×

Operads and chain rules for the calculus of functors. (English) Zbl 1239.55004

Astérisque 338. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-308-9/pbk). 158 p. (2011).
The authors study the structure of the sequence of Goodwillie derivatives based at the trivial object \(\ast\) of a reduced homotopy functor \(F:\mathcal{C}\to\mathcal{D}\), where \(\mathcal{C}\) and \(\mathcal{D}\) are either \({ sSets}_\ast\), the category of based simplicial sets, or \({ Spec}\), the category of Spectra. Goodwillie constructed for such a functor a tower of functors and natural transformations \[ F\to \dots\to P_nF\to P_{n-1}F\to\dots\to P_0F \] approximating \(F\), and a sequence \(\partial_\ast F=\{\partial_1 F,\partial_2 F,\dots\}\) of spectra, called the derivatives of \(F\). For each \(n\) the spectrum \(\partial_nF\) has a natural \(\Sigma_n\)-action so that \(\partial_\ast F\) is a symmetric sequence, and Goodwillie showed that the layer \(D_nF={ hofib}(P_nF\to P_{n-1}F)\) of the tower is determined by \(\partial_nF\). E.g. if \(\mathcal{D}={ Spec}\), then \[ D_nF(X)\simeq (\partial_nF\wedge X^{\wedge n})_{h\Sigma_n}. \] In an earlier paper, the second author has proved that the symmetric sequence \(\partial_\ast(I_{{ sSets}_\ast})\) of the identity functor \({ sSets}_\ast\to{ sSets}_\ast\) is an operad in \({ Spec}\). The first main result of the present paper is that \(\partial_\ast F\) has the structure of a right \(\partial_\ast\) \((I_{{ sSets}_\ast})\)-module if \(\mathcal{C}={ sSets}_\ast\), of a left \(\partial_\ast(I_{{ sSets}_n})\)-module if \(\mathcal{D}={ sSets}_\ast\), and of a \(\partial_\ast(I_{{ sSets}_n})\)-bimodule if \(\mathcal{C}=\mathcal{D}={ sSets}_\ast\). The derivatives of the identity functor \({ Spec}\to{ Spec}\) also form an operad which is equivalent to the initial operad. The authors use these structures to prove a chain rule for higher Goodwillie derivatives: Given reduced homotopy functors \(G:\mathcal{C}{\rightarrow}\mathcal{D}\) and \(F:\mathcal{D}{\rightarrow}\mathcal{E}\), where \(\mathcal{C}, \mathcal{D}, \mathcal{E}\) are either \({ sSets}_\ast\) or \({ Spec}\), and suppose that \(F\) preserves filtered homotopy colimits, then there is a natural equivalence of \((\partial_\ast I_{\mathcal{E}},\partial_\ast I_{\mathcal{C}})\)-bimodules \[ \partial_\ast(FG)\simeq\partial_\ast F\circ_{\partial_\ast I_\mathcal{D}}\partial_\ast G \] where the right-hand side is a derived composition product “over” \(\partial_\ast I_\mathcal{D}\). This result extends the work of Klein and Rognes to higher derivatives in the special case of reduced functors.
While this is the main result of the paper the authors also use the structures to attempt to reconstruct the Goodwillie tower from the derivatives: they construct a second tower \[ F\to\dots\to\Phi_nF \to\Phi_{n-1}F\to\dots\to \Phi_0F \] with layers \(\Delta_nF\) and a map of towers \(\varphi:P_\ast F\to\Phi_\ast F\), which is degreewise a weak equivalence if the induced map \(D_\ast F\to\Delta_\ast F\) is degreewise a weak equivalence. These maps fit into fibre sequences \[ D_nF\to\Delta_nF\to{ Tate}_{\Sigma_n} (\partial_n F\wedge(\Sigma^\infty X)^{\wedge n}) \] so that the Tate spectra measure the difference of these two towers.
The main ideas of the proofs are the following: First consider functors \(F:{ Spec}\to{ Spec}\) and define \(\partial^n(F)={ Nat}(FX,X^{\wedge n})\) where \({ Nat}(F,G)\) is the spectrum of natural transformations \(F\to G\). The symmetric sequence \(\partial^\ast F\) behaves nicely with respect to composition. There are weak equivalences \[ \partial^\ast F\circ\partial^\ast G\to\partial^\ast(FG) \quad \quad 1\to\partial^\ast I_{{ Spec}}, \] which are associative and unital. Here \(\circ\) is the composition product of symmetric sequences. If \(F\) is cofibrant in the category of functors then \(\partial^n F\) is naturally equivalent to the Spanier-Whitehead dual of \(\partial_nF\). So if \(F\) has homotopy finite derivatives we can deduce the chain rule. For more general \(F\) one has to detour through appropriate pro-spectra.
To extend the results for functors \(F:{ Spec}\to{ Spec}\) to functors \(\mathcal{C}\to\mathcal{D}\) where \(\mathcal{C}\) and \(\mathcal{D}\) are either \({ sSets}_\ast\) or \({ Spec}\) the authors work with the adjunction \[ \Sigma^\infty:{ sSets}_\ast\leftrightarrows{ Spec}:\Omega^\infty. \] The paper is well-organized and carefully written. In the first two sections the authors collect the basics: details about the categories \({ sSets}_\ast\) and \({ Spec}\), the Goodwillie tower and Goodwillie’s derivatives, functor categories and pro-spectra. They recall the category of symmetric sequences, the model categories of operads and modules over an operad, pro-symmetric sequences and Spanier-Whitehead duality. In Sections 3 and 4 they study derivatives and their chain rule of functors \({ Spec}\to{ Spec}\) and the general case \(\mathcal{C}\to\mathcal{D}\) respectively.

MSC:

55P65 Homotopy functors in algebraic topology
55P48 Loop space machines and operads in algebraic topology
55P42 Stable homotopy theory, spectra
18D50 Operads (MSC2010)
PDFBibTeX XMLCite
Full Text: arXiv Link