A CHAIN RULE IN THE CALCULUS

OF HOMOTOPY FUNCTORS

John R. Klein and John Rognes

25.03.02

Abstract. We formulate and prove a chain rule for the derivative, in the sense of Goodwillie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum ∂F (X) of such a functor F at a simplicial set X can be equipped with a right action by the loop group of its domain X, and a free left action by the loop group of its codomain Y = F (X). The derivative spectrum ∂(G ∘ F)(X) of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives ∂G(Y) and ∂F (X), with respect to the two actions of the loop group of Y. As an application we provide a non-manifold computation of the derivative of the functor F (X) = Q(Map(K, X)_{+}).

Introduction

The calculus of functors was introduced by Goodwillie in [Go90] as a language to keep track of stable range calculations of certain geometrically defined homotopy functors, such as stable pseudo-isotopy theory. The input for the theory is a homotopy functor

f: U ⟶ T

from spaces to based spaces. At an object X ∈ U it is then possible to associate the “best excisive approximation” to f near X. This so-called linearization of f at X is a functor

P_{X}f: U/X ⟶ T

from spaces over X to based spaces, which maps homotopy pushout squares to homotopy pullback squares. The associated reduced functor is called the differential of f at X, and is denoted by D_{X}f. Choosing a base point x ∈ X, the composite functor

L: T ⟶^{i} U/X ⟶^{D_{X}f} T

taking a based space T to D_{X}f(X ∨_{x} T) = hofib(P_{X}f(X ∨_{x} T) → P_{X}f(X)), is a linear functor, whose homotopy groups L_{∗}(T) = π_{∗}(L(T)) define a generalized homology theory. Each such homology theory is represented by a spectrum, and the spectrum associated to this particular homology theory L_{∗} is called the derivative ∂f(X) of f at (X, x).

The goal of this paper is to establish a chain rule for the derivative of a composite functor. This is a reasonable goal, since many naturally occurring functors have this form. For example, the topological Hochschild homology THH(X) of a space

Typeset by AMS-TEX

X has the homotopy type of Q(ΛX_{+}), where ΛX = Map(S^{1}, X) is the free loop space (see [BHM93, 3.7]). We can view this as the composite of the two functors f(X) = ΛX and g(Y) = Q(Y_{+}) = colim_{n} Ω^{n}Σ^{n}(Y_{+}).

In order to even state a chain rule, some modification has to be made to the above set-up. In particular we will relax the condition that the functor f takes values in based spaces, considering instead weak homotopy functors

f: U ⟶ U

from spaces to spaces. (All our spaces will be compactly generated.) Then for any space X we let Y = f(X), and choose base points x ∈ X and y ∈ Y. We then study the derivative ∂_{y}^{x}f(X) of f at X, with respect to the base points x and y. Of course, if f(X) naturally comes equipped with a base point, then we may take that point as y.

Thus consider functors f, g: U → U, with composite g ∘ f: U → U. Let X be a space, and set Y = f(X), Z = g(Y). Choose base points x ∈ X, y ∈ Y and z ∈ Z. Suppose that f and g are bounded below, stably excisive functors (section 2), that g satisfies the colimit axiom (section 2), and that Y is path-connected. Let Ω_{y}(Y) denote the geometric realization of the Kan loop group (section 7) of the total singular simplicial set of Y. This is a topological group, weakly homotopy equivalent to the usual loop space of (Y, y). (More precisely, Ω_{y}(Y) is a group object in the category of compactly generated topological spaces.) Then it turns out that, by choosing the models right (section 8, see also remark 11.4), the derivative ∂_{y}^{x}f(X) admits a left Ω_{y}(Y)-action and the derivative ∂_{z}^{y}g(Y) admits a right Ω_{y}(Y)-action. It thus makes sense to form the homotopy orbit spectrum for the diagonal Ω_{y}(Y)- action on the smash product of spectra ∂_{z}^{y}g(Y) ∧ ∂_{y}^{x}f(X).

Theorem (Chain Rule). Let f, g: U → U be bounded below, stably excisive functors, with Y = f(X) and Z = g(Y), and suppose that g satisfies the colimit axiom. Suppose that Y is path-connected, and choose base points x ∈ X, y ∈ Y and z ∈ Z. Then the composite g ∘ f is bounded below and stably excisive, and its derivative spectrum at X with respect to x and z is stably equivalent to

∂_{z}^{x}(g ∘ f)(X) ≃ ∂_{z}^{y}g(Y) ∧_{hΩ_{y}(Y)} ∂_{y}^{x}f(X) .

The subscript hΩ_{y}(Y) denotes homotopy orbits with respect to the diagonal action of the topological group Ω_{y}(Y).

This is theorem 11.3 specialized to the case when Y is path-connected.

If X_{α} is the path-component of x in X, and Z_{γ} is the path-component of z in Z, then the topological group Ω_{x}(X_{α}) acts on ∂_{z}^{x}(g ∘ f)(X) and ∂_{y}^{x}f(X) from the right, the topological group Ω_{z}(Z_{γ}) acts on ∂_{z}^{x}(g ∘ f)(X) and ∂_{z}^{y}g(Y) from the left, and the chain rule gives a stable equivalence of spectra with left Ω_{z}(Z_{γ})-action and right Ω_{x}(X_{α})-action.

Technically, it is easier to discuss these group actions on spectra that are formed in the category S_{∗} of based simplicial sets, rather than in T . This is because the definition of the right action by Ω_{y}(Y) on ∂_{z}^{y}g(Y) basically requires that g is a continuous functor. This is awkward to achieve from a weak homotopy functor in the topological context, while for functors between simplicial sets it is easy to promote a weak homotopy functor to a simplicial functor, which suffices to define the right action in the simplicial context. See proposition 8.6. We therefore choose to develop the whole theory for weak homotopy functors F: S → S from the category S of simplicial sets to itself, rather than for functors f: U → U. In this case, the chain rule appears as theorem 10.4.

It is also possible to start with functors Φ: S/X → S and Ψ: S/Y → S, with X a simplicial set and Y = Φ(X), that may or may not factor through the forgetful functors u: S/X → S and u: S/Y → S, respectively. This is the most convenient general framework, and the body of the paper is written in this context. Thus theorem 10.3 is really our main theorem, from which the other forms of the chain rule are easily deduced.

In outline, the contents of the paper are as follows. We set up the categories in which we work in section 1, and fix terminology like “bounded below” and “stably excisive” in section 2. Then in section 3 we start with a stably excisive weak homotopy functor Φ: S/X → S and construct its “best excisive approximation” P′Φ, adapting [Go90, §1]. Some modification is needed, since we want P′Φ to take values in S/Y in order to be able to compose with Ψ. In section 4 we recall the Goodwillie derivative ∂Φ(X), and show in section 5 that to prove a chain rule expressing ∂(Ψ ∘ Φ)(X) in terms of ∂Ψ(Y) and ∂Φ(X), we may replace Φ and Ψ by their respective best excisive approximations (see proposition 5.2). This leads us to study linear functors Φ: R(X) → R(Y) and Ψ: R(Y) → R(Z), where R(X) is the category of retractive spaces over X, and Z = Ψ(Y). In section 6 we reduce further to the case where X, Y and Z are all connected.

When (Y, y) is based and connected, there is a natural equivalence R(Y) ≃ R(∗, Ω_{y}(Y)) (see proposition 7.1), where R(∗, Ω_{y}(Y)) is the category of based, free Ω_{y}(Y)-spaces, which we study in section 7. We are thus led to study linear functors ϕ: R(∗, Ω_{x}(X)) → R(∗, Ω_{y}(Y)) and ψ: R(∗, Ω_{y}(Y)) → R(∗, Ω_{z}(Z)), and their composite ψ ∘ϕ. The Goodwillie derivative ∂ϕ of such a functor ϕ has a natural free left Ω_{y}(Y)-action. Using a canonical enrichment of ϕ to a simplicial functor ϕ̌, we show in section 8 that ∂ϕ also has a natural right Ω_{x}(X)-action (see proposition 8.6). Then, in section 9 we establish a version of the Brown–Whitehead representability theorem (see [Go90, 1.3]) that represents a linear functor like ϕ above in terms of its Goodwillie derivative ∂ϕ, equipped with these left and right actions, under the assumption that ϕ satisfies a “colimit axiom”. See propositions 9.1 and 9.4. In section 10 we bring these structures and representations together, to prove the chain rule for bounded below, excisive functors Φ and Ψ in proposition 10.1, and for bounded below, stably excisive functors Φ and Ψ in theorem 10.3. The translation to functors to and from topological spaces goes via the usual equivalence S ≃ U, and is found in section 11.

We give a list of examples in section 12, including a purely homotopy-theoretic derivation in example 12.4 of the “stable homotopy of mapping spaces” functor X ↦ Q(Map(K, X)_{+}), that was previously investigated in [Go90, §2], [He92], and [Ar99] using manifold- or configuration space techniques. Our answer apparently takes a different form from that given in the cited papers, but in [Kl95] the first author shows that the two answers are indeed equivalent.

The paper is written using fairly strong explicit hypotheses on the functors, such as being bounded below and stably excisive, in line with the style of [Go90] and [Go92]. Yet, many of the functors one typically considers satisfy these hypotheses. Technically the main point of this is to ensure that applying the second functor respects certain stable equivalences of spectra arising from the first functor, at the end of the proof of proposition 10.1. A side effect is that all proofs become explicit, appealing directly to homotopy excision rather than to closed model category theory. Conceivably some of these conditions could be relaxed by reference to the framework of simplicial functors, as in [Ly98], but the work leading to the present paper precedes that preprint. Likewise, the present work can be incorporated into the more general language of pointed simplicial algebraic theories, as in [Sch01]. The second author’s Master student H. Fausk [Fa95] proved a version of the chain rule in the special case when Y = f(X) is contractible.

This paper was first written in 1995, following a visit by the first author to Oslo as a guest of the Norwegian Academy of Sciences. The authors thank the referees for their constructive feedback, and apologize for the long delay in finalizing the manuscript.

1. Categories of simplicial sets

Let S be the category of simplicial sets, and let X be a (fixed) simplicial set. The category S/X of simplicial sets over X has objects the simplicial sets X′ equipped with a map X′ → X, and morphisms the maps X′ → X′′ commuting with the structure maps to X. The category S/X has the identity map X → X as a terminal object.

The category R(X) of retractive simplicial sets over X has objects the simplicial sets X′ with maps r: X′ → X and s: X → X′ such that rs: X → X is the identity map, and morphisms the maps X′ → X′′ that commute with both structure maps, to and from X. The category R(X) has the identity map X → X as an initial and terminal object. In the case X = ∗ (a one-point simplicial set), R(∗) is equivalent to the category S_{∗} of based simplicial sets.

