A CHAIN RULE FOR GOODWILLIE DERIVATIVES OF FUNCTORS ...
A CHAIN RULE FOR GOODWILLIE DERIVATIVES OF FUNCTORS FROM SPECTRA TO SPECTRA
arXiv:0710.5567v2 [math.AT] 31 Oct 2007
MICHAEL CHING
Abstract. We prove a chain rule in the Goodwillie calculus of functors from spectra to spectra. We show that the derivatives of a composite F G at a base object X are given by taking the composition product of the derivatives of F at G(X) with the derivatives of G at X. Our proof also allows us to say something about the Taylor tower of F G (and not just its layers) in terms of the Taylor towers of F and G.
Introduction In this paper we prove a version of the chain rule in the Goodwillie calculus of functors from spectra to spectra. Let Spec be a model for the stable homotopy category and let F : Spec → Spec be a homotopy functor (i.e. F preserves stable weak equivalences). Fix a base object X ∈ Spec and let Spec/X denote the category of spectra over X. Then the methods of Goodwillie [3] can be used to construct a Taylor tower for F analogous to the Taylor series of a function of a real variable, based at the object X. This tower is a sequence of functors Spec/X → Spec: X F (Y ) → · · · → PnX F (Y ) → Pn−1 F (Y ) → · · · → P0X F (Y ) = F (X)
that, for each map Y → X, interpolates between F (Y ) and F (X). The layers of the Taylor tower are the homotopy fibres X DnX F := hofib(PnX F → Pn−1 F)
and they can in general be written in the form DnX F (Y ) ≃ (∂n F (X) ∧ hofib(Y → X)∧n )hΣn . This formula holds when Y is a finite cell object in Spec and for all Y → X if F preserves filtered homotopy colimits. The object ∂n F (X) is a spectrum with an action of the symmetric group Σn and we refer to it as the nth derivative of F at X. The derivatives of F at X together form a symmetric sequence in Spec, that is a sequence of objects with actions of the symmetric groups. We write ∂∗ F (X) for this symmetric sequence. Also recall that there is a monoidal product on the category of symmetric sequences, called the composition product which we write ◦. (See Definition 1.11 below.) The main result of this paper is the following. 1
2
MICHAEL CHING
Theorem 1.13. Let F, G : Spec → Spec be homotopy functors and suppose that F preserves filtered homotopy colimits. Then we have the following formula: ∂∗ (F G)(X) ≃ ∂∗ F (GX) ◦ ∂∗ G(X). We call this a ‘chain rule’ since it expresses the derivatives of a composite of functors in terms of the derivatives of the individual functors. It bears a striking similarity to the corresponding result for functions of real-variables [6]. The derivatives of a functor determine the layers in the Taylor tower, but there are still extension problems in recovering the whole tower. We therefore consider also the problem of expressing PnX (F G)(Y ) in terms of the Taylor towers of F and G. Our method gives an answer to this question in the case that the map Y → X has a section, but there is no nice formula and it is not clear that the answer can be used in practice to calculate Taylor towers. We however give explicit descriptions of P2 (F G) and P3 (F G) where the calculations are not so hard. We should remark that our proof is non-constructive in the sense that we do not define specific models for the derivatives of a functor that satisfy the equivalence of Theorem 1.13. This has the downside that we cannot use our proof to obtain explicit operad or cooperad structures on the derivatives of monads and comonads. In separate work between the author and Greg Arone, an alternative proof of the Theorem is being developed that does give specific models for the derivatives and produces interesting operad and cooperad structures on the derivatives of certain functors. The current paper, on the other hand, has the advantage that its methods reveal something about a chain rule for the Taylor towers and not just for the derivatives. Finally, we remark that while we write out the proof for functors of spectra, most of the paper applies to functors between any stable model categories. Outline of the paper. In §1, we define what we mean by the derivatives of a functor of spectra at a general base object and state our main result in terms of the composition product of symmetric sequences. We also show that the general result follows from the special case where the base object is ∗ and the functors involved are reduced. In §2 we describe a map that on nth derivatives induces the equivalence of the theorem and show that the derivatives involved are correct. In §3 we prove the theorem in the case that F is homogeneous, obtaining along the way a simple formula for the full Taylor tower of F G under this condition. Then in §4 we complete the proof of Theorem 1.13 in general. In §5, we see how our results give us information about the full Taylor tower of F G and give explicit descriptions of the lowest few terms in this tower. Finally §6 contains a useful calculus lemma. Acknowledgements. This work owes a considerable debt to conversations between the author and Greg Arone, and is part of a joint project to prove chain rules for derivatives in a much wider context. More specifically, the underlying idea of this paper, that we can use the diagonal maps and co-cross-effects of McCarthy to study the chain rule for spectra, was suggested by Arone.
CHAIN RULE FOR FUNCTORS OF SPECTRA
3
1. Preliminaries Definition 1.1. Let Spec be a symmetric monoidal proper simplicial cofibrantly generated model category whose homotopy category is equivalent to the usual stable homotopy category. We shall refer to the objects of Spec as spectra. Any of the standard examples (such as EKMM S-modules [1], symmetric spectra [4] or orthogonal spectra [8]) can be used. Remark 1.2. We only use the symmetric monoidal structure on Spec, and the fact that it has the ‘usual’ stable homotopy category to express our chain rule in terms of derivatives. The construction of the map ∆ (Definition 2.4 below) and all the results of §§3-5 require only that Spec be a (proper, simplicial) stable model category (that is, the suspension functor is an equivalence on the homotopy category). Convention 1.3. From now on we assume any functor F : Spec → Spec that we use is simplicial (that is, F respects the simplicial enrichment in Spec) and a homotopy functor (that is, F preserves weak equivalences). Definition 1.4. (Taylor tower for functors from spectra to spectra) Goodwillie explicitly describes calculus for functors of topological spaces, but his ideas extend easily to a more general setting, for example, see Kuhn [7]. In particular, a functor F satisfying the conditions of Convention 1.3 has, for each X ∈ Spec, a Taylor tower at X, that is a sequence of functors X F → · · · → PnX F → Pn−1 F → . . . P0X F = F (X)
in which PnX F is an n-excisive functor Spec/X → Spec, that is takes strongly homotopy cocartesian (n + 1)-cubes to homotopy cartesian cubes. The nth layer in the tower is the homotopy fibre X DnX F := hofib(PnX F → Pn−1 F ). and this is n-homogeneous. We refer to [3, §1] for the precise definitions and properties of the Pn and Dn constructions. When the base object X is the trivial spectrum ∗, we write simply Pn F and Dn F . These are functors Spec → Spec. Remark 1.5. We now want to define the derivatives of a functor from spectra to spectra at an arbitrary base object. This case is not explicitly covered in [3] so we give a careful description. First notice that derivatives at the trivial spectrum ∗ work in the same way as for functors of spaces. Specifically, given F : Spec → Spec, there is a spectrum ∂n F with Σn -action such that Dn F (Y ) ≃ (∂n F ∧ Y ∧n )hΣn for finite cell spectra Y and for all Y ∈ Spec if F is finitary (i.e. preserves filtered homotopy colimits). The spectra ∂n F are the derivatives of F at ∗. To be precise we can define ∂n F by ∂n F := crn (Dn F )(S, . . . , S) where crn is the nth cross-effect, and S denotes the sphere spectrum. Definition 1.6. (Derivatives) Let F : Spec → Spec be a homotopy functor. The derivatives of F at X are the derivatives at ∗ of the functor FX (Z) = hofib(F (Z ∨ X) → F (X))
4
MICHAEL CHING
that is, ∂n F (X) := ∂n (FX ). This should be compared with Goodwillie’s definition of the derivatives of a functor of spaces at a general base object [3, 5.6]. Proposition 1.7. Let F : Spec → Spec be a homotopy functor and let Y → X be a map of spectra. Then DnX F (Y ) ≃ (∂n F (X) ∧ hofib(Y → X)∧n )hΣn . This holds whenever hofib(Y → X) is equivalent to a finite cell spectrum, and for all Y → X if F is finitary. The following lemma will be useful in proving this proposition: Lemma 1.8. The functor Spec/X → Spec given by (Y → X) 7→ hofib(Y → X) preserves homotopy cocartesian squares. Proof. Suppose we have a commutative square of spectra A
B //
B
′
//
C
together with a map C → X (which makes all the spectra in this diagram into objects in Spec/X). The forgetful functor Spec/X → Spec reflects homotopy colimits, and homotopy cartesian and cocartesian are the same for spectra, so it is enough to show that if the above square is homotopy cartesian, then so is the corresponding square of homotopy fibres. This follows easily using the commutativity of homotopy limits. Proof of Proposition 1.7. By construction, the right-hand side of the claimed equivalence is Dn (FX )(hofib(Y → X)). By Lemma 1.8, this is equivalent to DnX (FX (hofib(Y → X))) = DnX (Y 7→ hofib(F (hofib(Y → X) ∨ X) → F X)). Now we consider the category SpecX of spectra over and under X, that is maps X → Y → X whose composite is the identity. Goodwillie examines the relationship between homogeneous functors on SpecX and homogeneous functors on Spec/X. He shows in [3, 4.1] that two homogeneous functors on Spec/X are equivalent if and only if they are equivalent when precomposed with the forgetful functor SpecX → Spec/X. Now if X → Y → X is the identity, then we have an equivalence hofib(Y → X) ∨ X −→Y ˜ and so hofib(F (hofib(Y → X) ∨ X) → F X) ≃ hofib(F Y → F X).
CHAIN RULE FOR FUNCTORS OF SPECTRA
5
But DnX (hofib(F Y → F X)) ≃ hofib(DnX F (Y ) → ∗) ≃ DnX (F ). Therefore, by Goodwillie’s result, the right-hand side of the proposition is equivalent to DnX (F ). Remark 1.9. Proposition 1.7 should be viewed as the spectrum-level equivalent of Goodwillie’s analysis of how the layers of the Taylor tower of a functor of spaces is built from the derivative of that space (viewed as a spectrum parametrized by the chosen base object). See [3, §5]. We now introduce the composition product of symmetric sequences, making sure we are precise about the way partitions are handled. Definition 1.10 (Partitions). For a positive integer n, let P (n) denote the set of unordered partitions of n. An element of P (n) can then be written uniquely as an increasing sequence of positive integers whose sum is n. Thus, for example, P (3) = {(3), (1, 2), (1, 1, 1)}, and so on. We can also specify an element λ of P (n) uniquely by writing n = k1 l1 + · · · + kr lr for some ki ≥ 1 and 1 ≤ l1 < · · · < lr . The integers li are the numbers that appear in the partition λ and ki is the number of times that li appears. For such a partition λ ∈ P (n) we write H(λ) = (Σk1 ≀ Σl1 ) × · · · × (Σkr ≀ Σlr ) ≤ Σn . Here the wreath product Σki ≀ Σli is the subgroup of Σki li generated by permutations of ki blocks of size li , and by the permutations within each block. We identify their product with a subgroup of Σn using the decomposition n = k1 l1 + · · · + kr lr . We will not use this fact explicitly, but note that the subgroup H(λ) is the stabilizer of the transitive action of Σn on the set of partitions of a set of n elements that are of ‘type’ λ. The cosets of H(λ) thus correspond bijectively with such partitions. Definition 1.11. (Composition product of symmetric sequences) A symmetric sequence in Spec is a sequence of spectra A = (A1 , A2 , . . . ) with an action of Σn on An . Thus the collection of all derivatives of a functor F at a fixed X forms a symmetric sequence ∂∗ F (X). Let A, B be two symmetric sequences in Spec. The composition product A ◦ B is the symmetric sequence given by a 1 ∧ . . . ∧ Blkrr . (Σn )+ ∧H(λ) Ak ∧ Bl∧k (A ◦ B)n := 1 λ∈P (n)
Here H(λ) acts on Σn by (right) multiplication and on the right-hand side in the natural way. The Σn -action on (A ◦ B)n is then determined by the obvious action on the (Σn )+ factor in each term of the coproduct. Remark 1.12. A more common way to define the composition product is to take a coproduct over ordered partitions of the integer n and induct up to a Σn -action from the action of Σn1 × · · · × Σnk on objects of the form Ak ∧ Bn1 ∧ . . . ∧ Bnk . Our definition is equivalent and having a decomposition in terms of unordered partitions will be helpful later.
