no. 4, 295 – 332. · Zbl 0776.55008
[2] Thomas G. Goodwillie, Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645 – 711. · Zbl 1067.55006
[3] Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149 – 208. · Zbl 0931.55006
[4] Warren P. Johnson, The curious history of Faà di Bruno’s formula, Amer. Math. Monthly 109 (2002), no. 3, 217 – 234. · Zbl 1024.01010
[5] John R. Klein and John Rognes, A chain rule in the calculus of homotopy functors, Geom. Topol. 6 (2002), 853 – 887. · Zbl 1066.55009
[6] Nicholas J. Kuhn, Goodwillie towers and chromatic homotopy: An overview, Proceedings of Nishida Fest (Kinosaki 2003), Geometry and Topology Monographs, vol. 10, 2007, pp. 245-279. · Zbl 1105.55002
[7] Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs, vol. 96, American Mathematical Society, Providence, RI, 2002. · Zbl 1017.18001
[8] Randy McCarthy, Dual calculus for functors to spectra, Homotopy methods in algebraic topology (Boulder, CO, 1999) Contemp. Math., vol. 271, Amer. Math. Soc., Providence, RI, 2001, pp. 183 – 215. · Zbl 0996.19005
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