Let G be a simplicial group. A G-simplicial set W is a simplicial set with an action G × W → W. For any G-simplicial set W let R(W, G) be the category of relatively free, retractive G-simplicial sets over W. It has objects (W′, r, s), where W′ is a G-simplicial set, r: W′ → W and s: W → W′ are maps of G-simplicial sets, rs: W → W is the identity map, and W′ may be obtained from W by attaching free G-cells, i.e., by repeated pushouts along the inclusions G × ∂Δ^{n} ⊂ G × Δ^{n}. When G = 1 is the trivial group, R(W, 1) = R(W) as before. When W = ∗, the objects of R(∗, G) are based, free G-simplicial sets. (Cf. [Wa85, p. 378].)

Let u: S/X → S, v: R(X) → S/X and w: R(W, G) → R(W) be the obvious forgetful functors.

Consider any functor Φ: S/X → S. Let Y = Φ(X) be its value at the terminal object X (equipped with the identity map X → X). Then there is a canonical lift of Φ: S/X → S over u: S/Y → S to a functor S/X → S/Y, which we also denote by Φ. Furthermore, there is a canonical lift of Φ ∘ v: R(X) → S/Y over v: R(Y) → S/Y to a functor R(X) → R(Y), which we again denote by Φ.

Functors Φ: S/X → S sometimes arise from functors F: S → S as composites Φ = F ∘ u, but will in general depend on the structure map to X. We have a commutative diagram:

(1.1)  \xymatrix{R(X) \ar[r]^{v} \ar[d]^{Φ} & S/X \ar[r]^{u} \ar[d]^{Φ} \ar[dr]^{Φ} & S \ar@{-->}[d]^{F} \\ R(Y) \ar[r]^{v} & S/Y \ar[r]^{u} & S}

In this paper, a spectrum L is a sequence {n ↦ L_{n}} of based simplicial sets L_{n}, and based structure maps ΣL_{n} = L_{n} ∧ S^{1} → L_{n+1} for n ≥ 0, as in [BF78, 2.1]. Here it will be convenient to interpret S^{1} as Δ^{1} ∪_{∂Δ^{1}} Δ^{1}, rather than as Δ^{1}/∂Δ^{1}. We write Sp for the category of spectra.

A map of spectra f: M → L is a strict equivalence if each map f_{n}: L_{n} → M_{n} is a weak equivalence. It will be called a meta-stable equivalence if there exist integers c and κ such that f_{n}: L_{n} → M_{n} is (2n−c)-connected for all n ≥ κ (cf. section 2). And f is a stable equivalence if it induces an isomorphism π_{∗}(f): π_{∗}(L) → π_{∗}(M) on all homotopy groups. Clearly strict equivalences are meta-stable, and meta-stable equivalences are stable.

Let G be a simplicial group, as above. A spectrum with G-action L is a sequence {n ↦ L_{n}} of G-simplicial sets with a G-fixed base point, and based G-maps ΣL_{n} = L_{n} ∧ S^{1} → L_{n+1} for n ≥ 0, where G acts trivially on S^{1}. Let Sp^{G} be the category of spectra with G-action.

A free G-spectrum L is a sequence {n ↦ L_{n}} of based, free G-simplicial sets, and based G-maps ΣL_{n} = L_{n} ∧S^{1} → L_{n+1}. Here G acts trivially on S^{1}. Let Sp(G) be the category of free G-spectra.

There are obvious forgetful functors Sp(G) → Sp^{G} and Sp^{G} → Sp. A map of free G-spectra, or of spectra with G-action, is said to be a “strict”, “meta-stable”, or “stable equivalence”, if the underlying map of spectra has the corresponding property.

2. Excision conditions

A morphism f: X_{0} → X_{1} in S is k-connected if for every choice of base point x ∈ X_{0} the induced map π_{n}(f): π_{n}(X_{0}, x) → π_{n}(X_{1}, f(x)) is injective for 0 ≤ n < k and surjective for 0 ≤ n ≤ k. (No choice of base point is needed for n = 0, taking care of the case when X_{0} is empty.) A weak equivalence is a map that is k-connected for every integer k.

Let c and κ be integers. A functor F: S → S is said to satisfy condition E_{1}(c, κ) if for every k-connected map X_{0} → X_{1} with k ≥ κ the map F (X_{0}) → F (X_{1}) is (k − c)-connected. A functor satisfying condition E_{1}(c, κ) for some c and κ will be called bounded below. Such a functor necessarily takes weak equivalences to weak equivalences, i.e., is a weak homotopy functor.

We form functorial homotopy limits and homotopy colimits of diagrams of simplicial sets as in [BK72]. A commutative square of simplicial sets

(2.1)  \xymatrix{X_{0} \ar[r] \ar[d] & X_{1} \ar[d] \\ X_{2} \ar[r] & X_{3}}

is k-cartesian if the induced map a: X_{0} → holim(X_{1} → X_{3} ← X_{2}) is k-connected. It is cartesian if a is a weak equivalence. The square is k-cocartesian if the induced map b: hocolim(X_{1} ← X_{0} → X_{2}) → X_{3} is k-connected. It is cocartesian if b is a weak equivalence. (Cf. [Go90, 1.2].)

A functor F: S → S is said to satisfy condition E_{2}(c, κ) if, for every cocartesian square as above for which X_{0} → X_{i} is k_{i}-connected and k_{i} ≥ κ for i = 1, 2, the resulting square

\xymatrix{F (X_{0}) \ar[r] \ar[d] & F (X_{1}) \ar[d] \\ F (X_{2}) \ar[r] & F (X_{3})}

is (k_{1} + k_{2} − c)-cartesian. The functor F is called stably excisive if it satisfies condition E_{2}(c, κ) for some integers c and κ. F is called excisive if it takes all cocartesian squares to cartesian squares. (Cf. [Go90, 1.8].)

A morphism in one of the categories S/X, R(X) or R(W, G) is said to be “kconnected”, or a “weak equivalence”, if the underlying morphism in S has that property. Similarly for k-cartesian, cartesian, k-cocartesian and cocartesian squares. The conditions E_{1}(c, κ), “bounded below”, “weak homotopy functor”, E_{2}(c, κ), “stably excisive” and “excisive” then also make sense for functors S/X → S, S/X → S/Y, R(X) → R(Y), R(∗, H) → R(∗, G), etc.

Proposition 2.2. Let X be a simplicial set, Φ: S/X → S a functor, Y = Φ(X) a simplicial set, and Ψ: S/Y → S a functor. Suppose that Φ and Ψ are bounded below and stably excisive. Then the composite functor Ψ ∘ Φ: S/X → S is also bounded below and stably excisive.

Proof. Suppose that Φ and Ψ satisfy E_{1}(c, κ) and E_{2}(c, κ), where we may assume that c ≥ 1 and κ ≥ 0. We claim that Ψ∘Φ satisfies E_{1}(2c, κ+c) and E_{2}(3c+1, κ+c). The first claim is clear. For the second, consider a cocartesian diagram as in (2.1), with X_{0} → X_{i} k_{i}-connected for i = 1, 2, and k_{i} ≥ κ + c. Apply Φ to get a (k_{1} + k_{2} − c)-cartesian square

(2.3)  \xymatrix{Φ(X_{0}) \ar[r] \ar[d] & Φ(X_{1}) \ar[d] \\ Φ(X_{2}) \ar[r] & Φ(X_{3})}

with Φ(X_{0}) → Φ(X_{i}) (k_{i} − c)-connected for i = 1, 2. Let

P O = hocolim(Φ(X_{1}) ← Φ(X_{0}) → Φ(X_{2}))

be the homotopy pushout in this square. By homotopy excision (cf. [Ha02, 4.23]) the cocartesian square

(2.4)  \xymatrix{Φ(X_{0}) \ar[r] \ar[d] & Φ(X_{1}) \ar[d] \\ Φ(X_{2}) \ar[r] & P O}

is (k_{1} + k_{2} − 2c − 1)-cartesian. It follows by comparison of (2.3) and (2.4) that the canonical map P O → Φ(X_{3}) is (k_{1}+k_{2}−2c)-connected (when 2c+1 ≥ c). Applying Ψ to (2.4), we obtain a (k_{1} +k_{2} −3c)-cartesian square. The map Ψ(P O) → ΨΦ(X_{3}) is (k_{1} + k_{2} − 3c)-connected (when 2κ ≥ κ), so the square obtained by applying Ψ to (2.3) is (k_{1} + k_{2} − 3c − 1)-cartesian. □

A weak homotopy functor F: S → S satisfies the colimit axiom if it preserves filtered homotopy colimits up to weak homotopy. This means that for any filtered diagram X: D → S the canonical map

(2.5)  hocolim_{d∈D} F (X_{d}) ⟶ F (hocolim_{d∈D} X_{d})

is a weak equivalence. Any simplicial set is weakly equivalent to the homotopy colimit of the filtered diagram of its finite sub-objects, where a simplicial set is finite if it has only finitely many non-degenerate simplices. Thus a functor satisfying the colimit axiom is determined by its restriction to the subcategory of finite simplicial sets. Such functors are therefore also said to be finitary.

Similarly, a functor Φ: R(W, G) → S satisfies the colimit axiom if it preserves filtered homotopy colimits up to weak equivalence. An object of R(W, G) is said to be finite if it can be obtained from W by attaching finitely many free G-cells. Again a functor satisfying the colimit axiom is determined by its restriction to the finite objects in R(W, G).

The forgetful functor u: S/X → S preserves filtered homotopy colimits. Hence if F: S → S satisfies the colimit axiom, then so does the composite functor Φ = F ∘ u: S/X → S.

Remark 2.6. Let S^{κ}/X be the full subcategory of S/X with objects the κ-connected maps X′ → X. The conditions E_{1}(c, κ) and E_{2}(c, κ) then make sense for functors Φ: S^{κ}/X → S, and all of the results of this paper also apply to functors with such a restricted domain of definition. One could even consider germs of functors S/X → S, i.e., equivalence classes of functors Φ: S^{κ}/X → S defined for some integer κ, with two such functors Φ and Φ′: S^{κ′}/X → S considered to be equivalent if there is a κ′′ such that Φ|S^{κ′′}/X = Φ′|S^{κ′′}/X. For simplicity we will not include this extra generality in our notation.

3. Excisive approximation

If X′ → X is an object of S/X, its fiberwise (unreduced) cone C_{X}X′ is the mapping cylinder X′ × Δ^{1} ∪_{X′} X, and its fiberwise (unreduced) suspension S_{X}X′ is the union of two such mapping cylinders along X′. There is a cocartesian square of simplicial sets over X:

\xymatrix{X′ \ar[r] \ar[d] & C_{X}X′ \ar[d] \\ C_{X}X′ \ar[r] & S_{X}X′}

Consider a weak homotopy functor Φ: S/X → S. To Φ, we associate the functor T Φ: S/X → S given by

(T Φ)(X′) = holim(Φ(C_{X}X′) → Φ(S_{X}X′) ← Φ(C_{X}X′)) .