6
MICHAEL CHING
With this definition of composition product, we are in position to state the main result of this paper. Theorem 1.13. Let F, G : Spec → Spec be homotopy functors and suppose that F preserves filtered homotopy colimits. Then ∂∗ (F G)(X) ≃ ∂∗ F (GX) ◦ ∂∗ G(X). Remark 1.14. The proof of Theorem 1.13 will occupy the next three sections, but we conclude this section by showing that the general statement follows from the special case in which X = ∗ and F and G are reduced functors, that is F (∗) ≃ ∗ and G(∗) ≃ ∗. To see this, recall that ∂∗ F (GX) := ∂∗ (FGX );
∂∗ G(X) := ∂∗ (GX );
∂∗ (F G)(X) := ∂∗ ((F G)X )
where FGX , GX and (F G)X are as in Definition 1.6. Note that FGX and GX are reduced. Also observe that FGX (GX (Z)) = FGX (hofib(G(X ∨ Z) → GX)) F (hofib(G(X ∨ Z) → GX) ∨ GX) . = hofib F GX
But the inclusion X → X ∨ Z determines an equivalence
hofib(G(X ∨ Z) → GX) ∨ GX −→G(X ˜ ∨ Z) and so the above is equivalent to hofib(F G(X ∨ Z) → F GX) = (F G)X (Z). Therefore we have FGX ◦ GX ≃ (F G)X . The claim of Theorem 1.13 can then be written as ∂∗ (FGX ◦ GX ) ≃ ∂∗ (FGX ) ◦ ∂∗ (GX ) which is just the Theorem again applied to the derivatives at ∗ of the reduced functors FGX and GX . For the remainder of the proof then, we will assume that F and G are reduced and consider only the Taylor tower at the trivial spectrum ∗. 2. The map that induces the chain rule equivalence We prove Theorem 1.13 in the reduced case by constructing, for each n, a natural transformation that gives the nth derivative part of the statement of the theorem. To define these maps, we recall the definition of the co-cross-effect of a homotopy functor. Definition 2.1 (Co-cross-effects). Let F : Spec → Spec be a reduced homotopy functor. The rth co-cross-effect of F is the functor of r variables (i.e. from Specr to Spec) defined by ! Y crr (F )(X1 , . . . , Xr ) := hocofib hocolim F ( Xj ) → F (X1 × · · · × Xr ) J {1,...,r}
j∈J
CHAIN RULE FOR FUNCTORS OF SPECTRA
7
In the language of [2] this is the total homotopy cofibre of the r-cube given by applying F to products of subsets of (X1 , . . . , Xr ), and is dual to the notion of cross-effect. Note that for functors from spectra to spectra (or on any stable model category), the rth co-cross-effect is equivalent to the rth cross-effect. Definition 2.2. The natural transformations we use to prove Theorem 1.13 are based on the following construction: ∆r : F (X)
∆
//
F (X × · · · × X) → crr (F )(X, . . . , X).
The first map here is given by the diagonal map X → X × · · · × X. The second map is the natural map from F (X × · · · × X), which is the terminal object of the cube defining the co-cross-effect, to the total cofibre of that cube. Now let λ be an element of P (n), that is, an unordered partition λ of n into positive integers, and recall that we can express λ uniquely by writing n as the sum n = k1 l1 + · · · + kr lr where ki ≥ 1 and l1 < · · · < lr . We then define a map ∆λ : F G
∆r
//
crr (F )(G, . . . , G) //
[Pk1 . . . Pkr crr (F )](Pl1 G, . . . , Plr G)
where we compose ∆r with the application of Pki to the multivariable functor crr (F ) in the ith position, and of Pli to the ith copy of G. For the sake of notation, we write (F G)λ for the target of this map. Remark 2.3. The duals of the maps ∆r (i.e. maps from the cross-effect of F to F ) form the basis for an alternative formulation of calculus of functors due to Johnson and McCarthy [5]. The maps ∆r themselves appear in the ‘dual calculus’ of McCarthy [9] which he used to study Tate cohomology obstructions to the splitting of the Taylor tower of a functor from spectra to spectra. It is no surprise that these maps appear in the current work, and it would be worth exploring if there is a deeper relationship with the work of McCarthy. Definition 2.4. Putting together the maps ∆λ for all λ ∈ P (n), we get a map Y ∆ : FG → (F G)λ . λ∈P (n)
In the next section, we will show that the map ∆ determines an equivalence on nth derivatives. Theorem 1.13 will then follow from the following calculation of the nth derivative of (F G)λ . Proposition 2.5. ∂n ((F G)λ ) ≃ (Σn )+ ∧H(λ) ∂k (F ) ∧ ∂l1 G∧k1 ∧ . . . ∧ ∂lr G∧kr where the action of H(λ) on the right-hand side is as in the definition of the composition product (Definition 1.11). The proof of the proposition depends on the following two lemmas: Lemma 2.6. Let H : Specr → Spec be a finitary functor of r variables that is (k1 , . . . , kr )excisive (that is, ki-excisive in the ith variable). Let G1 , . . . , Gr : Spec → Spec be a sequence of reduced functors such that Gj is lj -excisive. Then the composite functor H(G1, . . . , Gr ) : Spec → Spec
8
MICHAEL CHING
is (k1 l1 + · · · + kr lr )-excisive. Proof. Induction on k1 using the fibre sequences [Dk1 Pk2 . . . Pkr ]H → [Pk1 Pk2 . . . Pkr ]H → [Pk1 −1 Pk2 . . . Pkr ]H reduces to the case where H is k1 -homogeneous in its first variable. Then induction on k2 reduces to the case where H is also k2 -homogeneous in the second variable. Successive inductions on the remaining ki reduce to the case where H is (k1 , . . . , kr )-homogenous. Since H is assumed to be finitary, we can then write H(X1 , . . . , Xr ) ≃ [E ∧ (X1 )∧k1 ∧ . . . ∧ (Xr )kr ]h(Σk1 ×···×Σkr ) for some spectrum E with Σk1 × · · · × Σkr -action. But now it is sufficient to show that if G1 , . . . , Gk : Spec → Spec are functors such that Gj is lj -excisive, then G1 ∧ . . . ∧ Gk is l1 + · · · + lk -excisive. Induction on l1 using the fibre sequences (Dl1 G1 ) ∧ G2 ∧ . . . ∧ Gk → (Pl1 G1 ) ∧ G2 ∧ . . . ∧ Gk → (Pl1 −1 G1 ) ∧ G2 ∧ . . . ∧ Gk reduces to the case where G1 is l1 -homogeneous. Successive inductions on the remaining lj allow us to reduce to the case that all Gi are homogeneous. If we then write Gj (X) ≃ (Cj ∧ X ∧lj )hΣlj for suitable spectra Cj with Σlj -actions, then we can explicitly see that G1 ∧ . . . ∧ Gk is l1 + · · · + lk -excisive (and in fact homogeneous). Lemma 2.7. Let F : Spec → Spec be a finitary homotopy functor. Then [Dk1 . . . Dkr ] crr (F )(X1 , . . . , Xr ) ≃ ∂k F ∧ X1∧k1 ∧ . . . ∧ Xr∧kr h(Σ
k1 ×···×Σkr )
where k = k1 + · · · + kr .
Proof. We start by induction on the Taylor tower of F . Notice that the fibration sequence Di F → Pi F → Pi−1 F induces fibration sequences on both sides of this equation. If F is i-homogeneous, then, since it is assumed finitary, we can write F (X) ≃ (∂i F ∧ X ∧i )hΣi . We can explicitly calculate the rth (co-)cross-effect of this functor: Y (∂i F ∧ X1∧i1 ∧ . . . ∧ Xr∧ir )h(Σi1 ×···×Σir ) crr (F )(X1 , . . . , Xr ) ≃ i1 +···+ir =i ∗
if r ≤ i; otherwise.
Now apply [Dk1 . . . Dkr ]. The only time we get something non-trivial is when ij = kj for all j. This proves the Lemma in the case that F is i-homogeneous for some i. By induction then, it is true when F is n-excisive for some n.
It remains to show that it is true for all F . Clearly the natural transformation F → Pk F induces an equivalence on the right-hand side of the statement of the proposition, so it is enough to show that it induces an equivalence also on the left-hand side. We leave this as
CHAIN RULE FOR FUNCTORS OF SPECTRA
9
an exercise for the reader, noting that in order to prove Theorem 1.13 it is sufficient (by Lemma 6.1) to replace F with Pn F throughout this paper, so that the full strength of this result is not needed. Proof of Proposition 2.5. Lemma 2.6 tells us the functor (F G)λ is n-excisive. It also tells us that [Dk1 Pk2 . . . Pkr ] crr (F )(Pl1 G, . . . , Plr G) → [Pk1 Pk2 . . . Pkr ] crr (F )(Pl1 G, . . . , Plr G) is an equivalence on nth derivatives since it fits in a fibre sequence with [Pk1 −1 Pk2 . . . Pkr ] crr (F )(Pl1 G, . . . , Plr G) which is (n − l1 )-excisive (by Lemma 2.6). Repeating for k2 and so on, we see that the map [Dk1 . . . Dkr ] crr (F )(Pl1 G, . . . , Plr G) → [Pk1 . . . Pkr ] crr (F )(Pl1 G, . . . , Plr G) is an equivalence on nth derivatives. Now the map [Dk1 . . . Dkr ] crr (F )(Dl1 G, Pl2 G, . . . , Plr G) → [Dk1 . . . Dkr ] crr (F )(Pl1 G, Pl2 G, . . . , Plr G) is an equivalence of nth derivatives because it sits in a fibre sequence with [Dk1 . . . Dkr ] crr (F )(Pl1 −1 G, Pl2 G, . . . , Plr G) which is (n − k1 )-excisive (again by Lemma 2.6). Repeating for l2 and so on, we deduce that Dn ((F G)λ ) ≃ [Dk1 . . . Dkr ] crr (F )(Dl1 G, . . . , Dlr G). which by Lemma 2.7 is equivalent to ∂k F ∧ (Dl1 G)∧k1 ∧ . . . ∧ (Dlr G)∧kr
h(Σk1 ×···×Σkr )
.