There is a natural map tΦ: Φ → T Φ. Define T^{n}Φ: S/X → S for n ≥ 0 by iteration, and let P Φ: S/X → S be the homotopy colimit

(P Φ)(X′) = hocolim_{n}(T^{n}Φ)(X′) .

Again there is a natural map pΦ: Φ → P Φ, as functors S/X → S. (Cf. [Go90, 1.10].)

We know that P Φ lifts to a functor S/X → S/P Φ(X), but we wish to modify it to a functor S/X → S/Y, with Y = Φ(X). There is a commutative square

\xymatrix{Φ(X′) \ar[r]^{pΦ(X′)} \ar[d] & P Φ(X′) \ar[d] \\ Φ(X) \ar[r]^{pΦ(X)}_{≃} & P Φ(X)}

induced by the unique morphism X′ → X in S/X. The lower horizontal map is a weak equivalence by inspection of the construction of P Φ, using that Φ was assumed to preserve weak equivalences. We set

P′Φ(X′) = holim(P Φ(X′) → P Φ(X) ← Φ(X))

equal to the homotopy limit (pullback) of the lower right hand part of the diagram. The diagram then extends to

\xymatrix{Φ(X′) \ar[r]^{p′Φ(X′)} \ar[d] & P′Φ(X′) \ar[r]^{≃} \ar[d] & P Φ(X′) \ar[d] \\ Φ(X) \ar[r]^{=} & Φ(X) \ar[r]^{pΦ}_{≃} & P Φ(X) ,}

where the right hand square only commutes up to homotopy. Thus we can view Φ and P′Φ as functors S/X → S/Y, in which case there is a natural map p′Φ: Φ → P′Φ. Viewing Φ and P′Φ as functors to S, the natural map pΦ factors as

(3.1)  pΦ: Φ ⟶^{p′Φ} P′Φ ⟶^{≃} P Φ ,

where the right hand map is a natural weak equivalence.

Remark 3.2. Note that P′Φ(X) is typically not equal to Φ(X) = Y, although the canonical map P′Φ(X) → Y is a weak equivalence, so P′Φ: S/X → S/Y is not the canonical lift of its forgetful version u ∘ P′Φ: S/X → S.

Suppose now that Φ: S/X → S satisfies condition E_{2}(c, κ), hence is stably excisive. When the structure map X′ → X is k-connected, for k ≥ κ, it follows that the maps tΦ(X′): Φ(X′) → T Φ(X′), pΦ(X′): Φ(X′) → P Φ(X′) and p′Φ(X′): Φ(X′) → P′Φ(X′) are all (2k − c)-connected.

We say that two functors Φ, Φ′: S/X → S satisfy condition O(c, κ) along a natural map f: Φ → Φ′ if whenever X′ → X is k-connected and k ≥ κ, then f(X′): Φ(X′) → Φ′(X′) is (2k −c)-connected. If Φ and Φ′ satisfy condition O(c, κ) for some integers c and κ, then we say that Φ and Φ′ agree to first order along f. (Cf. [Go90, 1.13].)

Proposition 3.3. Let X be a simplicial set, Φ: S/X → S a stably excisive weak homotopy functor, and let Y = Φ(X). Then P′Φ: S/X → S/Y is excisive, and Φ and P′Φ: S/X → S/Y agree to first order along p′Φ. If Φ is bounded below, then so is P′Φ. If Φ satisfies the colimit axiom, then so does P′Φ.

Proof. The functor P Φ is excisive, and Φ agrees with P Φ to first order along pΦ, by [Go90, 1.14]. In view of the weak equivalence in (3.1), the same applies to P′Φ. If Φ satisfies E_{1}(c, κ), then T Φ satisfies E_{1}(c, κ − 1), hence P Φ and P′Φ satisfy E_{1}(c, κ′) for all κ′. So P′Φ is bounded below. If Φ satisfies the colimit axiom, then so does T Φ, because filtered homotopy colimits commute with homotopy pullbacks, up to weak equivalence. Hence also P Φ (and P′Φ) satisfies the colimit axiom, since the order of two homotopy colimits can be commuted. □

4. Goodwillie derivatives

Let X be a simplicial set, let Φ: S/X → S be a weak homotopy functor, and let Y = Φ(X). As before, we may view Φ as a functor S/X → S/Y or R(X) → R(Y), without change in notation. Choose base points x ∈ X and y ∈ Y.

There is a functor i_{0} = i_{0}(X, x): S_{∗} → R(X) that takes a based simplicial set T to the retractive simplicial set

i_{0}(T) = X ∨_{x} T ,

where r: X ∨_{x} T → X takes T to the base point x ∈ X, and s: X → X ∨_{x} T is the standard inclusion. This functor preserves cocartesian squares.

There is a second functor j_{0} = j_{0}(Y, y): R(Y) → S_{∗} that takes a retractive simplicial set (Y′, r, s) to the homotopy fiber

j_{0}(Y′) = hofib_{y}(r: Y′ → Y) ,

(with the natural base point that maps to s(y) ∈ Y′). This functor preserves k-cartesian squares for all k, hence also cartesian squares.

We shall later consider equivariant improvements i and j of i_{0} and j_{0}, respectively, which may justify the notation.

If Φ is an excisive weak homotopy functor, then the composite functor

L: S_{∗} ⟶^{i_{0}} R(X) ⟶^{Φ} R(Y) ⟶^{j_{0}} S_{∗}

is an excisive weak homotopy functor that takes ∗ to L(∗) = hofib_{y}(Y → Y), which is contractible. We say that L is a linear functor. It corresponds to a generalized (reduced) homology theory given by L_{∗}(T) = π_{∗}(L(T)), with an associated coefficient spectrum L = {n ↦ L(S^{n})} (modulo a technical rectification, as in [Go90, 0.1]). There is a natural weak equivalence Ω^{∞}(L ∧ T) → L(T), at least for finite simplicial sets T.

Even if Φ is not excisive, we can still form the composite functor j_{0} ∘ Φ ∘ i_{0} and assemble the based simplicial sets (j_{0} ∘ Φ ∘ i_{0})(S^{n}) into a spectrum. Thus for any weak homotopy functor Φ: S/X → S let

(4.1)  ∂_{y}^{x}Φ(X)_{n} = hofib_{y}(Φ(X ∨_{x} S^{n}) → Φ(X))

for n ≥ 0, and let

(4.2)  ∂_{y}^{x}Φ(X) = {n ↦ ∂_{y}^{x}Φ(X)_{n}}

be the spectrum obtained (using [Go90, 0.1]) from the indicated sequence of based simplicial sets. By definition, ∂_{y}^{x}Φ(X) is the Goodwillie derivative of Φ at X, with respect to the base points x ∈ X and y ∈ Y = Φ(X). (Cf. [Go90, 1.16].)

A natural map f: Φ → Φ′ of functors S/X → S induces maps ∂f: ∂_{y}^{x}Φ(X)_{n} → ∂_{y′}^{x}Φ′(X)_{n} for all n ≥ 0, and a spectrum map ∂f: ∂_{y}^{x}Φ(X) → ∂_{y′}^{x}Φ′(X). This presupposes that Φ′(X) is given the base point y′ = f(X)(y), where y is the chosen base point in Φ(X) and f(X): Φ(X) → Φ′(X).

Proposition 4.3. If Φ and Φ′: S/X → S agree to first order along f: Φ → Φ′, then f induces a meta-stable equivalence of spectra ∂f: ∂_{y}^{x}Φ(X) → ∂_{y′}^{x}Φ′(X).

Proof. This is basically [Go90, 1.17]. Suppose that Φ and Φ′ satisfy O(c, κ). The retraction X ∨_{x} S^{n} → X is n-connected, so for n ≥ κ the map f(X ∨_{x} S^{n}): Φ(X ∨_{x} S^{n}) → Φ′(X ∨_{x} S^{n}) is (2n − c)-connected. Similarly f(X): Φ(X) → Φ′(X) is a weak equivalence. Hence the map of homotopy fibers ∂f: ∂_{y}^{x}Φ(X)_{n} → ∂_{y′}^{x}Φ′(X)_{n} is (2n − c)-connected. □

Let X be a simplicial set, Φ: S/X → S a weak homotopy functor, Y = Φ(X), and choose base points x ∈ X and y ∈ Y. Give P′Φ(X) the base point y′ = p′Φ(X)(y).

Corollary 4.4. If Φ is stably excisive, then p′Φ induces a meta-stable equivalence of spectra ∂(p′Φ): ∂_{y}^{x}Φ(X) → ∂_{y′}^{x}(P′Φ)(X).

When F: S → S is a weak homotopy functor, X a simplicial set, Y = F (X), x ∈ X, y ∈ Y and Φ = F ∘ u, we let ∂_{y}^{x}F (X)_{n} = ∂_{y}^{x}Φ(X)_{n} = hofib_{y}(F (X ∨_{x} S^{n}) → F (X)) and ∂_{y}^{x}F (X) = ∂_{y}^{x}Φ(X) = {n ↦ ∂_{y}^{x}F (X)_{n}}.

5. Composite functors

Let X be a simplicial set, Φ: S/X → S a functor, Y = Φ(X), Ψ: S/Y → S, and Z = Ψ(Y). Suppose that Φ and Ψ are weak homotopy functors. Let Y′ = P′Φ(X), Z′ = Ψ(Y′) and Z′′ = P′Ψ(Y′). Choose base points x ∈ X, y ∈ Y z ∈ Z, and let y′ = p′Φ(X)(x) ∈ Y′, z′ = Ψ(p′Φ)(X)(z) ∈ Z′ and z′′ = p′Ψ(Y′)(z′) ∈ Z′′.

[figure: X; Φ; P′Φ; Y = Φ(X); Ψ; p′Φ(X); ≃; Z = Ψ(Y); Ψ(p′Φ(X)); ≃; Y′ = P′Φ(X); Ψ; P′Ψ; Z′ = Ψ(Y′); p′Ψ(Y′); ≃; Z′′ = P′Ψ(Y′)]

Proposition 5.1. Suppose that Φ and Ψ are bounded below, stably excisive functors. Then the composite functors Ψ ∘ Φ and P′Ψ ∘ P′Φ: S/X → S agree to first order along p′Ψ ∘ Ψ(p′Φ) = P′Ψ(p′Φ) ∘ p′Ψ.

Proof. Assume that Φ and Ψ satisfy E_{1}(c, κ) and E_{2}(c, κ), for some sufficiently large integers c and κ. Let X′ → X be a k-connected map, with k ≥ κ + c. Then p′Φ(X′): Φ(X′) → P′Φ(X′) is (2k − c)-connected, so Ψ(p′Φ)(X′): (Ψ ∘ Φ)(X′) → (Ψ ∘ P′Φ)(X′) is (2k − 2c)-connected. Furthermore, P′Φ(X′) → P′Φ(X) is (k − c)- connected by proposition 3.3, so p′Ψ(P′Φ(X′)): (Ψ ∘ P′Φ)(X′) → (P′Ψ ∘ P′Φ)(X′) is (2k − 3c)-connected. Thus p′Ψ ∘ Ψ(p′Φ) satisfies O(3c, κ + c), and Ψ ∘ Φ and P′Ψ ∘ P′Φ: S/X → S agree to first order. □

Recall also that for Φ and Ψ bounded below and stably excisive the composite functor Ψ ∘ Φ: S/X → S is bounded below and stably excisive (proposition 2.2), hence agrees to first order with P′(Ψ ∘ Φ): S/X → S (proposition 3.3). We are therefore legitimately interested in its derivative ∂_{z}^{x}(Ψ ∘ Φ)(X).

Proposition 5.2. Suppose that Φ and Ψ are bounded below, stably excisive functors, with notation as above. Then there are natural meta-stable equivalences

(1) ∂(p′Φ): ∂_{y}^{x}Φ(X) ≃ ∂_{y′}^{x}P′Φ(X),

(2) ∂(p′Ψ): ∂_{z′}^{y′}Ψ(Y′) ≃ ∂_{z′′}^{y′}(P′Ψ)(Y′),

(3) ∂(p′Ψ ∘ Ψ(p′Φ)): ∂_{z}^{x}(Ψ ∘ Φ)(X) ≃ ∂_{z′′}^{x}(P′Ψ ∘ P′Φ)(X) and

(4) ∂_{z}^{y}Ψ(Y) ≃ ∂_{z′}^{y′}Ψ(Y′).

Proof. By propositions 3.3 and 5.1, the pairs of functors Φ and P′Φ, Ψ and P′Ψ, and the functors Ψ ∘ Φ and P′Ψ ∘ P′Φ agree to first order, respectively. Hence their derivatives are meta-stably equivalent by proposition 4.3.

Case (4) remains. There is a commutative square in S/Y

\xymatrix{Y ∨_{y} S^{n} \ar[d] \ar[r]^{≃} & Y′ ∨_{y′} S^{n} \ar[d] \\ Y \ar[r]_{p′Φ(X)}^{≃} & Y′}

where the vertical maps take S^{n} to the respective base points, and the upper horizontal map is the identity on S^{n}. The lower horizontal map is a weak equivalence, as in remark 3.2, hence so is the upper horizontal map. Applying Ψ and taking vertical homotopy fibers yields a weak equivalence of n-th spaces

∂_{z}^{y}Ψ(Y)_{n} → ∂_{z′}^{y′}Ψ(Y′)_{n} .

Thus the associated spectra are strictly equivalent. □

Remark 5.3. It follows that for the purpose of expressing the derivative of Ψ ∘ Φ in terms of the derivatives of Φ and Ψ, we are free to replace the bounded below, stably excisive functors Φ and Ψ by their bounded below, excisive approximations u ∘ P′Φ: S/X → S and u ∘ P′Ψ: S/Y → S, respectively. If Ψ satisfies the colimit axiom, then so does its replacement.

Equivalently, we may assume that Φ and Ψ are themselves bounded below, excisive functors. Furthermore, the derivatives only depend on the associated functors Φ: R(X) → R(Y), Ψ: R(X) → R(Y) with composite Ψ ∘ Φ: R(X) → R(Z).

6. Multiple connected components

We now reduce to the case when X, Y and Z are connected.

Let Φ: R(X) → R(Y) and Ψ: R(Y) → R(Z) be weak homotopy functors, with Y = Φ(X) and Z = Ψ(Y). Write X = ∐_{α∈A} X_{α}, Y = ∐_{β∈B} Y_{β} and Z = ∐_{γ∈C} Z_{γ}, where each X_{α}, Y_{β} and Z_{γ} is connected. So A = π_{0}(X), B = π_{0}(Y) and C = π_{0}(Z). Choose base points x_{α} ∈ X_{α}, y_{β} ∈ Y_{β} and z_{γ} ∈ Z_{γ} for all α, β and γ.

Let in_{α}: R(X_{α}) → R(X) be given by pushout along X_{α} ⊂ X, so in_{α}(X′_{α}) = X ∪_{X_{α}} X′_{α}. Similarly let pr_{β}: R(Y) → R(Y_{β}) be given by pullback along Y_{β} ⊂ Y, so pr_{β}(Y′) = Y_{β} ×_{Y} Y′. Let Φ_{β}^{α} = pr_{β} ∘Φ ∘ in_{α}: R(X_{α}) → R(Y_{β}). Clearly in_{α} preserves k-connected maps and (k-)cocartesian squares, while pr_{β} preserves kconnected maps and (k-)cartesian squares. So if Φ is bounded below, excisive, stably excisive or satisfies the colimit axiom, then the same applies to Φ_{β}^{α}.

Lemma 6.1. There is a natural strict equivalence

∂_{y_{β}}^{x_{α}}Φ(X) ≃ ∂_{y_{β}}^{x_{α}}Φ_{β}^{α}(X_{α}) .

Proof. Let X′_{α} = X_{α} ∨_{x_{α}} S^{n}, X′ = in_{α}(X′_{α}) ≅ X ∨_{x_{α}} S^{n}, Y′ = Φ(X′) and Y′_{β} = pr_{β}(Y′), so that Y′_{β} = Φ_{β}^{α}(X′_{α}). The pullback square

\xymatrix{Y′_{β} \ar[r] \ar[d] & Y′ \ar[d] \\ Y_{β} \ar[r] & Y}

is cartesian, so there is a weak equivalence

∂_{y_{β}}^{x_{α}}Φ(X)_{n} ≅ hofib_{y_{β}}(Y′ → Y) ≃ hofib_{y_{β}}(Y′_{β} → Y_{β}) = ∂_{y_{β}}^{x_{α}}Φ_{y_{β}}^{x_{α}}(X_{α})_{n} .

□

Let Ψ^{β} = Ψ ∘ in_{β}: R(Y_{β}) → R(Z) and consider a base point z ∈ Z. For each Y′ in R(Y) and β ∈ B let Y′_{β} = pr_{β}(Y′) in R(Y_{β}).

Proposition 6.2. Let Ψ: R(Y) → R(Z) be a bounded below, excisive functor that satisfies the colimit axiom. Then the functors Y′ ↦ hofib_{z}(Ψ(Y′) → Z) and

Y′ ↦ ⋁_{β∈B} hofib_{z}(Ψ^{β}(Y′_{β}) → Z)

agree to first order along a natural chain of maps.

Proof. The retraction Y′ → in_{β}(Y′_{β}) induces a retraction Ψ(Y′) → Ψ^{β}(Y′_{β}). More generally, for each finite subset S ⊂ B let

Y_{S}^{∗} = ∐_{β∈S} Y′_{β} ∪ ∐_{β∉S} Y_{β} .

There is then a strongly cocartesian (cf. [Go92, 2.1]) cubical diagram

(T ⊂ S) ⟼ Y_{S\T}^{∗}

in R(Y). Applying the excisive functor Ψ yields a strongly cartesian cubical diagram, where each map admits a section. Hence there is a weak equivalence

hofib_{z}(Ψ(Y_{S}^{∗}) → Z) ⟶^{≃} ∏_{β∈S} hofib_{z}(Ψ^{β}(Y′_{β}) → Z) .

Passing to homotopy colimits over S ⊂ B, and using that Ψ satisfies the colimit axiom, yields a weak equivalence

hofib_{z}(Ψ(Y′) → Z) ⟶^{≃} hocolim_{S⊂B} ∏_{β∈S} hofib_{z}(Ψ^{β}(Y′_{β}) → Z) .

When Ψ satisfies E_{1}(c, κ) and Y′ → Y is k-connected, with k ≥ κ, then in_{β}(Y′_{β}) → Y is k-connected and Ψ^{β}(Y′_{β}) → Z is (k − c)-connected, for each β. So the inclusion

hocolim_{S⊂B} ⋁_{β∈S} hofib_{z}(Ψ^{β}(Y′_{β}) → Z) ⟶ hocolim_{S⊂B} ∏_{β∈S} hofib_{z}(Ψ^{β}(Y′_{β}) → Z)

is (2k − 2c − 1)-connected. The source of this map is naturally equivalent to ⋁_{β∈B} hofib_{z}(Ψ^{β}(Y′_{β}) → Z). □

Let Φ^{α} = Φ ∘ in_{α}: R(X_{α}) → R(Y) and Ψ_{γ} = pr_{γ} ∘Ψ: R(Y) → R(Z_{γ}).

Proposition 6.3. Let Φ: R(X) → R(Y) and Ψ: R(Y) → R(Z) be bounded below, and suppose that Ψ is excisive and satisfies the colimit axiom. With notation as above, there is a natural chain of meta-stable equivalences

∂_{z_{γ}}^{x_{α}}(Ψ ∘ Φ)(X) ≃ ∂_{z_{γ}}^{x_{α}}(Ψ_{γ} ∘ Φ^{α})(X_{α}) ≃ ⋁_{β∈B} ∂_{z_{γ}}^{x_{α}}(Ψ_{γ}^{β} ∘ Φ_{β}^{α})(X_{α}) .

Proof. The first strict equivalence is a case of lemma 6.1. To establish the second meta-stable equivalence we keep the notation introduced in the last two proofs. Then

∂_{z_{γ}}^{x_{α}}(Ψ_{γ} ∘ Φ^{α})(X_{α})_{n} = hofib_{z_{γ}} ((Ψ_{γ} ∘ Φ^{α})(X′_{α}) → Z_{γ})

= hofib_{z_{γ}} (Ψ_{γ}(Y′) → Z_{γ})

∼ ⋁_{β∈B} hofib_{z_{γ}} (Ψ_{γ}^{β}(Y′_{β}) → Z_{γ})

= ⋁_{β∈B} hofib_{z_{γ}} ((Ψ_{γ}^{β} ∘ Φ_{β}^{α})(X′_{α}) → Z_{γ})

= ⋁_{β∈B} ∂_{z_{γ}}^{x_{α}}(Ψ_{γ}^{β} ∘ Φ_{β}^{α})(X_{α})_{n}

where ∼ denotes agreement up to first order as functors of Y′, by proposition 6.2. Since Φ is bounded below, the map Y′ → Y is (n−c)-connected for some constant c, and so this chain of maps is (2n − c)-connected for some (other) c. Hence the associated spectrum map is a meta-stable equivalence. □

7. The Kan loop group

By the results of section 5 we need only consider bounded below, excisive functors Φ: R(X) → R(Y) and Ψ: R(Y) → R(Z), where X is a simplicial set, Y = Φ(X) and Z = Ψ(Y), and by section 6 we may assume that X, Y and Z are all connected. Choose base points x ∈ X, y ∈ Y and z ∈ Z, and recall the functors i_{0} = i_{0}(X, x) and j_{0} = j_{0}(Z, z). We are interested in the coefficient spectrum of the composite functor:

S_{∗} ⟶^{i_{0}} R(X) ⟶^{Φ} R(Y) ⟶^{Ψ} R(X) ⟶^{j_{0}} S_{∗} .