Now writing Dj G(X) ≃ (∂j G ∧ X ∧j )hΣj , we deduce that i h ∧lr ∧kr ∧l1 ∧k1 Dn ((F G)λ )(X) ≃ ∂k F ∧ (∂l1 G ∧hΣl1 X ) ∧ . . . ∧ (∂lr G ∧hΣlr X )
h(Σk1 ×···×Σkr )
.
This is equivalent to
∂k F ∧ (∂l1 G)∧k1 ∧ . . . ∧ (∂lr G)∧kr ∧ X ∧n
hH(λ)
where we recall that H(λ) is the subgroup (Σl1 ≀ Σk1 × · · · × Σlr ≀ Σkr ) of Σn . From this it follows that the nth derivative of (F G)λ is as claimed. 3. The Taylor tower of F G when F is homogeneous We now prove that the map ∆ is an equivalence on nth derivatives. Our method is to prove this when F is k-homogeneous for some k, and then use induction on the Taylor tower to prove it for all F . In fact, when F is homogeneous, we will do something much stronger, namely we will describe the entire Taylor tower of F G. Goodwillie shows that a k-homogeneous functor F is equivalent to the functor X 7→ L(X, . . . , X)hΣk
10
MICHAEL CHING
where L is the symmetric multilinear functor of k variables given by the kth cross-effect (or equivalently co-cross-effect) of F . Since homotopy orbits commutes with Pn and Dn for spectrum-valued functors, it will be sufficient to consider functors of the form X 7→ L(X, . . . , X) for L a multilinear functor. Definition 3.1. Let L be a multilinear finitary homotopy functor of k variables and let G : Spec → Spec be a reduced homotopy functor. To connect with the previous notation, we also write F (X) := L(X, . . . , X). Let Pk (n) be the poset of ordered k-term partitions of the integer n. (This is in contrast to P (n) which was the set of unordered partitions.) That is, Pk (n) is the category whose objects are ordered k-tuples (r1 , . . . , rk ) of positive integers with the property that r1 + · · · + rk ≤ n and such that there is a unique morphism from (r1 , . . . , rk ) to (s1 , . . . , sk ) if and only if ri ≥ si for all i. Define functors pn (F G) : Spec → Spec by taking the following homotopy limit over Pk (n): pn (F G)(X) :=
holim
(r1 ,...,rk )∈Pk (n)
L(Pr1 G(X), . . . , Prk G(X)).
The maps in this diagram for this homotopy limit come from the structure maps in the Taylor tower for G. The inclusion Pk (n − 1) → Pk (n) determines a restriction pn (F G) → pn−1 (F G) and the natural maps F G = L(G, . . . , G) → L(Pr1 G, . . . , Prk G) assemble to form maps F G → pn (F G) that commute with the restrictions. Our main aim now is to show that the sequence {pn (F G)} is equivalent to the Taylor tower of the functor F G = L(G, . . . , G). The key to this is a calculation of the homotopy fibres of the maps pn (F G) → pn−1 (F G). Let dn (F G) denote this homotopy fibre. Then we have the following result. Proposition 3.2. There is a natural equivalence Y ˜ n (F G). α: L(Dr1 G, . . . , Drk G)−→d r1 +···+rk =n
Proof. To construct this equivalence, it is convenient to assume that the maps Pr G → Pr−1 G in the Taylor tower for G are fibrations, and the layers Dr G are taken to be the strict fibres of these maps. This means that the composite Dr G → Pr G → Pr−1 G is the trivial map. Now consider the diagram indexed on Pk (n) that sends (r1 , . . . , rk ) to L(Dr1 , . . . , Drk ) if r1 + · · · + rk = n and to ∗ otherwise. The homotopy limit of this diagram is isomorphic to
CHAIN RULE FOR FUNCTORS OF SPECTRA
11
the product in the statement of the proposition. There is a natural transformation from this diagram to that defining pn (F G) and so we get an induced map Y L(Dr1 G, . . . , Drk G) → pn (F G) r1 +···+rk =n
Composing with the restriction pn (F G) → pn−1 (F G) we get the trivial map and so this factors via dn (F G), giving us the map α.
We construct the inverse equivalence for α as follows. If r1 + · · · + rk = n, consider the following diagram dn (F G) //
pn (F G)
RRR RRR RRR RRR RRR β RRR R))
//
γ
L(Pr1 G, . . . , Prk G)
pn−1 (F G)
L(Pr1 G, . . . , Prk −1 G) //
where the vertical maps are the obvious projections from the homotopy limits. The square commutes up to homotopy which implies that γβ is nullhomotopic. Therefore, in the homotopy category, β factors via the homotopy fibre of γ. Since L is linear in each variable, that fibre is equivalent to L(Pr1 G, . . . , Prk−1 G, Drk G). By a similar argument, the composite dr (F G) → L(Pr1 G, . . . , Prk−1 G, Drk G) → L(Pr1 G, . . . , Prk−1 −1 G, Drk G) is nullhomotopic, and so factors via L(Pr1 G, . . . , Drk−1 G, Drk G), and so on. By induction, we can factor via all the Dri G and putting the resulting maps together for all k-tuples with r1 + · · · + rk = n, we get the required map (in the homotopy category) Y α′ : dr (F G) → L(Dr1 G, . . . , Drk G). r1 +···+rk =n
We now claim that α′ is an inverse weak equivalence to α. It is easy to check from the definitions that α′ α is the identity. That αα′ is homotopic to the identity on dn (F G) follows from the commutativity (in the homotopy category) of the diagrams dn (F G)
//
L(Dr1 G, . . . , Drk G) This completes the proof.
pn (F G)
//
L(Pr1 G, . . . , Prk G)
Corollary 3.3. For each n, dn (F G) is n-homogeneous and pn (F G) is n-excisive. Moreover, Pr (pn (F G)) ≃ pr (F G) for r ≤ n. Proof. Since L is finitary, L(Dr1 G, . . . , Drk G) is n-homogeneous. By induction on n, we may assume that pn−1 (F G) is (n − 1)-excisive, and hence n-excisive. But then the fibration
12
MICHAEL CHING
sequence dn (F G) → pn (F G) → pn−1 (F G) implies that pn (F G) is n-excisive too. It also follows from the induction that Pr (pn (F G)) ≃ pr (F G) for r ≤ n. Proposition 3.4. The sequence F G → · · · → pn (F G) → pn−1 (F G) → . . . is equivalent to the Taylor tower of F G. Proof. The maps Pn (F G) → Pn (pn (F G)) ← pn (F G) commute with the tower restriction maps and so it is sufficient to show that these are equivalences. The right-hand map is an equivalence by Corollary 3.3. Suppose first that G is m-excisive for some m. Consider the diagram indexed by Pk (n) whose homotopy limit is pn (F G) and suppose n ≥ mk. Many of the maps in this diagram are equivalences and the homotopy limit is equivalent to the homotopy limit of the restriction of the diagram to the subcategory of k-tuples (r1 , . . . , rk ) such that ri ≤ m for all i. This restricted diagram has initial object L(Pm G, . . . , Pm G), so its homotopy limit is just this initial object which is also equivalent to F G. Therefore, we have F G−→p ˜ n (F G) for all n ≥ mk. It follows immediately that Pn (F G)−→P ˜ n (pn (F G))←−p ˜ n (F G) for n ≥ mk. For n < mk, we then have Pn (F G)−→P ˜ n (pmk (F G))−→P ˜ n (pn (F G))←−p ˜ n (F G). Now suppose that G is arbitrary. Then we have a commutative diagram Pn (F G)
∼
//
Pn (F Pn G)
pn (F G)
∼
//
∼
Pn (pn (F G))
∼
//
Pn (pn (F Pn G))
We have just shown that the right-hand map is an equivalence. The bottom-right map is an equivalence since Pr G → Pr (Pn G) is an equivalence for r ≤ n so that the diagrams defining the homotopy limits in pn (F G) → pn (F Pn G) are termwise equivalent. The bottom-left map is an equivalence by Corollary 3.3. Finally, the top map is an equivalence by a general calculus argument (see Lemma 6.1) and we are done. Corollary 3.5. Dn (F G) ≃
Y
L(Dr1 G, . . . , Drk G)
r1 +···+rk =n
Proof. This follows from Propositions 3.4 and 3.2.
CHAIN RULE FOR FUNCTORS OF SPECTRA
13
Recall that we can now take the homotopy orbits to get formulas for the Taylor tower of F G for any homogeneous functor F . Theorem 3.6. Let F : Spec → Spec be a k-homogeneous finitary homotopy functor, and let G : Spec → Spec be any reduced homotopy functor. Then we have Pn (F G) ≃ holim crk (F )(Pr1 G, . . . , Prk G) r1 +···+rk ≤n
hΣk
and Dn (F G) ≃
"
Y
crk (F )(Dr1 G, . . . , Drk G)
r1 +···+rk =n
#
hΣk
where Σk acts on the homotopy limit (respectively, product) by permuting (r1 , . . . , rk ) using the symmetry isomorphisms of the symmetric multilinear functor crk (F ) (the kth cross-effect of F ). 4. Proof of the chain rule Theorem 3.6 is of interest in its own right and can easily be used to prove the chain rule for derivatives in the case that F is k-homogenous, but our main point is to show that the map ∆ of Definition 2.4 induces an equivalence on nth derivatives. Proposition 4.1. Let F : Spec → Spec be a k-homogeneous finitary homotopy functor and let G : Spec → Spec be any reduced homotopy functor. Then the map Y Dn ∆ : Dn (F G)−→ ˜ Dn ((F G)λ ) λ∈P (n)
is an equivalence. Proof. We start by taking F to be of the form F (X) = L(X, . . . , X) for a multilinear functor L of k variables and calculate (F G)λ . Since L is linear (and since finite products and coproducts are equivalent in Spec), we have Y L(X1 × · · · × Xr , . . . , X1 × · · · × Xr ) ≃ L(Xs(1) , . . . , Xs(k) ). s:{1,...,k}→{1,...,r}
where the product is taken over all functions s from {1, . . . , k} to {1, . . . , r}. The co-crosseffect is given by Y crr (F )(X1 , . . . , Xr ) ≃ L(Xs(1) , . . . , Xs(k) ) s:{1,...,k}։{1,...,r}
where now the product is over all surjective functions. Applying Pk1 ,...,kr to this we get Y L(Xs(1) , . . . , Xs(k) ) s:{1,...,k}→{1,...,r} 1≤|s−1 (j)|≤kj
14
MICHAEL CHING
and so if the partition λ is represented by n = k1 l1 + · · · + kr lr then Y L(Pls(1) G, . . . , Pls(k) G) (F G)λ = s:{1,...,k}→{1,...,r} |s−1 (j)|≤kj
with the map ∆λ : F G → (F G)λ given by (4.2)
F G = L(G, . . . , G) → L(Pls(1) G, . . . , Pls(k) G)
induced by the maps in the Taylor tower for G. We want to apply Dn to this. To help simplify this, notice the following facts: • ls(1) + · · · + ls(k) ≤ k1 l1 + · · · + kr lr = n because |s−1 (j)| ≤ kj , with equality if and only if |s−1 (j)| = kj for j = 1, . . . , r • if ls(1) + · · · + ls(k) < n then Dn (L(Pls(1) G, . . . , Pls(k) G)) ≃ ∗ • if ls(1) + · · · + ls(k) = n then Dn (L(Pls(1) G, . . . , Pls(k) G)) ≃ L(Dls(1) G, . . . , Dls(k) G) Therefore:
Y
Dn ((F G)λ) ≃
L(Dls(1) G, . . . , Dls(k) G)
s:{1,...,k}→{1,...,r} |s−1 (j)|=kj
Now given a function s : {1, . . . , k} → {1, . . . , r} as in the indexing set of the product, define ri := ls(i) . Then (r1 , . . . , rk ) is an ordered k-tuples such that r1 + · · · + rk = n. This k-tuple is of type λ in the sense that precisely kj of the terms are equal to lj . Conversely, given such a k-tuple, define a function s : {1, . . . , k} → {1, . . . , r} by s(i) := j where lj = ri . Such a j exists because (r1 , . . . , rk ) is of type λ and is unique because all the lj are distinct. These constructions set up a 1-1 correspondence between the indexing set of the above product, and the set of ordered k-tuples (r1 , . . . , rk ) of ‘type’ λ (i.e. ki of the terms are equal to li for i = 1, . . . , r). Therefore, we can write Y Dn ((F G)λ ) ≃ L(Dr1 G, . . . , Drk G). (r1 ,...,rk ) of type λ
But now by Corollary 3.5 we have Y Dn (F G) ≃
r1 +···+rk =n
L(Dr1 G, . . . , Drk G) ≃
Y
Dn ((F G)λ )
λ∈P (n)
and since the maps ∆λ are defined as in (4.2) in terms of the projections of the Taylor tower of G, the map ∆ expresses the above equivalence, as required. Finally, to deduce the result for a general F , we take L to be the kth cross-effect of F and apply homotopy orbits with respect to the action of Σk determined by the symmetry isomorphisms for the cross-effect. Taking homotopy orbits commutes with all the relevant constructions, namely Dn , Pki and taking the co-cross-effect, so we are done. A simple induction now completes the proof that Dn ∆ is an equivalence for all F .