We would like to be able to express this in terms of the coefficient spectra of the composites j_{0}Φi_{0} : S_{∗} → R(X) → R(Y) → S_{∗} and j_{0}Ψi_{0} : S_{∗} → R(Y) → R(Z) → S_{∗}. But, as consideration of the two special cases Φ = P_{X}(id) and Ψ = P_{Y} (id) indicates, where id: S → S is the identity functor, this is not likely to be possible. Composition with the functors j_{0} : R(Y) → S_{∗} and i_{0} : S_{∗} → R(Y) does not retain enough information.

We shall instead replace the category S_{∗} = R(∗) by the category R(∗, Ω_{y}(Y)), where Ω_{y}(Y) is the Kan loop group of (Y, y), in which case there are suitable inverse weak equivalences i: R(∗, Ω_{y}(Y)) → R(Y) and j: R(Y) → R(∗, Ω_{y}(Y)). The Goodwillie derivative ∂_{y}^{x}Φ(X) then becomes a free left Ω_{y}(Y)-spectrum. Furthermore, there is a model for the Goodwillie derivative ∂_{z}^{y}Ψ(Y) that becomes a spectrum with right Ω_{y}(Y)-action. It turns out that these extra simplicial group actions suffice to express the derivative ∂_{z}^{x}(Ψ ∘ Φ)(X) in terms of these equivariant Goodwillie derivatives, leading to the chain rule.

Suppose now that Y is a connected simplicial set, with a chosen base point y ∈ Y. The Kan loop group of Y is then a functorially defined simplicial group Ω_{y}(Y). (Cf. [Wa96], where the Kan loop group Ω_{y}(Y) is denoted G(Y).) There is a principal Ω_{y}(Y)-bundle

Ω_{y}(Y) ⟶ Γ̃(Y) ⟶^{π} Γ(Y)

with Γ̃(Y) weakly contractible, and a natural inclusion γ: Y → Γ(Y) which is a weak equivalence. There are natural base points in Γ(Y) and Γ̃(Y), and π and γ preserve these points. Let ϵ: Γ̃(Y) → ∗ be the (unique) weak equivalence.

Proposition 7.1. There are natural weak homotopy functors

i: R(∗, Ω_{y}(Y)) → R(Y)
j: R(Y) → R(∗, Ω_{y}(Y)) ,

such that i = i(Y, y) preserves k-connected maps and (k-)cocartesian squares, and j = j(Y, y) preserves k-connected maps and (k-)cartesian squares. There is a natural weak equivalence from the identity functor on R(∗, Ω_{y}(Y)) to the composite j ∘ i: R(∗, Ω_{y}(Y)) → R(∗, Ω_{y}(Y)).

Remark 7.2. This is closely related to [Wa85, 2.1.4]. There is also a natural chain of weak equivalences from the composite i∘j: R(Y) → R(Y) to the identity functor on R(Y), but we shall not make any use of it in this paper.

Proof. We construct i as a composite

i: R(∗, Ω_{y}(Y)) ⟶^{ϵ^{∗}} R(Γ̃(Y), Ω_{y}(Y)) ⟶^{π_{!}} R(Γ(Y)) ⟶^{Rγ^{∗}} R(Y)

and j as a composite

j: R(Y) ⟶^{γ_{∗}} R(Γ(Y)) ⟶^{π^{!}} R(Γ̃(Y), Ω_{y}(Y)) ⟶^{ϵ_{∗}} R(∗, Ω_{y}(Y)) .

The maps ϵ_{∗} and ϵ^{∗} are given by pushout and pullback along ϵ: Γ̃(Y) → ∗, respectively. These are weak homotopy functors because the structural section s is always a cofibration, and the map ϵ is a weak equivalence. On an object (E, r, s) of R(Γ̃(Y), Ω_{y}(Y)) the composite ϵ^{∗}ϵ_{∗}(E) equals Γ̃(Y) × (∗ ∪_{Γ̃(Y)} E), and there is a natural weak equivalence from the identity on R(Γ̃(Y), Ω_{y}(Y)) to ϵ^{∗} ∘ ϵ_{∗} in view of the commutative diagram

\xymatrix{E \ar[rr]^{≃} \ar[dr]_{≃} &  & Γ̃(Y) × (∗ ∪_{Γ̃(Y)} E) \ar@{-->}[dl]^{≃}_{_{pr_{2}}} \\  & ∗ ∪_{Γ̃(Y)} E .}

Here the lower left arrow collapses the image of s: Γ̃(Y) → E to a point, while the upper horizontal arrow has components r: E → Γ̃(Y) and the map just mentioned.

The maps π_{!} and π^{!} are given by passage to Ω_{y}(Y)-orbits and pullback along π: Γ̃(Y) → Γ(Y), respectively. These are weak homotopy functors because objects of R(Γ̃(Y), Ω_{y}(Y)) are free Ω_{y}(Y)-simplicial sets, and the bundle projection π is a Kan fibration. On an object (W, r, s) of R(Γ(Y)) the composite π_{!}π^{!}(W) = (Γ̃(Y) ×_{Γ(Y)} W)/Ω_{y}(Y) is naturally isomorphic to W, so there is a natural isomorphism from the identity on R(Γ(Y)) to π_{!} ∘ π^{!}.

The maps γ_{∗} and Rγ^{∗} are given by pushout and homotopy pullback along γ: Y → Γ(Y). These are weak homotopy functors because γ is a cofibration, and because we take homotopy pullback (rather than pullback) along γ to define Rγ^{∗}. On an object (Y′, r, s) of R(Y) the composite Rγ^{∗}γ_{∗}(Y′) equals the homotopy pullback of Y and Γ(Y)∪_{Y} Y′ over Γ(Y). There is a natural weak equivalence from the identity on R(Y) to Rγ^{∗} ∘ γ_{∗}, in view of the commutative diagram

\xymatrix{Y′ \ar[rr]^{≃} \ar[dr]_{≃} &  & holim(Y → Γ(Y) ← Γ(Y) ∪_{Y} Y′) \ar[dl]^{≃} \\  & Γ(Y) ∪_{Y} Y′ .}

Here the lower left arrow is the canonical inclusion, the lower right arrow is the canonical projection from the homotopy pullback, and the upper horizontal arrow is induced by r: Y′ → Y and the inclusion just mentioned.

It is clear by inspection that ϵ^{∗}, π_{!} and Rγ^{∗} preserve k-connected maps and cartesian squares, and that ϵ_{∗}, π^{!} and γ_{∗} preserve k-connected maps and cocartesian squares. □

Proposition 7.3. The functor i_{0} : S_{∗} → R(Y) factors up to a natural chain of weak equivalences as the composite

i_{0} : S_{∗} ⟶^{Ω_{y}(Y)_{+}∧(−)} R(∗, Ω_{y}(Y)) ⟶^{i} R(Y) ,

where the left hand functor takes T to Ω_{y}(Y)_{+} ∧ T.

The functor j_{0} : R(Y) → S_{∗} factors up to a natural chain of weak equivalences as the composite

j_{0} : R(Y) ⟶^{j} R(∗, Ω_{y}(Y)) ⟶^{w} S_{∗} ,

where the right hand functor forgets the Ω_{y}(Y)-action.

Proof. Let T be a based simplicial set. We interpret Ω_{y}(Y)_{+} ∧ T as the pushout of

∗ ← Ω_{y}(Y) × ∗ → Ω_{y}(Y) × T

and apply π_{!}ϵ^{∗} to obtain the pushout square

\xymatrix{Γ̃(Y) \ar[r] \ar[d]^{π} & Γ̃(Y) × T \ar[d] \\ Γ(Y) \ar[r] & (Γ̃(Y) × (Ω_{y}(Y)_{+} ∧ T))/Ω_{y}(Y) .}

Using the weak equivalence γ: Y → Γ(Y), and the inclusion of the canonical base point into Γ̃(Y), we obtain a map from the pushout square

\xymatrix{∗ \ar[r] \ar[d] & T \ar[d] \\ Y \ar[r] & i_{0}(T)}

to the square above, which is a weak equivalence at each corner. This yields a natural weak equivalence

i_{0}(T) = Y ∨_{y} T ⟶^{≃} π_{!}ϵ^{∗}(Ω_{y}(Y)_{+} ∧ T) .

The canonical projection from the homotopy pullback defining Rγ^{∗} gives another natural map

i(Ω_{y}(Y)_{+} ∧ T) = Rγ^{∗}π_{!}ϵ^{∗}(Ω_{y}(Y)_{+} ∧ T) ⟶^{≃} π_{!}ϵ^{∗}(Ω_{y}(Y)_{+} ∧ T) ,

which is a weak equivalence because γ is one. Taken together, these two maps define the required natural chain of weak equivalences.

Next, let (Y′, r, s) be a retractive simplicial set in R(Y). We evaluate j = ϵ_{∗}π^{!}γ_{∗} on Y′ and obtain a commutative diagram in S_{∗}:

\xymatrix{Y′ \ar[d] \ar[r]^{≃} & Γ(Y) ∪_{Y} Y′ \ar[d] & Γ̃(Y) ×_{Γ(Y)} (Γ(Y) ∪_{Y} Y′) \ar@{->>}[l] \ar[d] \ar[r]^{≃} & j(Y′) \ar[d] \\ Y \ar[r]^{≃}_{γ} & Γ(Y) & Γ̃(Y) \ar@{->>}[l]^{π} \ar[r]^{≃}_{ϵ} & ∗}

The left and right hand squares have horizontal weak equivalences, while the middle square is cartesian, using again that π is a Kan fibration. Hence the induced maps of vertical homotopy fibers at the canonical base points of Y, Γ(Y), Γ̃(Y) and ∗ define a natural chain of weak equivalences linking j_{0}(Y′) = hofib_{y}(Y′ → Y) to hofib(j(Y′) → ∗), and thus to j(Y′), as based simplicial sets. □

Lemma 7.4. Let Φ: R(X) → R(Y) be a weak homotopy functor, and let ϕ = j ∘ Φ ∘ i: R(∗, Ω_{x}(X)) → R(∗, Ω_{y}(Y)). If Φ satisfies the colimit axiom, then so does ϕ.