CHAIN RULE FOR FUNCTORS OF SPECTRA
15
Proposition 4.3. The conclusion of Proposition 4.1 holds for any homotopy functor F . Proof. We first note that by Lemma 6.1 we have Dn (F G) ≃ Dn ((Pn F )G) and Dn ((F G)λ ) ≃ Dn (((Pn F )G)λ) so it is sufficient to prove it when F is n-excisive. We do this by induction on the Taylor tower of F . Consider the fibration sequence Dm F → Pm F → Pm−1 F. This induces a fibration sequence Dn ((Dm F )G) → Dn ((Pm F )G) → Dn ((Pm−1 F )G). It also induces a fibration sequence crr (Dm F ) → crr (Pm F ) → crr (Pm−1 F ) because the co-cross-effects are equivalent to the cross-effects which are defined by total homotopy fibres which commute with homotopy fibres. Therefore, we also have a fibration sequence Dn (((Dm F )G)λ ) → Dn (((Pm F )G)λ) → Dn (((Pm−1 F )G)λ ). But Dn ∆ is an equivalence for Dm F by Proposition 4.1, and for Pm−1 F by the induction hypothesis. Hence it is an equivalence for Pm F . This completes the proof. Finally, we deduce our chain rule. Theorem 1.13. Let F, G : Spec → Spec be homotopy functors and suppose that F preserves filtered homotopy colimits. Then ∂∗ (F G)(X) ≃ ∂∗ (F )(GX) ◦ ∂∗ (G)(X). Proof. The case where F and G are reduced and X = ∗ follows from Propositions 4.3 and 2.5 using the definition of the composition product (1.11) and the fact that finite products and coproducts of spectra are equivalent. The general case then follows from the argument of Remark 1.14. 5. Chain rule for Taylor towers of functors of spectra We now use the results on derivatives to say something about the full Taylor tower for a composed functor F G. We should caution the reader that as far as we can tell our results only apply to the Taylor tower when evaluated for a map Y → X that has a section, that is an object in SpecX . While we used to Goodwillie’s general theory to pass from this to the general case on the level of derivatives, this is not possible for the actual terms in the tower. First note that again it is sufficient to study the reduced case because when Y → X has a section, we have PnX (F )(Y ) ≃ Pn (FX )(hofib(Y → X)) where FX is as in Definition 1.6 and is reduced. We therefore assume from here on that F and G are reduced and consider only the Taylor tower at X = ∗.
16
MICHAEL CHING
We then have the following commutative diagram Dn (F G) //
Pn (F G)
Y
Pn−1 (F G)
∼
//
Dn ((F G)λ) //
Y
λ∈P(n)
Pn ((F G)λ )
Y
//
λ∈P(n)
Pn−1 ((F G)λ)
λ∈P(n)
The rows in this diagram are fibre sequences and the left-hand vertical map is an equivalence by Proposition 4.3. Therefore the right-hand square is a homotopy pullback. Thus we can express Pn (F G) as a homotopy pullback of Pn−1 (F G) over the product of the maps Pn ((F G)λ ) → Pn−1 ((F G)λ ). Since (F G)λ is already n-excisive, the Pn is unnecessary here. The usefulness (or otherwise) of this result depends on how easy it is to express the maps and objects in this diagram in terms of the Taylor towers of F and G. We give calculations for small n. Examples. For n = 1, we get the following simple expression P1 (F G) ≃ P1 (F ) ◦ P1 (G) which is clearly analogous to the chain rule for first derivatives in ordinary calculus. The case n = 2 is also manageable. There are two partitions of n = 2, corresponding to 2 = 2 and 2 = 1 + 1. We therefore have ∂2 (F G) ≃ (∂2 F ∧ ∂1 G∧2 ) ∨ (∂1 F ∧ ∂2 G). We have (F G)2 ≃ P1 F ◦ P2 G and (F G)11 ≃ P2 F ◦ P1 G. Each of these has P1 ((F G)λ ) ≃ P1 F ◦ P1 G. Therefore, we get a homotopy pullback P2 (F G) //
P1 F ◦ P1 G
∆
P2 F ◦ P1 G × P1 F ◦ P2 G
P1 F ◦ P1 G × P1 F ◦ P1 G //
This is equivalent to the existence of a homotopy pullback P2 (F G)
//
P1 F ◦ P2 G
P2 F ◦ P1 G
//
P1 F ◦ P1 G
CHAIN RULE FOR FUNCTORS OF SPECTRA
17
For n = 3, there are three partition types: 3, 12 and 111. This gives a pullback square P3 (F G) //
(P3 F ◦ P1 G) × (P1 F ◦ P3 G) × (P1 P1 cr2 (F )(P1 G, P2 G))
P2 (F G) //
(P2 F ◦ P1 G) × (P1 F ◦ P2 G) × (P1 P1 cr2 (F )(P1 G, P1 G))
We can identify the cross-effects in this case as P1 P1 cr2 (F )(P1 G, P2 G) ≃ ∂2 F ∧ P1 G ∧ P2 G The only real question mark then is over the map P2 (F G) → ∂2 F ∧ P1 G ∧ P1 G that forms part of the bottom map in the above diagram. This map clearly factors as: P2 (F G) → P2 F ◦ P1 G → cr2 (P2 F )(P1 G, P1 G) ≃ ∂2 F ∧ P1 G ∧ P1 G. where the second map is essentially ∆2 , but it is not at all obvious how to identify this map in a practical form in general. In order to calculate with a given functor F , understanding the corresponding maps ∆r is essential. In any case, we can write P3 (F G) as the homotopy limit of the following diagram: P1 F ◦ P3 G
P3 F ◦ P1 G
P1 F ◦ P2 G
P2 F ◦ P1 G
JJ JJ JJ JJ JJ JJ JJ JJ J$$
∂2 F ∧ P1 G ∧ P2 G
∆2
//
∂2 F ∧ P1 G∧2
P1 F ◦ P1 G Theoretically, this process could be used to get ‘formulas’ for all Pn (F G) as homotopy limits. In practice, these diagrams seem to quickly get rather complicated and identifying the maps involved depends on a lot of information about the functor F , including the multivariable Taylor towers of all the functors Pk1 . . . Pkr crr (F ). 6. A calculus lemma Lemma 6.1. Let F, G : Spec → Spec be reduced simplicial homotopy functors with F finitary. Then the following maps are equivalences. (1) Pr (F G) → Pr ((Pr F )G); (2) Pr (F G) → Pr (F (Pr G)).
18
MICHAEL CHING
Proof. (1) As with many of the basic results of calculus, there is a simple proof of this in the case where F, G are analytic functors, based on connectivity. We give a diagrammatic proof, inspired by various proofs of Goodwillie of similar results, that works for all F, G. Since Pn commutes with filtered homotopy colimits [3, 1.7], it is enough to show that Pn (F G) → Pn ((Tn F )G) is an equivalence. This is the map of homotopy colimits induced by the following (solidarrowed) map of filtered diagrams: FG
//
t
(Tn F )G
t
t
t
t
t
t
t
Tn (F G)
//
t
t
Tn ((Tn F )G)
t
t
t
t
t
t
t::
//
// . . . t::
Tn2 (F G) //
t::
t
t
Tn2 ((Tn F )G)
t
t
t
t
t
t
t
//
...
In order to show that the induced map on homotopy colimits is a weak equivalence, it is enough to construct the shown diagonal (dashed-arrow) maps that make all the triangles in the resulting diagram commute. The natural transformation (Tn F )G(X) → Tn (F G)(X) is given by the map holim F (U ∗ G(X)) → holim F G(U ∗ X) U 6=∅
U 6=∅
induced by the maps U ∗ GX → G(U ∗ X) which exist because G is a simplicial functor. (Note that U ∗ X denotes the ‘join’ of an object ` X ∈ C with the finite set U. This can be described as the homotopy cofibre of the map U X → X.) The other diagonal maps are given by applying Tn to the map just constructed and it is easy to check that all the resulting triangles commute. (2) The proof is virtually identical, except that the inverse equivalence is based on a natural transformation F (Tn G)(X) → Tn (F G)(X) given by F (holim U ∗ G(X)) → holim U ∗ F G(X) U 6=∅
U 6=∅
which comes from the fact that F is simplicial. We also here require the condition that F is finitary to pass from F (Tn G) to F (Pn G). References 1. A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, With an appendix by M. Cole. MR 97h:55006 2. Thomas G. Goodwillie, Calculus. II. Analytic functors, K-Theory 5 (1991/92), no. 4, 295–332. MR 93i:55015 , Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711 (electronic). MR 2 026 544 3. 4. Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. MR MR1695653 (2000h:55016) 5. B. Johnson and R. McCarthy, Deriving calculus with cotriples, Trans. Amer. Math. Soc. 356 (2004), no. 2, 757–803 (electronic). MR 2 022 719
CHAIN RULE FOR FUNCTORS OF SPECTRA
19
6. Warren P. Johnson, The curious history of Fa` a di Bruno’s formula, Amer. Math. Monthly 109 (2002), no. 3, 217–234. MR MR1903577 (2003d:01019) 7. Nicholas J. Kuhn, Goodwillie towers and chromatic homotopy: an overview. 8. M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512. MR MR1806878 (2001k:55025) 9. 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. MR MR1831354 (2002c:18009)
arXiv:0710.5567v2 [math.AT] 31 Oct 2007
MICHAEL CHING
Abstract. We prove a chain rule in the Goodwillie calculus of functors from spectra to spectra. We show that the derivatives of a composite F G at a base object X are given by taking the composition product of the derivatives of F at G(X) with the derivatives of G at X. Our proof also allows us to say something about the Taylor tower of F G (and not just its layers) in terms of the Taylor towers of F and G.
Introduction In this paper we prove a version of the chain rule in the Goodwillie calculus of functors from spectra to spectra. Let Spec be a model for the stable homotopy category and let F : Spec → Spec be a homotopy functor (i.e. F preserves stable weak equivalences). Fix a base object X ∈ Spec and let Spec/X denote the category of spectra over X. Then the methods of Goodwillie [3] can be used to construct a Taylor tower for F analogous to the Taylor series of a function of a real variable, based at the object X. This tower is a sequence of functors Spec/X → Spec: X F (Y ) → · · · → PnX F (Y ) → Pn−1 F (Y ) → · · · → P0X F (Y ) = F (X)
that, for each map Y → X, interpolates between F (Y ) and F (X). The layers of the Taylor tower are the homotopy fibres X DnX F := hofib(PnX F → Pn−1 F)
and they can in general be written in the form DnX F (Y ) ≃ (∂n F (X) ∧ hofib(Y → X)∧n )hΣn . This formula holds when Y is a finite cell object in Spec and for all Y → X if F preserves filtered homotopy colimits. The object ∂n F (X) is a spectrum with an action of the symmetric group Σn and we refer to it as the nth derivative of F at X. The derivatives of F at X together form a symmetric sequence in Spec, that is a sequence of objects with actions of the symmetric groups. We write ∂∗ F (X) for this symmetric sequence. Also recall that there is a monoidal product on the category of symmetric sequences, called the composition product which we write ◦. (See Definition 1.11 below.) The main result of this paper is the following. 1
2
MICHAEL CHING
Theorem 1.13. Let F, G : Spec → Spec be homotopy functors and suppose that F preserves filtered homotopy colimits. Then we have the following formula: ∂∗ (F G)(X) ≃ ∂∗ F (GX) ◦ ∂∗ G(X). We call this a ‘chain rule’ since it expresses the derivatives of a composite of functors in terms of the derivatives of the individual functors. It bears a striking similarity to the corresponding result for functions of real-variables [6]. The derivatives of a functor determine the layers in the Taylor tower, but there are still extension problems in recovering the whole tower. We therefore consider also the problem of expressing PnX (F G)(Y ) in terms of the Taylor towers of F and G. Our method gives an answer to this question in the case that the map Y → X has a section, but there is no nice formula and it is not clear that the answer can be used in practice to calculate Taylor towers. We however give explicit descriptions of P2 (F G) and P3 (F G) where the calculations are not so hard. We should remark that our proof is non-constructive in the sense that we do not define specific models for the derivatives of a functor that satisfy the equivalence of Theorem 1.13. This has the downside that we cannot use our proof to obtain explicit operad or cooperad structures on the derivatives of monads and comonads. In separate work between the author and Greg Arone, an alternative proof of the Theorem is being developed that does give specific models for the derivatives and produces interesting operad and cooperad structures on the derivatives of certain functors. The current paper, on the other hand, has the advantage that its methods reveal something about a chain rule for the Taylor towers and not just for the derivatives. Finally, we remark that while we write out the proof for functors of spectra, most of the paper applies to functors between any stable model categories. Outline of the paper. In §1, we define what we mean by the derivatives of a functor of spectra at a general base object and state our main result in terms of the composition product of symmetric sequences. We also show that the general result follows from the special case where the base object is ∗ and the functors involved are reduced. In §2 we describe a map that on nth derivatives induces the equivalence of the theorem and show that the derivatives involved are correct. In §3 we prove the theorem in the case that F is homogeneous, obtaining along the way a simple formula for the full Taylor tower of F G under this condition. Then in §4 we complete the proof of Theorem 1.13 in general. In §5, we see how our results give us information about the full Taylor tower of F G and give explicit descriptions of the lowest few terms in this tower. Finally §6 contains a useful calculus lemma. Acknowledgements. This work owes a considerable debt to conversations between the author and Greg Arone, and is part of a joint project to prove chain rules for derivatives in a much wider context. More specifically, the underlying idea of this paper, that we can use the diagonal maps and co-cross-effects of McCarthy to study the chain rule for spectra, was suggested by Arone.
CHAIN RULE FOR FUNCTORS OF SPECTRA
3
1. Preliminaries Definition 1.1. Let Spec be a symmetric monoidal proper simplicial cofibrantly generated model category whose homotopy category is equivalent to the usual stable homotopy category. We shall refer to the objects of Spec as spectra. Any of the standard examples (such as EKMM S-modules [1], symmetric spectra [4] or orthogonal spectra [8]) can be used. Remark 1.2. We only use the symmetric monoidal structure on Spec, and the fact that it has the ‘usual’ stable homotopy category to express our chain rule in terms of derivatives. The construction of the map ∆ (Definition 2.4 below) and all the results of §§3-5 require only that Spec be a (proper, simplicial) stable model category (that is, the suspension functor is an equivalence on the homotopy category). Convention 1.3. From now on we assume any functor F : Spec → Spec that we use is simplicial (that is, F respects the simplicial enrichment in Spec) and a homotopy functor (that is, F preserves weak equivalences). Definition 1.4. (Taylor tower for functors from spectra to spectra) Goodwillie explicitly describes calculus for functors of topological spaces, but his ideas extend easily to a more general setting, for example, see Kuhn [7]. In particular, a functor F satisfying the conditions of Convention 1.3 has, for each X ∈ Spec, a Taylor tower at X, that is a sequence of functors X F → · · · → PnX F → Pn−1 F → . . . P0X F = F (X)
in which PnX F is an n-excisive functor Spec/X → Spec, that is takes strongly homotopy cocartesian (n + 1)-cubes to homotopy cartesian cubes. The nth layer in the tower is the homotopy fibre X DnX F := hofib(PnX F → Pn−1 F ). and this is n-homogeneous. We refer to [3, §1] for the precise definitions and properties of the Pn and Dn constructions. When the base object X is the trivial spectrum ∗, we write simply Pn F and Dn F . These are functors Spec → Spec. Remark 1.5. We now want to define the derivatives of a functor from spectra to spectra at an arbitrary base object. This case is not explicitly covered in [3] so we give a careful description. First notice that derivatives at the trivial spectrum ∗ work in the same way as for functors of spaces. Specifically, given F : Spec → Spec, there is a spectrum ∂n F with Σn -action such that Dn F (Y ) ≃ (∂n F ∧ Y ∧n )hΣn for finite cell spectra Y and for all Y ∈ Spec if F is finitary (i.e. preserves filtered homotopy colimits). The spectra ∂n F are the derivatives of F at ∗. To be precise we can define ∂n F by ∂n F := crn (Dn F )(S, . . . , S) where crn is the nth cross-effect, and S denotes the sphere spectrum. Definition 1.6. (Derivatives) Let F : Spec → Spec be a homotopy functor. The derivatives of F at X are the derivatives at ∗ of the functor FX (Z) = hofib(F (Z ∨ X) → F (X))
4
MICHAEL CHING
that is, ∂n F (X) := ∂n (FX ). This should be compared with Goodwillie’s definition of the derivatives of a functor of spaces at a general base object [3, 5.6]. Proposition 1.7. Let F : Spec → Spec be a homotopy functor and let Y → X be a map of spectra. Then DnX F (Y ) ≃ (∂n F (X) ∧ hofib(Y → X)∧n )hΣn . This holds whenever hofib(Y → X) is equivalent to a finite cell spectrum, and for all Y → X if F is finitary. The following lemma will be useful in proving this proposition: Lemma 1.8. The functor Spec/X → Spec given by (Y → X) 7→ hofib(Y → X) preserves homotopy cocartesian squares. Proof. Suppose we have a commutative square of spectra A
B //
B
′
//
C
together with a map C → X (which makes all the spectra in this diagram into objects in Spec/X). The forgetful functor Spec/X → Spec reflects homotopy colimits, and homotopy cartesian and cocartesian are the same for spectra, so it is enough to show that if the above square is homotopy cartesian, then so is the corresponding square of homotopy fibres. This follows easily using the commutativity of homotopy limits. Proof of Proposition 1.7. By construction, the right-hand side of the claimed equivalence is Dn (FX )(hofib(Y → X)). By Lemma 1.8, this is equivalent to DnX (FX (hofib(Y → X))) = DnX (Y 7→ hofib(F (hofib(Y → X) ∨ X) → F X)). Now we consider the category SpecX of spectra over and under X, that is maps X → Y → X whose composite is the identity. Goodwillie examines the relationship between homogeneous functors on SpecX and homogeneous functors on Spec/X. He shows in [3, 4.1] that two homogeneous functors on Spec/X are equivalent if and only if they are equivalent when precomposed with the forgetful functor SpecX → Spec/X. Now if X → Y → X is the identity, then we have an equivalence hofib(Y → X) ∨ X −→Y ˜ and so hofib(F (hofib(Y → X) ∨ X) → F X) ≃ hofib(F Y → F X).
CHAIN RULE FOR FUNCTORS OF SPECTRA
5
But DnX (hofib(F Y → F X)) ≃ hofib(DnX F (Y ) → ∗) ≃ DnX (F ). Therefore, by Goodwillie’s result, the right-hand side of the proposition is equivalent to DnX (F ). Remark 1.9. Proposition 1.7 should be viewed as the spectrum-level equivalent of Goodwillie’s analysis of how the layers of the Taylor tower of a functor of spaces is built from the derivative of that space (viewed as a spectrum parametrized by the chosen base object). See [3, §5]. We now introduce the composition product of symmetric sequences, making sure we are precise about the way partitions are handled. Definition 1.10 (Partitions). For a positive integer n, let P (n) denote the set of unordered partitions of n. An element of P (n) can then be written uniquely as an increasing sequence of positive integers whose sum is n. Thus, for example, P (3) = {(3), (1, 2), (1, 1, 1)}, and so on. We can also specify an element λ of P (n) uniquely by writing n = k1 l1 + · · · + kr lr for some ki ≥ 1 and 1 ≤ l1 < · · · < lr . The integers li are the numbers that appear in the partition λ and ki is the number of times that li appears. For such a partition λ ∈ P (n) we write H(λ) = (Σk1 ≀ Σl1 ) × · · · × (Σkr ≀ Σlr ) ≤ Σn . Here the wreath product Σki ≀ Σli is the subgroup of Σki li generated by permutations of ki blocks of size li , and by the permutations within each block. We identify their product with a subgroup of Σn using the decomposition n = k1 l1 + · · · + kr lr . We will not use this fact explicitly, but note that the subgroup H(λ) is the stabilizer of the transitive action of Σn on the set of partitions of a set of n elements that are of ‘type’ λ. The cosets of H(λ) thus correspond bijectively with such partitions. Definition 1.11. (Composition product of symmetric sequences) A symmetric sequence in Spec is a sequence of spectra A = (A1 , A2 , . . . ) with an action of Σn on An . Thus the collection of all derivatives of a functor F at a fixed X forms a symmetric sequence ∂∗ F (X). Let A, B be two symmetric sequences in Spec. The composition product A ◦ B is the symmetric sequence given by a 1 ∧ . . . ∧ Blkrr . (Σn )+ ∧H(λ) Ak ∧ Bl∧k (A ◦ B)n := 1 λ∈P (n)
Here H(λ) acts on Σn by (right) multiplication and on the right-hand side in the natural way. The Σn -action on (A ◦ B)n is then determined by the obvious action on the (Σn )+ factor in each term of the coproduct. Remark 1.12. A more common way to define the composition product is to take a coproduct over ordered partitions of the integer n and induct up to a Σn -action from the action of Σn1 × · · · × Σnk on objects of the form Ak ∧ Bn1 ∧ . . . ∧ Bnk . Our definition is equivalent and having a decomposition in terms of unordered partitions will be helpful later.