Proof. By an easy inspection of the definitions, the functors i = Rγ^{∗} ∘ π_{!} ∘ ϵ^{∗} and j = ϵ_{∗} ∘ π^{!} ∘ γ_{∗} preserve filtered homotopy colimits up to weak equivalence. □

8. Equivariant derivatives

We now recast the Goodwillie derivative in an equivariant setting.

Let X be a simplicial set, let Φ: S/X → Y be a weak homotopy functor, and let Y = Φ(X). We now view Φ as a functor Φ: R(X) → R(Y). First assume that X and Y are connected, and choose base points x ∈ X and y ∈ Y. Let H = Ω_{x}(X) and G = Ω_{y}(Y) be the Kan loop groups of (X, x) and (Y, y), respectively.

Recall the functors i = i(X, x): R(∗, H) → R(X) and j = j(Y, y): R(Y) → R(∗, G). Let ϕ: R(∗, H) → R(∗, G) be the composite functor

(8.1)  ϕ: R(∗, H) ⟶^{i} R(X) ⟶^{Φ} R(Y) ⟶^{j} R(∗, G) .

By proposition 7.1, ϕ is also a weak homotopy functor. By inspection, it is pointed in the strong sense that ϕ(∗) = ∗.

The categories R(∗, H) and R(∗, G) are in fact enriched in the category S_{∗} of based simplicial sets. Given objects U and V in R(∗, H), the (based) simplicial mapping space Map_{H}(U, V) is the based simplicial set with p-simplices the set of morphisms Δ_{+}^{p} ∧U → V in R(∗, H). The usual set of morphisms U → V in R(∗, H) can be recovered as the 0-simplices of Map_{H}(U, V). Similarly, Map_{G}(−, −) is the simplicial mapping space in the category R(∗, G). A pointed simplicial functor ϕ: R(∗, H) → R(∗, G) comes equipped with a based map

Map_{H}(U, V) ⟶^{ϕ} Map_{G}(ϕ(U), ϕ(V))

which on 0-simplices takes a map f: U → V to the usual image ϕ(f): ϕ(U) → ϕ(V).

Any (pointed) weak homotopy functor ϕ can be promoted to a weakly equivalent (pointed) simplicial weak homotopy functor ϕ̌, following [Wa85, 3.1]. For a based, free H-simplicial set W, let W^{Δ^{q}} = Map_{H}(H_{+} ∧ Δ_{+}^{q}, W). Then W^{Δ^{q}} is again a based, free H-simplicial set. The functor [q] ↦ ϕ(W^{Δ^{q}}) is a simplicial object in based, free G-simplicial sets. Let

ϕ̌(W) = diag([q] ↦ ϕ(W^{Δ^{q}})) = ∐_{q≥0}(ϕ(W^{Δ^{q}}) × Δ^{q})/ ∼

be the associated diagonal based, free G-simplicial set. Then ϕ̌ is naturally a simplicial functor. The required map

Map_{H}(U, V) ⟶^{ϕ̌} Map_{G}(ϕ̌(U), ϕ̌(V))

takes a p-simplex f: Δ_{+}^{p} ∧ U → V to the p-simplex ϕ̌(f): Δ_{+}^{p} ∧ ϕ̌(U) → ϕ̌(V) that in simplicial degree q is given by the map

ϕ̌(f)_{q}: ⋁_{α: Δ^{q}→Δ^{p}} ϕ(U^{Δ^{q}}) ≅ (Δ_{q}^{p})_{+} ∧ ϕ(U^{Δ^{q}}) ⟶ ϕ(V^{Δ^{q}})

that on the wedge summand indexed by α is induced by

U^{Δ^{q}} ⟶^{(α,id)} (Δ_{+}^{p} ∧ U)^{Δ^{q}} ⟶^{f^{Δ^{q}}} V^{Δ^{q}}

under ϕ.

Each projection Δ^{q} → ∗ induces a weak equivalence W → W^{Δ^{q}}, and thus each q-fold degeneracy map ϕ(W) → ϕ(W^{Δ^{q}}) is a weak equivalence. Thus the inclusion of 0-simplices is a natural weak equivalence

(8.2)  ϕ ⟶^{≃} ϕ̌

of weak homotopy functors R(∗, H) → R(∗, G), by the realization lemma. We can therefore replace ϕ = j ∘ Φ ∘ i by ϕ̌ without changing its weak homotopy type.

To each based, free H-simplicial set W we associate its cone CW = W ∧ Δ^{1}, and its suspension ΣW = CW ∪_{W} CW = W ∧ S^{1}, where S^{1} = Δ^{1} ∪_{∂Δ^{1}} Δ^{1}. By iteration, Σ^{n}W = W ∧ S^{n}, where S^{n} = S^{1} ∧ ⋯ ∧ S^{1} (n copies of S^{1}).

Using that ϕ̌ is a pointed simplicial functor, there is a natural based map

(8.3)  σ: Σϕ̌(W) → ϕ̌(ΣW)

in R(∗, G). For the identity map W ∧ S^{1} = ΣW is left adjoint to a based map

S^{1} → Map_{H}(W, ΣW) .

Since ϕ̌ is pointed simplicial there is a natural based map

Map_{H}(W, ΣW) ⟶^{ϕ̌} Map_{G}(ϕ̌(W), ϕ̌(ΣW)) .

The composite of these two maps is then right adjoint to the desired map σ.

We consider the sequence of based, free H-simplicial sets Σ^{n}H_{+} = H_{+} ∧ S^{n} in R(∗, H). Applying ϕ̌ we obtain a sequence of based, free G-simplicial sets

(8.4)  ∂ϕ_{n} = ϕ̌(Σ^{n}H_{+}) .

The natural map σ in the case of W = Σ^{n}H_{+} then defines the structure map from Σ∂ϕ_{n} to ∂ϕ_{n+1} in the free G-spectrum

(8.5)  ∂ϕ = {n ↦ ∂ϕ_{n}} .

This is the equivariant Goodwillie derivative of ϕ. By proposition 7.3 and the weak equivalence (8.2), there is a natural chain of weak equivalences linking the underlying based simplicial set of ∂ϕ_{n} to ∂_{y}^{x}Φ(X)_{n}. Similarly the underlying nonequivariant spectrum of ∂ϕ is strictly equivalent to the (non-equivariant) Goodwillie derivative ∂_{y}^{x}Φ(X). Hence ∂ϕ provides a model for the Goodwillie derivative as a free left G-spectrum.

The simplicial enrichment ensures that the Goodwillie derivative ∂ϕ also admits another simplicial group action, this time by H acting from the right. For the multiplication on H defines a map (Σ^{n}H_{+}) ∧ H_{+} → (Σ^{n}H_{+}), which is left adjoint to a based map

H_{+} ⟶ Map_{H}(Σ^{n}H_{+}, Σ^{n}H_{+}) .

Using that ϕ̌: R(∗, H) → R(∗, G) is a pointed simplicial functor, we obtain a map of based simplicial sets

Map_{H}(Σ^{n}H_{+}, Σ^{n}H_{+}) ⟶^{ϕ̌} Map_{G}(ϕ̌(Σ^{n}H_{+}), ϕ̌(Σ^{n}H_{+})) .

The composite of these two based maps is then right adjoint to a map

ϕ̌(Σ^{n}H_{+}) ∧ H_{+} ⟶ ϕ̌(Σ^{n}H_{+}) ,

that defines a right action of H on ϕ̌(Σ^{n}H_{+}) = ∂ϕ_{n} in the category of based, free G-simplicial sets.

The structure maps of ∂ϕ are likewise natural with respect to this right H-action, hence ∂ϕ is also a spectrum with right H-action.

If X or Y are not connected, with base points x ∈ X and y ∈ Y, let X_{α} be the connected component of x in X and let Y_{β} be the connected component of y in Y. Then let Φ_{β}^{α}: R(X_{α}) → R(Y_{β}) be as in section 6, and let ϕ = j ∘ Φ_{β}^{α} ∘ i: R(∗, Ω_{x}(X_{α})) → R(∗, Ω_{y}(Y_{β})). Then we have already seen that ∂ϕ ≃ ∂_{y}^{x}Φ_{β}^{α}(X_{α}), while ∂_{y}^{x}Φ_{β}^{α}(X_{α}) ≃ ∂_{y}^{x}Φ(X) by lemma 6.1.

We summarize this discussion in:

Proposition 8.6. Let X be a simplicial set, Φ: S/X → S a weak homotopy functor, and Y = Φ(X). Let (X_{α}, x) and (Y_{β}, y) be based, connected components of X and Y, with Kan loop groups Ω_{x}(X_{α}) and Ω_{y}(Y_{β}), respectively. Let

ϕ = j ∘ Φ_{β}^{α} ∘ i: R(∗, Ω_{x}(X_{α})) → R(∗, Ω_{y}(Y_{β})) ,

with i = i(X_{α}, x) and j = j(Y_{β}, y), and let

ϕ̌(W) = |[q] ↦ ϕ(W^{Δ^{q}})|

be the simplicial enrichment of ϕ. The equivariant Goodwillie derivative of ϕ is the free left Ω_{y}(Y_{β})-spectrum with right Ω_{x}(X_{α})-action:

∂ϕ = {n ↦ ∂ϕ_{n} = ϕ̌(Σ^{n}Ω_{x}(X_{α})_{+})} .

Its underlying spectrum is strictly equivalent to the (non-equivariant) Goodwillie derivative ∂_{y}^{x}Φ(X). □

9. Equivariant Brown–Whitehead representability

Let H and G be simplicial groups. We say that a functor ϕ: R(∗, H) → R(∗, G) is linear if it is a pointed, excisive, weak homotopy functor. We show in this section that linear functors that satisfy the colimit axiom are classified by their equivariant Goodwillie derivative.

Proposition 9.1. Let ϕ: R(∗, H) → R(∗, G) be a bounded below, linear functor. There is a free left G-spectrum

∂ϕ = {n ↦ ∂ϕ_{n} = ϕ̌(Σ^{n}H_{+})}

with right H-action, and a natural map

θ(W): ∂ϕ ∧_{H} W ⟶ {n ↦ ϕ̌(Σ^{n}W)}

of free G-spectra, which is a meta-stable equivalence for all finite W in R(∗, H). If ϕ satisfies the colimit axiom, then the map is a stable equivalence for all W in R(∗, H).

Proof. The group action map H_{+}∧W → W suspends to a map Σ^{n}H_{+}∧W → Σ^{n}W, which is left adjoint to a based map

W ⟶ Map_{H}(Σ^{n}H_{+}, Σ^{n}W)

of H-simplicial sets, where H acts on the simplicial mapping space by right multiplication in the domain. Using that ϕ̌ is pointed simplicial, there is a based map

Map_{H}(Σ^{n}H_{+}, Σ^{n}W) ⟶^{ϕ̌} Map_{G}(ϕ̌(Σ^{n}H_{+}), ϕ̌(Σ^{n}W))

of H-simplicial sets. The composite map is right adjoint to a map

(9.2)  ϕ̌(Σ^{n}H_{+}) ∧_{H} W ⟶ ϕ̌(Σ^{n}W)

of based, free G-simplicial sets. These maps are compatible with the spectrum structure maps for varying n, hence define the natural map θ(W) of free G-spectra.

To finish the proof, we will use the following lemma.

Lemma 9.3. Let

\xymatrix{W_{0} \ar[r] \ar[d] & W_{1} \ar[d] \\ W_{2} \ar[r] & W_{3}}

be a cocartesian square in R(∗, H). If θ(W_{0}), θ(W_{1}) and θ(W_{2}) are meta-stable equivalences, then so is θ(W_{3}).

Proof. Applying (9.2) to the given cocartesian square yields a map from the square

\xymatrix{ϕ̌(Σ^{n}H_{+}) ∧_{H} W_{0} \ar[r] \ar[d] & ϕ̌(Σ^{n}H_{+}) ∧_{H} W_{1} \ar[d] \\ ϕ̌(Σ^{n}H_{+}) ∧_{H} W_{2} \ar[r] & ϕ̌(Σ^{n}H_{+}) ∧_{H} W_{3}}

to the square

\xymatrix{ϕ̌(Σ^{n}W_{0}) \ar[r] \ar[d] & ϕ̌(Σ^{n}W_{1}) \ar[d] \\ ϕ̌(Σ^{n}W_{2}) \ar[r] & ϕ̌(Σ^{n}W_{3}) .}

The first square is cocartesian with (n − c)-connected maps for some c, since each W_{i} is H-free and ϕ is bounded below. By homotopy excision it is (2n − c)-cartesian for some (other) c. The second square is cartesian since ϕ ≃ ϕ̌ is excisive. By hypothesis the maps in the upper left, upper right and lower left hand corners are (2n − c)-connected, for some c. It follows that the map in the lower right hand corner is also (2n − c)-connected, for some c. Hence θ(W_{3}) is a meta-stable equivalence. □

The map θ(W) is the identity when W = H_{+} and when W = ∗. Hence it is a strict equivalence whenever W is weakly contractible. It follows that θ(Σ^{k}H_{+}) is a meta-stable equivalence for all k ≥ 0, by induction on k, using lemma 9.3 applied to the pushout square

\xymatrix{Σ^{k}H_{+} \ar[r] \ar[d] & CΣ^{k}H_{+} \ar[d] \\ CΣ^{k}H_{+} \ar[r] & Σ^{k+1}H_{+} .}

Likewise it follows that θ(W) is a meta-stable equivalence for all finite objects W in R(∗, H), by induction on the number of free H-cells in W, using lemma 9.3 applied to the pushout square associated to the attachment of a free H-cell. If ϕ satisfies the colimit axiom, then this implies that θ(W) is a stable equivalence for any based, free H-simplicial set W. This completes the proof of 9.1. □

The following form of the Brown–Whitehead representability theorem is perhaps more familiar, although we shall not use it directly.

Proposition 9.4. Let ϕ: R(∗, H) → R(∗, G) be a bounded below, linear functor that satisfies the colimit axiom. There is a natural chain of weak equivalences

Ω^{∞}(∂ϕ ∧_{H} W) ≃ ϕ(W)

for all W in R(∗, H).

Proof. The chain consists of the weak equivalence

Ω^{∞}(∂ϕ ∧_{H} W) ⟶^{≃} Ω^{∞}{n ↦ ϕ̌(Σ^{n}W)}

of proposition 9.1, the weak equivalence

Ω^{∞}{n ↦ ϕ(Σ^{n}W)} ⟶^{≃} Ω^{∞}{n ↦ ϕ̌(Σ^{n}W)

of (8.2), and the weak equivalence

ϕ(W) ⟶^{≃} Ω^{∞}{n ↦ ϕ(Σ^{n}W)}

that follows since ϕ is linear. □

10. The chain rule

Let Φ: R(X) → R(Y) and Ψ: R(Y) → R(Z) be bounded below, excisive functors, with Y = Φ(X) and Z = Ψ(Y), such that (X, x), (Y, y) and (Z, z) are based connected simplicial sets. We form ϕ = j ∘ Φ ∘ i and ψ = j ∘ Ψ ∘ i, as in (8.1).

\xymatrix{R(X) \ar[r]^{Φ} & R(Y) \ar[r]^{Ψ} \ar[d]_{j} & R(Z) \ar[d]^{j} \\ R(∗, Ω_{x}(X)) \ar[u]^{i} \ar[r]^{ϕ} & R(∗, Ω_{y}(Y)) \ar[u]_{i} \ar[r]^{ψ} & R(∗, Ω_{z}(Z)) .}

We let Ξ = Ψ ∘ Φ and ξ = j ∘ Ψ ∘ Φ ∘ i, so that ∂ξ ≃ ∂_{z}^{x}(Ξ)(X) = ∂_{z}^{x}(Ψ ∘ Φ)(X) is the equivariant Goodwillie derivative of the composite functor.

Proposition 10.1. With notation as above, suppose that Ψ satisfies the colimit axiom. Then there is a natural chain of stable equivalences of free left Ω_{z}(Z)-spectra with right Ω_{x}(X)-action

∂_{z}^{x}(Ψ ∘ Φ)(X) ≃ ∂_{z}^{y}Ψ(Y) ∧_{Ω_{y}(Y)} ∂_{y}^{x}Φ(X) .

Proof. By proposition 7.1 there is a natural weak equivalence ξ = j ∘ Ψ ∘ Φ ∘ i ≃ j ∘Ψ∘i∘j ∘Φ∘i = ψ ∘ϕ. Hence there is also a natural weak equivalence of simplicial functors ξ̌ ≃ ψ̌ ∘ ϕ̌, and a strict equivalence of equivariant Goodwillie derivatives ∂ξ ≃ ∂(ψ ∘ ϕ).

For brevity, let H = Ω_{x}(X) and G = Ω_{y}(Y).

The smash product of spectra is so constructed that in order to produce a stable equivalence of spectra ∂ψ ∧_{G} ∂ϕ ⟶ ∂(ψ ∘ϕ) it suffices to define a stable equivalence of bi-spectra

{m, n ↦ ∂ψ_{m} ∧_{G} ∂ϕ_{n}} ⟶ {m, n ↦ ∂(ψ ∘ ϕ)_{m,n}}

where ∂ψ_{m} ∧_{G} ∂ϕ_{n} = ψ̌(Σ^{m}G_{+})∧_{G} ϕ̌(Σ^{n}H_{+}) and ∂(ψ ∘ϕ)_{m,n} = (ψ̌∘ϕ̌)(Σ^{m}Σ^{n}H_{+}). Such a stable equivalence is provided by the following composite:

(10.2)  ψ̌(Σ^{m}G_{+}) ∧_{G} ϕ̌(Σ^{n}H_{+}) ⟶^{a} ψ̌(Σ^{m}ϕ̌(Σ^{n}H_{+})) ⟶^{b} (ψ̌ ∘ ϕ̌)(Σ^{m}Σ^{n}H_{+}) .

Here the first map a is a case of (9.2), which induces a stable equivalence (as m → ∞) by proposition 9.1, in view of lemma 7.4. The second map b is ψ̌ applied to the map

Σ^{m}ϕ̌(Σ^{n}H_{+}) ⟶ ϕ̌(Σ^{m}Σ^{n}H_{+}) ,

which is (9.2) applied to the case W = Σ^{m}H_{+}. We proved just after lemma 9.3 that this is a meta-stable equivalence (as n → ∞). Since ψ̌ is bounded below, the second map b is also a meta-stable equivalence. □

Theorem 10.3. Let Φ: S/X → S and Ψ: S/Y → S be bounded below, stably excisive functors, with Y = Φ(X) and Z = Ψ(Y), and suppose that Ψ satisfies the colimit axiom. Write Y = ∐_{β∈B} Y_{β} with each Y_{β} connected, and choose base points x ∈ X, y_{β} ∈ Y_{β} and z ∈ Z. Let X_{α} be the connected component of x in X, and let Z_{γ} be the connected component of z in Z. Then Ψ ∘ Φ is also bounded below and stably excisive, and there is a natural chain of stable equivalences

∂_{z}^{x}(Ψ ∘ Φ)(X) ≃ ⋁_{β∈B} ∂_{z}^{y_{β}}Ψ(Y) ∧_{Ω_{y_{β}} (Y_{β})} ∂_{y_{β}}^{x}Φ(X)

of free left Ω_{z}(Z_{γ})-spectra with right Ω_{x}(X_{α})-action.

Proof. The composite Ψ ∘ Φ is bounded below and stably excisive by proposition 2.2. By propositions 3.3 and 5.2 we can replace Φ and Ψ by the bounded below, excisive functors P′Φ and P′Ψ, respectively, without changing the derivatives of Φ, Ψ and Ψ ∘ Φ by more than a stable equivalence, and such that P′Ψ satisfies the colimit axiom. Hence we can assume from the beginning that Φ and Ψ are excisive. By proposition 6.3 there is a stable equivalence ∂_{z}^{x}(Ψ ∘ Φ)(X) ≃ ⋁_{β∈B} ∂_{z}^{x}(Ψ_{γ}^{β} ∘ Φ_{β}^{α})(X_{α}). By proposition 10.1 each summand here is stably equivalent to ∂_{z}^{y_{β}}(Ψ_{γ}^{β})(Y_{β}) ∧_{Ω_{y_{β}} (Y_{β})} ∂_{y_{β}}^{x}(Φ_{β}^{α})(X_{α}). By lemma 6.1 each such term can be rewritten as ∂_{z}^{y_{β}}Ψ(Y) ∧_{Ω_{y_{β}} (Y_{β})} ∂_{y_{β}}^{x}Φ(X). □

Theorem 10.4. Let F, G: S → S be bounded below, stably excisive functors, with Y = F (X) and Z = G(Y), and suppose that G satisfies the colimit axiom. Write Y = ∐_{β∈B} Y_{β} with each Y_{β} connected, and choose base points x ∈ X, y_{β} ∈ Y_{β} and z ∈ Z. Let X_{α} be the connected component of x in X, and let Z_{γ} be the connected component of z in Z. Then the composite G∘F is bounded below and stably excisive, and there is a natural chain of stable equivalences

∂_{z}^{x}(G ∘ F)(X) ≃ ⋁_{β∈B} ∂_{z}^{y_{β}}G(Y) ∧_{Ω_{y_{β}} (Y_{β})} ∂_{y_{β}}^{x}F (X)

of free left Ω_{z}(Z_{γ})-spectra with right Ω_{x}(X_{α})-action.