6
MICHAEL CHING
With this definition of composition product, we are in position to state the main result of this paper. Theorem 1.13. Let F, G : Spec → Spec be homotopy functors and suppose that F preserves filtered homotopy colimits. Then ∂∗ (F G)(X) ≃ ∂∗ F (GX) ◦ ∂∗ G(X). Remark 1.14. The proof of Theorem 1.13 will occupy the next three sections, but we conclude this section by showing that the general statement follows from the special case in which X = ∗ and F and G are reduced functors, that is F (∗) ≃ ∗ and G(∗) ≃ ∗. To see this, recall that ∂∗ F (GX) := ∂∗ (FGX );
∂∗ G(X) := ∂∗ (GX );
∂∗ (F G)(X) := ∂∗ ((F G)X )
where FGX , GX and (F G)X are as in Definition 1.6. Note that FGX and GX are reduced. Also observe that FGX (GX (Z)) = FGX (hofib(G(X ∨ Z) → GX)) F (hofib(G(X ∨ Z) → GX) ∨ GX) . = hofib F GX
But the inclusion X → X ∨ Z determines an equivalence
hofib(G(X ∨ Z) → GX) ∨ GX −→G(X ˜ ∨ Z) and so the above is equivalent to hofib(F G(X ∨ Z) → F GX) = (F G)X (Z). Therefore we have FGX ◦ GX ≃ (F G)X . The claim of Theorem 1.13 can then be written as ∂∗ (FGX ◦ GX ) ≃ ∂∗ (FGX ) ◦ ∂∗ (GX ) which is just the Theorem again applied to the derivatives at ∗ of the reduced functors FGX and GX . For the remainder of the proof then, we will assume that F and G are reduced and consider only the Taylor tower at the trivial spectrum ∗. 2. The map that induces the chain rule equivalence We prove Theorem 1.13 in the reduced case by constructing, for each n, a natural transformation that gives the nth derivative part of the statement of the theorem. To define these maps, we recall the definition of the co-cross-effect of a homotopy functor. Definition 2.1 (Co-cross-effects). Let F : Spec → Spec be a reduced homotopy functor. The rth co-cross-effect of F is the functor of r variables (i.e. from Specr to Spec) defined by ! Y crr (F )(X1 , . . . , Xr ) := hocofib hocolim F ( Xj ) → F (X1 × · · · × Xr ) J {1,...,r}
j∈J
CHAIN RULE FOR FUNCTORS OF SPECTRA
7
In the language of [2] this is the total homotopy cofibre of the r-cube given by applying F to products of subsets of (X1 , . . . , Xr ), and is dual to the notion of cross-effect. Note that for functors from spectra to spectra (or on any stable model category), the rth co-cross-effect is equivalent to the rth cross-effect. Definition 2.2. The natural transformations we use to prove Theorem 1.13 are based on the following construction: ∆r : F (X)
∆
//
F (X × · · · × X) → crr (F )(X, . . . , X).
The first map here is given by the diagonal map X → X × · · · × X. The second map is the natural map from F (X × · · · × X), which is the terminal object of the cube defining the co-cross-effect, to the total cofibre of that cube. Now let λ be an element of P (n), that is, an unordered partition λ of n into positive integers, and recall that we can express λ uniquely by writing n as the sum n = k1 l1 + · · · + kr lr where ki ≥ 1 and l1 < · · · < lr . We then define a map ∆λ : F G
∆r
//
crr (F )(G, . . . , G) //
[Pk1 . . . Pkr crr (F )](Pl1 G, . . . , Plr G)
where we compose ∆r with the application of Pki to the multivariable functor crr (F ) in the ith position, and of Pli to the ith copy of G. For the sake of notation, we write (F G)λ for the target of this map. Remark 2.3. The duals of the maps ∆r (i.e. maps from the cross-effect of F to F ) form the basis for an alternative formulation of calculus of functors due to Johnson and McCarthy [5]. The maps ∆r themselves appear in the ‘dual calculus’ of McCarthy [9] which he used to study Tate cohomology obstructions to the splitting of the Taylor tower of a functor from spectra to spectra. It is no surprise that these maps appear in the current work, and it would be worth exploring if there is a deeper relationship with the work of McCarthy. Definition 2.4. Putting together the maps ∆λ for all λ ∈ P (n), we get a map Y ∆ : FG → (F G)λ . λ∈P (n)
In the next section, we will show that the map ∆ determines an equivalence on nth derivatives. Theorem 1.13 will then follow from the following calculation of the nth derivative of (F G)λ . Proposition 2.5. ∂n ((F G)λ ) ≃ (Σn )+ ∧H(λ) ∂k (F ) ∧ ∂l1 G∧k1 ∧ . . . ∧ ∂lr G∧kr where the action of H(λ) on the right-hand side is as in the definition of the composition product (Definition 1.11). The proof of the proposition depends on the following two lemmas: Lemma 2.6. Let H : Specr → Spec be a finitary functor of r variables that is (k1 , . . . , kr )excisive (that is, ki-excisive in the ith variable). Let G1 , . . . , Gr : Spec → Spec be a sequence of reduced functors such that Gj is lj -excisive. Then the composite functor H(G1, . . . , Gr ) : Spec → Spec
8
MICHAEL CHING
is (k1 l1 + · · · + kr lr )-excisive. Proof. Induction on k1 using the fibre sequences [Dk1 Pk2 . . . Pkr ]H → [Pk1 Pk2 . . . Pkr ]H → [Pk1 −1 Pk2 . . . Pkr ]H reduces to the case where H is k1 -homogeneous in its first variable. Then induction on k2 reduces to the case where H is also k2 -homogeneous in the second variable. Successive inductions on the remaining ki reduce to the case where H is (k1 , . . . , kr )-homogenous. Since H is assumed to be finitary, we can then write H(X1 , . . . , Xr ) ≃ [E ∧ (X1 )∧k1 ∧ . . . ∧ (Xr )kr ]h(Σk1 ×···×Σkr ) for some spectrum E with Σk1 × · · · × Σkr -action. But now it is sufficient to show that if G1 , . . . , Gk : Spec → Spec are functors such that Gj is lj -excisive, then G1 ∧ . . . ∧ Gk is l1 + · · · + lk -excisive. Induction on l1 using the fibre sequences (Dl1 G1 ) ∧ G2 ∧ . . . ∧ Gk → (Pl1 G1 ) ∧ G2 ∧ . . . ∧ Gk → (Pl1 −1 G1 ) ∧ G2 ∧ . . . ∧ Gk reduces to the case where G1 is l1 -homogeneous. Successive inductions on the remaining lj allow us to reduce to the case that all Gi are homogeneous. If we then write Gj (X) ≃ (Cj ∧ X ∧lj )hΣlj for suitable spectra Cj with Σlj -actions, then we can explicitly see that G1 ∧ . . . ∧ Gk is l1 + · · · + lk -excisive (and in fact homogeneous). Lemma 2.7. Let F : Spec → Spec be a finitary homotopy functor. Then [Dk1 . . . Dkr ] crr (F )(X1 , . . . , Xr ) ≃ ∂k F ∧ X1∧k1 ∧ . . . ∧ Xr∧kr h(Σ
k1 ×···×Σkr )
where k = k1 + · · · + kr .
Proof. We start by induction on the Taylor tower of F . Notice that the fibration sequence Di F → Pi F → Pi−1 F induces fibration sequences on both sides of this equation. If F is i-homogeneous, then, since it is assumed finitary, we can write F (X) ≃ (∂i F ∧ X ∧i )hΣi . We can explicitly calculate the rth (co-)cross-effect of this functor: Y (∂i F ∧ X1∧i1 ∧ . . . ∧ Xr∧ir )h(Σi1 ×···×Σir ) crr (F )(X1 , . . . , Xr ) ≃ i1 +···+ir =i ∗
if r ≤ i; otherwise.
Now apply [Dk1 . . . Dkr ]. The only time we get something non-trivial is when ij = kj for all j. This proves the Lemma in the case that F is i-homogeneous for some i. By induction then, it is true when F is n-excisive for some n.
It remains to show that it is true for all F . Clearly the natural transformation F → Pk F induces an equivalence on the right-hand side of the statement of the proposition, so it is enough to show that it induces an equivalence also on the left-hand side. We leave this as
CHAIN RULE FOR FUNCTORS OF SPECTRA
9
an exercise for the reader, noting that in order to prove Theorem 1.13 it is sufficient (by Lemma 6.1) to replace F with Pn F throughout this paper, so that the full strength of this result is not needed. Proof of Proposition 2.5. Lemma 2.6 tells us the functor (F G)λ is n-excisive. It also tells us that [Dk1 Pk2 . . . Pkr ] crr (F )(Pl1 G, . . . , Plr G) → [Pk1 Pk2 . . . Pkr ] crr (F )(Pl1 G, . . . , Plr G) is an equivalence on nth derivatives since it fits in a fibre sequence with [Pk1 −1 Pk2 . . . Pkr ] crr (F )(Pl1 G, . . . , Plr G) which is (n − l1 )-excisive (by Lemma 2.6). Repeating for k2 and so on, we see that the map [Dk1 . . . Dkr ] crr (F )(Pl1 G, . . . , Plr G) → [Pk1 . . . Pkr ] crr (F )(Pl1 G, . . . , Plr G) is an equivalence on nth derivatives. Now the map [Dk1 . . . Dkr ] crr (F )(Dl1 G, Pl2 G, . . . , Plr G) → [Dk1 . . . Dkr ] crr (F )(Pl1 G, Pl2 G, . . . , Plr G) is an equivalence of nth derivatives because it sits in a fibre sequence with [Dk1 . . . Dkr ] crr (F )(Pl1 −1 G, Pl2 G, . . . , Plr G) which is (n − k1 )-excisive (again by Lemma 2.6). Repeating for l2 and so on, we deduce that Dn ((F G)λ ) ≃ [Dk1 . . . Dkr ] crr (F )(Dl1 G, . . . , Dlr G). which by Lemma 2.7 is equivalent to ∂k F ∧ (Dl1 G)∧k1 ∧ . . . ∧ (Dlr G)∧kr
h(Σk1 ×···×Σkr )
.
Now writing Dj G(X) ≃ (∂j G ∧ X ∧j )hΣj , we deduce that i h ∧lr ∧kr ∧l1 ∧k1 Dn ((F G)λ )(X) ≃ ∂k F ∧ (∂l1 G ∧hΣl1 X ) ∧ . . . ∧ (∂lr G ∧hΣlr X )
h(Σk1 ×···×Σkr )
.
This is equivalent to
∂k F ∧ (∂l1 G)∧k1 ∧ . . . ∧ (∂lr G)∧kr ∧ X ∧n
hH(λ)
where we recall that H(λ) is the subgroup (Σl1 ≀ Σk1 × · · · × Σlr ≀ Σkr ) of Σn . From this it follows that the nth derivative of (F G)λ is as claimed. 3. The Taylor tower of F G when F is homogeneous We now prove that the map ∆ is an equivalence on nth derivatives. Our method is to prove this when F is k-homogeneous for some k, and then use induction on the Taylor tower to prove it for all F . In fact, when F is homogeneous, we will do something much stronger, namely we will describe the entire Taylor tower of F G. Goodwillie shows that a k-homogeneous functor F is equivalent to the functor X 7→ L(X, . . . , X)hΣk
10
MICHAEL CHING
where L is the symmetric multilinear functor of k variables given by the kth cross-effect (or equivalently co-cross-effect) of F . Since homotopy orbits commutes with Pn and Dn for spectrum-valued functors, it will be sufficient to consider functors of the form X 7→ L(X, . . . , X) for L a multilinear functor. Definition 3.1. Let L be a multilinear finitary homotopy functor of k variables and let G : Spec → Spec be a reduced homotopy functor. To connect with the previous notation, we also write F (X) := L(X, . . . , X). Let Pk (n) be the poset of ordered k-term partitions of the integer n. (This is in contrast to P (n) which was the set of unordered partitions.) That is, Pk (n) is the category whose objects are ordered k-tuples (r1 , . . . , rk ) of positive integers with the property that r1 + · · · + rk ≤ n and such that there is a unique morphism from (r1 , . . . , rk ) to (s1 , . . . , sk ) if and only if ri ≥ si for all i. Define functors pn (F G) : Spec → Spec by taking the following homotopy limit over Pk (n): pn (F G)(X) :=
holim
(r1 ,...,rk )∈Pk (n)
L(Pr1 G(X), . . . , Prk G(X)).
The maps in this diagram for this homotopy limit come from the structure maps in the Taylor tower for G. The inclusion Pk (n − 1) → Pk (n) determines a restriction pn (F G) → pn−1 (F G) and the natural maps F G = L(G, . . . , G) → L(Pr1 G, . . . , Prk G) assemble to form maps F G → pn (F G) that commute with the restrictions. Our main aim now is to show that the sequence {pn (F G)} is equivalent to the Taylor tower of the functor F G = L(G, . . . , G). The key to this is a calculation of the homotopy fibres of the maps pn (F G) → pn−1 (F G). Let dn (F G) denote this homotopy fibre. Then we have the following result. Proposition 3.2. There is a natural equivalence Y ˜ n (F G). α: L(Dr1 G, . . . , Drk G)−→d r1 +···+rk =n
Proof. To construct this equivalence, it is convenient to assume that the maps Pr G → Pr−1 G in the Taylor tower for G are fibrations, and the layers Dr G are taken to be the strict fibres of these maps. This means that the composite Dr G → Pr G → Pr−1 G is the trivial map. Now consider the diagram indexed on Pk (n) that sends (r1 , . . . , rk ) to L(Dr1 , . . . , Drk ) if r1 + · · · + rk = n and to ∗ otherwise. The homotopy limit of this diagram is isomorphic to
CHAIN RULE FOR FUNCTORS OF SPECTRA
11
the product in the statement of the proposition. There is a natural transformation from this diagram to that defining pn (F G) and so we get an induced map Y L(Dr1 G, . . . , Drk G) → pn (F G) r1 +···+rk =n
Composing with the restriction pn (F G) → pn−1 (F G) we get the trivial map and so this factors via dn (F G), giving us the map α.
We construct the inverse equivalence for α as follows. If r1 + · · · + rk = n, consider the following diagram dn (F G) //
pn (F G)
RRR RRR RRR RRR RRR β RRR R))
//
γ
L(Pr1 G, . . . , Prk G)
pn−1 (F G)
L(Pr1 G, . . . , Prk −1 G) //
where the vertical maps are the obvious projections from the homotopy limits. The square commutes up to homotopy which implies that γβ is nullhomotopic. Therefore, in the homotopy category, β factors via the homotopy fibre of γ. Since L is linear in each variable, that fibre is equivalent to L(Pr1 G, . . . , Prk−1 G, Drk G). By a similar argument, the composite dr (F G) → L(Pr1 G, . . . , Prk−1 G, Drk G) → L(Pr1 G, . . . , Prk−1 −1 G, Drk G) is nullhomotopic, and so factors via L(Pr1 G, . . . , Drk−1 G, Drk G), and so on. By induction, we can factor via all the Dri G and putting the resulting maps together for all k-tuples with r1 + · · · + rk = n, we get the required map (in the homotopy category) Y α′ : dr (F G) → L(Dr1 G, . . . , Drk G). r1 +···+rk =n
We now claim that α′ is an inverse weak equivalence to α. It is easy to check from the definitions that α′ α is the identity. That αα′ is homotopic to the identity on dn (F G) follows from the commutativity (in the homotopy category) of the diagrams dn (F G)
//
L(Dr1 G, . . . , Drk G) This completes the proof.
pn (F G)
//
L(Pr1 G, . . . , Prk G)
Corollary 3.3. For each n, dn (F G) is n-homogeneous and pn (F G) is n-excisive. Moreover, Pr (pn (F G)) ≃ pr (F G) for r ≤ n. Proof. Since L is finitary, L(Dr1 G, . . . , Drk G) is n-homogeneous. By induction on n, we may assume that pn−1 (F G) is (n − 1)-excisive, and hence n-excisive. But then the fibration
12
MICHAEL CHING
sequence dn (F G) → pn (F G) → pn−1 (F G) implies that pn (F G) is n-excisive too. It also follows from the induction that Pr (pn (F G)) ≃ pr (F G) for r ≤ n. Proposition 3.4. The sequence F G → · · · → pn (F G) → pn−1 (F G) → . . . is equivalent to the Taylor tower of F G. Proof. The maps Pn (F G) → Pn (pn (F G)) ← pn (F G) commute with the tower restriction maps and so it is sufficient to show that these are equivalences. The right-hand map is an equivalence by Corollary 3.3. Suppose first that G is m-excisive for some m. Consider the diagram indexed by Pk (n) whose homotopy limit is pn (F G) and suppose n ≥ mk. Many of the maps in this diagram are equivalences and the homotopy limit is equivalent to the homotopy limit of the restriction of the diagram to the subcategory of k-tuples (r1 , . . . , rk ) such that ri ≤ m for all i. This restricted diagram has initial object L(Pm G, . . . , Pm G), so its homotopy limit is just this initial object which is also equivalent to F G. Therefore, we have F G−→p ˜ n (F G) for all n ≥ mk. It follows immediately that Pn (F G)−→P ˜ n (pn (F G))←−p ˜ n (F G) for n ≥ mk. For n < mk, we then have Pn (F G)−→P ˜ n (pmk (F G))−→P ˜ n (pn (F G))←−p ˜ n (F G). Now suppose that G is arbitrary. Then we have a commutative diagram Pn (F G)
∼
//
Pn (F Pn G)
pn (F G)
∼
//
∼
Pn (pn (F G))
∼
//
Pn (pn (F Pn G))
We have just shown that the right-hand map is an equivalence. The bottom-right map is an equivalence since Pr G → Pr (Pn G) is an equivalence for r ≤ n so that the diagrams defining the homotopy limits in pn (F G) → pn (F Pn G) are termwise equivalent. The bottom-left map is an equivalence by Corollary 3.3. Finally, the top map is an equivalence by a general calculus argument (see Lemma 6.1) and we are done. Corollary 3.5. Dn (F G) ≃
Y
L(Dr1 G, . . . , Drk G)
r1 +···+rk =n
Proof. This follows from Propositions 3.4 and 3.2.
CHAIN RULE FOR FUNCTORS OF SPECTRA
13
Recall that we can now take the homotopy orbits to get formulas for the Taylor tower of F G for any homogeneous functor F . Theorem 3.6. Let F : Spec → Spec be a k-homogeneous finitary homotopy functor, and let G : Spec → Spec be any reduced homotopy functor. Then we have Pn (F G) ≃ holim crk (F )(Pr1 G, . . . , Prk G) r1 +···+rk ≤n
hΣk
and Dn (F G) ≃
"
Y
crk (F )(Dr1 G, . . . , Drk G)
r1 +···+rk =n
#
hΣk
where Σk acts on the homotopy limit (respectively, product) by permuting (r1 , . . . , rk ) using the symmetry isomorphisms of the symmetric multilinear functor crk (F ) (the kth cross-effect of F ). 4. Proof of the chain rule Theorem 3.6 is of interest in its own right and can easily be used to prove the chain rule for derivatives in the case that F is k-homogenous, but our main point is to show that the map ∆ of Definition 2.4 induces an equivalence on nth derivatives. Proposition 4.1. Let F : Spec → Spec be a k-homogeneous finitary homotopy functor and let G : Spec → Spec be any reduced homotopy functor. Then the map Y Dn ∆ : Dn (F G)−→ ˜ Dn ((F G)λ ) λ∈P (n)
is an equivalence. Proof. We start by taking F to be of the form F (X) = L(X, . . . , X) for a multilinear functor L of k variables and calculate (F G)λ . Since L is linear (and since finite products and coproducts are equivalent in Spec), we have Y L(X1 × · · · × Xr , . . . , X1 × · · · × Xr ) ≃ L(Xs(1) , . . . , Xs(k) ). s:{1,...,k}→{1,...,r}
where the product is taken over all functions s from {1, . . . , k} to {1, . . . , r}. The co-crosseffect is given by Y crr (F )(X1 , . . . , Xr ) ≃ L(Xs(1) , . . . , Xs(k) ) s:{1,...,k}։{1,...,r}
where now the product is over all surjective functions. Applying Pk1 ,...,kr to this we get Y L(Xs(1) , . . . , Xs(k) ) s:{1,...,k}→{1,...,r} 1≤|s−1 (j)|≤kj
14
MICHAEL CHING
and so if the partition λ is represented by n = k1 l1 + · · · + kr lr then Y L(Pls(1) G, . . . , Pls(k) G) (F G)λ = s:{1,...,k}→{1,...,r} |s−1 (j)|≤kj
with the map ∆λ : F G → (F G)λ given by (4.2)
F G = L(G, . . . , G) → L(Pls(1) G, . . . , Pls(k) G)
induced by the maps in the Taylor tower for G. We want to apply Dn to this. To help simplify this, notice the following facts: • ls(1) + · · · + ls(k) ≤ k1 l1 + · · · + kr lr = n because |s−1 (j)| ≤ kj , with equality if and only if |s−1 (j)| = kj for j = 1, . . . , r • if ls(1) + · · · + ls(k) < n then Dn (L(Pls(1) G, . . . , Pls(k) G)) ≃ ∗ • if ls(1) + · · · + ls(k) = n then Dn (L(Pls(1) G, . . . , Pls(k) G)) ≃ L(Dls(1) G, . . . , Dls(k) G) Therefore:
Y
Dn ((F G)λ) ≃
L(Dls(1) G, . . . , Dls(k) G)
s:{1,...,k}→{1,...,r} |s−1 (j)|=kj
Now given a function s : {1, . . . , k} → {1, . . . , r} as in the indexing set of the product, define ri := ls(i) . Then (r1 , . . . , rk ) is an ordered k-tuples such that r1 + · · · + rk = n. This k-tuple is of type λ in the sense that precisely kj of the terms are equal to lj . Conversely, given such a k-tuple, define a function s : {1, . . . , k} → {1, . . . , r} by s(i) := j where lj = ri . Such a j exists because (r1 , . . . , rk ) is of type λ and is unique because all the lj are distinct. These constructions set up a 1-1 correspondence between the indexing set of the above product, and the set of ordered k-tuples (r1 , . . . , rk ) of ‘type’ λ (i.e. ki of the terms are equal to li for i = 1, . . . , r). Therefore, we can write Y Dn ((F G)λ ) ≃ L(Dr1 G, . . . , Drk G). (r1 ,...,rk ) of type λ
But now by Corollary 3.5 we have Y Dn (F G) ≃
r1 +···+rk =n
L(Dr1 G, . . . , Drk G) ≃
Y
Dn ((F G)λ )
λ∈P (n)
and since the maps ∆λ are defined as in (4.2) in terms of the projections of the Taylor tower of G, the map ∆ expresses the above equivalence, as required. Finally, to deduce the result for a general F , we take L to be the kth cross-effect of F and apply homotopy orbits with respect to the action of Σk determined by the symmetry isomorphisms for the cross-effect. Taking homotopy orbits commutes with all the relevant constructions, namely Dn , Pki and taking the co-cross-effect, so we are done. A simple induction now completes the proof that Dn ∆ is an equivalence for all F .
CHAIN RULE FOR FUNCTORS OF SPECTRA
15
Proposition 4.3. The conclusion of Proposition 4.1 holds for any homotopy functor F . Proof. We first note that by Lemma 6.1 we have Dn (F G) ≃ Dn ((Pn F )G) and Dn ((F G)λ ) ≃ Dn (((Pn F )G)λ) so it is sufficient to prove it when F is n-excisive. We do this by induction on the Taylor tower of F . Consider the fibration sequence Dm F → Pm F → Pm−1 F. This induces a fibration sequence Dn ((Dm F )G) → Dn ((Pm F )G) → Dn ((Pm−1 F )G). It also induces a fibration sequence crr (Dm F ) → crr (Pm F ) → crr (Pm−1 F ) because the co-cross-effects are equivalent to the cross-effects which are defined by total homotopy fibres which commute with homotopy fibres. Therefore, we also have a fibration sequence Dn (((Dm F )G)λ ) → Dn (((Pm F )G)λ) → Dn (((Pm−1 F )G)λ ). But Dn ∆ is an equivalence for Dm F by Proposition 4.1, and for Pm−1 F by the induction hypothesis. Hence it is an equivalence for Pm F . This completes the proof. Finally, we deduce our chain rule. Theorem 1.13. Let F, G : Spec → Spec be homotopy functors and suppose that F preserves filtered homotopy colimits. Then ∂∗ (F G)(X) ≃ ∂∗ (F )(GX) ◦ ∂∗ (G)(X). Proof. The case where F and G are reduced and X = ∗ follows from Propositions 4.3 and 2.5 using the definition of the composition product (1.11) and the fact that finite products and coproducts of spectra are equivalent. The general case then follows from the argument of Remark 1.14. 5. Chain rule for Taylor towers of functors of spectra We now use the results on derivatives to say something about the full Taylor tower for a composed functor F G. We should caution the reader that as far as we can tell our results only apply to the Taylor tower when evaluated for a map Y → X that has a section, that is an object in SpecX . While we used to Goodwillie’s general theory to pass from this to the general case on the level of derivatives, this is not possible for the actual terms in the tower. First note that again it is sufficient to study the reduced case because when Y → X has a section, we have PnX (F )(Y ) ≃ Pn (FX )(hofib(Y → X)) where FX is as in Definition 1.6 and is reduced. We therefore assume from here on that F and G are reduced and consider only the Taylor tower at X = ∗.
16
MICHAEL CHING
We then have the following commutative diagram Dn (F G) //
Pn (F G)
Y
Pn−1 (F G)
∼
//
Dn ((F G)λ) //
Y
λ∈P(n)
Pn ((F G)λ )
Y
//
λ∈P(n)
Pn−1 ((F G)λ)
λ∈P(n)
The rows in this diagram are fibre sequences and the left-hand vertical map is an equivalence by Proposition 4.3. Therefore the right-hand square is a homotopy pullback. Thus we can express Pn (F G) as a homotopy pullback of Pn−1 (F G) over the product of the maps Pn ((F G)λ ) → Pn−1 ((F G)λ ). Since (F G)λ is already n-excisive, the Pn is unnecessary here. The usefulness (or otherwise) of this result depends on how easy it is to express the maps and objects in this diagram in terms of the Taylor towers of F and G. We give calculations for small n. Examples. For n = 1, we get the following simple expression P1 (F G) ≃ P1 (F ) ◦ P1 (G) which is clearly analogous to the chain rule for first derivatives in ordinary calculus. The case n = 2 is also manageable. There are two partitions of n = 2, corresponding to 2 = 2 and 2 = 1 + 1. We therefore have ∂2 (F G) ≃ (∂2 F ∧ ∂1 G∧2 ) ∨ (∂1 F ∧ ∂2 G). We have (F G)2 ≃ P1 F ◦ P2 G and (F G)11 ≃ P2 F ◦ P1 G. Each of these has P1 ((F G)λ ) ≃ P1 F ◦ P1 G. Therefore, we get a homotopy pullback P2 (F G) //
P1 F ◦ P1 G
∆
P2 F ◦ P1 G × P1 F ◦ P2 G
P1 F ◦ P1 G × P1 F ◦ P1 G //
This is equivalent to the existence of a homotopy pullback P2 (F G)
//
P1 F ◦ P2 G
P2 F ◦ P1 G
//
P1 F ◦ P1 G
CHAIN RULE FOR FUNCTORS OF SPECTRA
17
For n = 3, there are three partition types: 3, 12 and 111. This gives a pullback square P3 (F G) //
(P3 F ◦ P1 G) × (P1 F ◦ P3 G) × (P1 P1 cr2 (F )(P1 G, P2 G))
P2 (F G) //
(P2 F ◦ P1 G) × (P1 F ◦ P2 G) × (P1 P1 cr2 (F )(P1 G, P1 G))
We can identify the cross-effects in this case as P1 P1 cr2 (F )(P1 G, P2 G) ≃ ∂2 F ∧ P1 G ∧ P2 G The only real question mark then is over the map P2 (F G) → ∂2 F ∧ P1 G ∧ P1 G that forms part of the bottom map in the above diagram. This map clearly factors as: P2 (F G) → P2 F ◦ P1 G → cr2 (P2 F )(P1 G, P1 G) ≃ ∂2 F ∧ P1 G ∧ P1 G. where the second map is essentially ∆2 , but it is not at all obvious how to identify this map in a practical form in general. In order to calculate with a given functor F , understanding the corresponding maps ∆r is essential. In any case, we can write P3 (F G) as the homotopy limit of the following diagram: P1 F ◦ P3 G
P3 F ◦ P1 G
P1 F ◦ P2 G
P2 F ◦ P1 G
JJ JJ JJ JJ JJ JJ JJ JJ J$$
∂2 F ∧ P1 G ∧ P2 G
∆2
//
∂2 F ∧ P1 G∧2
P1 F ◦ P1 G Theoretically, this process could be used to get ‘formulas’ for all Pn (F G) as homotopy limits. In practice, these diagrams seem to quickly get rather complicated and identifying the maps involved depends on a lot of information about the functor F , including the multivariable Taylor towers of all the functors Pk1 . . . Pkr crr (F ). 6. A calculus lemma Lemma 6.1. Let F, G : Spec → Spec be reduced simplicial homotopy functors with F finitary. Then the following maps are equivalences. (1) Pr (F G) → Pr ((Pr F )G); (2) Pr (F G) → Pr (F (Pr G)).
18
MICHAEL CHING
Proof. (1) As with many of the basic results of calculus, there is a simple proof of this in the case where F, G are analytic functors, based on connectivity. We give a diagrammatic proof, inspired by various proofs of Goodwillie of similar results, that works for all F, G. Since Pn commutes with filtered homotopy colimits [3, 1.7], it is enough to show that Pn (F G) → Pn ((Tn F )G) is an equivalence. This is the map of homotopy colimits induced by the following (solidarrowed) map of filtered diagrams: FG
//
t
(Tn F )G
t
t
t
t
t
t
t
Tn (F G)
//
t
t
Tn ((Tn F )G)
t
t
t
t
t
t
t::
//
// . . . t::
Tn2 (F G) //
t::
t
t
Tn2 ((Tn F )G)
t
t
t
t
t
t
t
//
...
In order to show that the induced map on homotopy colimits is a weak equivalence, it is enough to construct the shown diagonal (dashed-arrow) maps that make all the triangles in the resulting diagram commute. The natural transformation (Tn F )G(X) → Tn (F G)(X) is given by the map holim F (U ∗ G(X)) → holim F G(U ∗ X) U 6=∅
U 6=∅
induced by the maps U ∗ GX → G(U ∗ X) which exist because G is a simplicial functor. (Note that U ∗ X denotes the ‘join’ of an object ` X ∈ C with the finite set U. This can be described as the homotopy cofibre of the map U X → X.) The other diagonal maps are given by applying Tn to the map just constructed and it is easy to check that all the resulting triangles commute. (2) The proof is virtually identical, except that the inverse equivalence is based on a natural transformation F (Tn G)(X) → Tn (F G)(X) given by F (holim U ∗ G(X)) → holim U ∗ F G(X) U 6=∅
U 6=∅
which comes from the fact that F is simplicial. We also here require the condition that F is finitary to pass from F (Tn G) to F (Pn G). References 1. A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, American Mathematical Society, Providence, RI, 1997, With an appendix by M. Cole. MR 97h:55006 2. Thomas G. Goodwillie, Calculus. II. Analytic functors, K-Theory 5 (1991/92), no. 4, 295–332. MR 93i:55015 , Calculus. III. Taylor series, Geom. Topol. 7 (2003), 645–711 (electronic). MR 2 026 544 3. 4. Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. MR MR1695653 (2000h:55016) 5. B. Johnson and R. McCarthy, Deriving calculus with cotriples, Trans. Amer. Math. Soc. 356 (2004), no. 2, 757–803 (electronic). MR 2 022 719
CHAIN RULE FOR FUNCTORS OF SPECTRA
19
6. Warren P. Johnson, The curious history of Fa` a di Bruno’s formula, Amer. Math. Monthly 109 (2002), no. 3, 217–234. MR MR1903577 (2003d:01019) 7. Nicholas J. Kuhn, Goodwillie towers and chromatic homotopy: an overview. 8. M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512. MR MR1806878 (2001k:55025) 9. 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. MR MR1831354 (2002c:18009)