Proof. This is the special case of theorem 10.3 when Φ = F ∘ u and Ψ = G ∘ u. □

11. Topological spaces

Let U be the category of compactly generated topological spaces. The geometric realization functor | − |: S → U is left adjoint to the total singular simplicial set functor Sing: U → S. There is a natural weak equivalence | Sing(X)| → X.

Given a weak homotopy functor f: U → U we get a weak homotopy functor F: S → S by setting F (X) = Sing(f(|X|)). Let Y = f(X), Y′ = Sing(f(X)) and Y′′ = F (Sing(X)). There are natural weak equivalences |Y′′| → |Y′| → Y.

Choose base points x ∈ X and y ∈ Y′′. Let x ∈ Sing(X), y ∈ Y′ and y ∈ Y also denote the corresponding base points, via the maps just mentioned.

The Goodwillie derivative of f at X with respect to the base points x ∈ X and y ∈ Y is the spectrum

(11.1)  ∂_{y}^{x}f(X) = {n ↦ hofib_{y}(f(X ∨_{x} S^{n}) → Y)} .

It receives a natural strict equivalence from the spectrum

|∂_{y}^{x}F (Sing(X))| = {n ↦ | hofib_{y}(F (Sing X ∨_{x} S^{n}) → Y′′|} .

The latter spectrum has a free left action by |Ω_{y}(Y′′_{β})|, and a right action by |Ω_{x}(Sing(X_{α}))|, where X_{α} is the path-component of x in X, and Y′′_{β} is the pathcomponent of y in Y′′.

It will be convenient to forgo the condition that the left action is free. We thus apply the forgetful functor Sp(G) → Sp^{G} from free G-spectra to spectra with Gaction, for G = Ω_{y}(Y′′_{β}). Hence we will consider ∂_{y}^{x}f(X) as a spectrum with left |Ω_{y}(Y′′_{β})|-action and right |Ω_{x}(Sing(X_{α}))|-action.

We can always recover a strictly equivalent free G-spectrum by smashing with EG_{+}, where EG is a free, contractible G-space. For example we may take EG = Γ̃(Y′′_{β}). Thus if L is a spectrum with right G-action and M is a free left G-spectrum, thought of as a spectrum with left G-action, the stable homotopy type of the G-orbit spectrum L ∧_{G} M can be recovered as the homotopy orbit spectrum

(11.2)  L ∧_{G} (EG_{+} ∧ M) = L ∧_{hG} M .

We now switch to this notation.

Suppose that f, g: U → U are bounded below, stably excisive functors and that g satisfies the colimit axiom. Define F (X) = Sing(f(|X|)) and G(Y) = Sing(g(|Y |)).

Let Y = f(X) and Z = g(Y). Let Y = ∐_{β∈B} Y_{β}, Y′ = ∐_{β∈B} Y′_{β} and Y′′ = ∐_{β∈B} Y′′_{β} be the decompositions into path-components. (These simplicial sets are weakly equivalent, so the indexing sets B are equal.) Choose base points x ∈ X, y_{β} ∈ Y′′_{β} and z ∈ G(Y′′) = (G ∘ F)(Sing(X)). Let x ∈ Sing(X), y_{β} ∈ Y′_{β}, y_{β} ∈ Y_{β} and z ∈ Z denote their images under the natural weak equivalences.

The chain rule 10.4 for G ∘ F at Sing(X) then asserts that there is a stable equivalence

|∂_{z}^{x}(G ∘ F)(Sing(X))| ≃ ⋁_{β∈B} |∂_{z}^{y_{β}}G(Y′′)| ∧_{h|Ω_{y_{β}} (Y′′_{β})|} |∂_{y_{β}}^{x}F (Sing(X))| .

There are natural weak equivalences

|∂_{y_{β}}^{x}F (Sing(X))| ⟶^{≃} ∂_{y_{β}}^{x}f(X)
|∂_{z}^{y_{β}}G(Y′′)| ⟶^{≃} ∂_{z}^{y_{β}}g(Y′′) ⟶^{≃} ∂_{z}^{y_{β}}g(Y)
|∂_{z}^{x}(G ∘ F)(Sing(X))| ⟶^{≃} ∂_{z}^{x}(g ∘ f)(X)
|Ω_{y_{β}}(Y′′_{β})| ⟶^{≃} |Ω_{y_{β}}(Y′_{β})| .

Hence we can summarize:

Theorem 11.3. Let f, g: U → U be bounded below, stably excisive functors, with Y = f(X) and Z = g(Y), and suppose that g satisfies the colimit axiom. Let Y = ∐_{β∈B} Y_{β} with each Y_{β} path-connected, and choose base points x ∈ X, y_{β} ∈ Y_{β} and z ∈ Z. Let X_{α} be the path-component of x in X, and let Z_{γ} be the path component of z in Z. Also set Ω_{y_{β}}(Y_{β}) = |Ω_{y_{β}}(Sing(Y_{β}))|. Then the composite g ∘ f is bounded below and stably excisive, and there is a stable equivalence

∂_{z}^{x}(g ∘ f)(X) ≃ ⋁_{β∈B} ∂_{z}^{y_{β}}g(Y) ∧_{hΩ_{y_{β}} (Y_{β})} ∂_{y_{β}}^{x}f(X)

of spectra with left Ω_{z}(Z_{γ})-action and right Ω_{x}(X_{α})-action.

Remark 11.4. To be precise, this formula needs to be interpreted in line with the weak equivalences above. In particular it is not Ω_{y_{β}}(Y_{β}) = |Ω_{y_{β}}(Y′_{β})| that really acts, but the naturally weakly equivalent topological group |Ω_{y_{β}}(Y′′_{β})|. And the action is not really on the spectra ∂_{z}^{y_{β}}g(Y) and ∂_{y_{β}}^{x}f(X), but on the naturally weakly equivalent spectra |∂_{z}^{y_{β}}G(Y′′)| and |∂_{y_{β}}^{x}F (Sing(X))|.

12. Examples

Example 12.1. Let id: U → U be the identity functor. It is clearly bounded below, and is stably excisive by homotopy excision. For a path-connected space X, choose non-degenerate base points x ∈ X and y ∈ id(X) = X. Let P_{y}(X) = {γ: I → X | γ(0) = y} and P_{y}^{x}(X) = {γ: I → X | γ(0) = y, γ(1) = x}. There is a natural map

hofib_{y}(X∨_{x}S^{n} → X) = P_{y}(X)∪_{P_{y}^{x}(X)}(P_{y}^{x}(X)×S^{n}) ⟶ P_{y}^{x}(X)_{+}∧S^{n} = Σ^{n}P_{y}^{x}(X)_{+}

which is a homotopy equivalence. Hence there is a stable equivalence

∂_{y}^{x}(id)(X) ≃ Σ^{∞}P_{y}^{x}(X)_{+} ,

with the natural left Ω_{y}(X)-action and right Ω_{x}(X)-action given by composition of paths.

Example 12.2. Let K be a finite CW complex and consider the mapping space functor f(X) = X^{K} = Map(K, X). Then f satisfies E_{1}(d, κ) and E_{2}(2d + 1, κ) for all κ, where d = dim(K). Choose a non-degenerate base point x ∈ X, and base Y = X^{K} at the constant map y taking K to {x}. There are homotopy equivalences

hofib_{y}((X ∨_{x} S^{n})^{K} → X^{K}) ≃ hofib_{x}(X ∨_{x} S^{n} → X)^{K} ≃ Map(K, Σ^{n}Ω_{x}(X)_{+})

and thus a stable equivalence

∂_{y}^{x}(X^{K}) ≃ Map(K, Σ^{∞}Ω_{x}(X)_{+}) .

When X^{K} is path-connected, the group Ω_{y}(X^{K}) = Map(K, Ω_{x}(X)) acts from the left by pointwise multiplication, while Ω_{x}(X) acts uniformly from the right.

Example 12.3. Let g(Y) = Q(Y_{+}) = colim_{n} Ω^{n}(Σ^{n}Y_{+}) be the (unreduced) stable homotopy functor. This functor is bounded below and excisive, and satisfies the colimit axiom. Choose a base point y ∈ Y, and take any base point z of Z = Q(Y_{+}). The pushout of S^{n} ← (Y ∨_{y} S^{n})_{+} → Y_{+} is ∗, so there is a natural cartesian square

\xymatrix{Q((Y ∨_{y} S^{n})_{+}) \ar[r] \ar[d] & Q(S^{n}) \ar[d] \\ Q(Y_{+}) \ar[r] & Q(∗)}

and a natural weak equivalence ∂_{z}^{y}Q(Y_{+})_{n} → Q(S^{n}). Thus there is a stable equivalence

∂_{z}^{y}Q(Y_{+}) ≃ S ,

where S = {n ↦ Q(S^{n})} is the sphere spectrum. The left action of Ω_{z}(Q(Y_{+})) on ∂_{z}^{y}Q(Y_{+})_{n} pulls back from ∗ in the cartesian square above. Likewise the right action of Ω_{y}(Y) pulls back from the trivial action on Q(S^{n}). Hence both of these actions are trivial, up to homotopy.

Example 12.4. Let (g ∘ f)(X) = Q(X_{+}^{K}) be the composite functor. By the chain rule 11.3, its derivative at X is

∂_{z}^{x}Q(X_{+}^{K}) ≃ ∂_{z}^{y}Q(Y_{+}) ∧_{Ω_{y}(Y)} ∂_{y}^{x}(X)
≃ S ∧_{Map(K,Ω_{x}(X))} Map(K, Σ^{∞}Ω_{x}(X)_{+})
≃ Map(K, Σ^{∞}Ω_{x}(X)_{+})_{h Map(K,Ω_{x}(X))} .

This assumes that X^{K} is path connected. The derivative of this functor X ↦ Q(X_{+}^{K}) was first computed as the spectrum of stable sections in a suitable Serre fibration in [Go90, §2]. In the paper [Kl95] the first author shows that the two descriptions of this derivative are indeed equivalent.

References

[Ar99] G. Arone, A generalization of Snaith-type filtration, Trans. Am. Math. Soc. 351 (1999), 1123–1150.

[BHM93] M. Bökstedt, W.C. Hsiang and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 465–539.

[BF78] A.K. Bousfield and E.M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 80–130.

[BK72] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, vol. 304, Springer-Verlag, Berlin-New York, 1972.

[Fa95] H. Fausk, Chain rule in functor calculus, Cand. Scient. thesis, University of Oslo (1995).

[Go90] T.G. Goodwillie, Calculus I: The first derivative of pseudoisotopy theory, K-Theory 4 (1990), 1–27.

[Go92] T.G. Goodwillie, Calculus II: Analytic functors, K-Theory 5 (1992), 295–332.

[Ha02] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.

[He92] L. Hesselholt, A homotopy theoretical derivation of QMap(K, −)_{+}, Math. Scand. 70 (1992), 193–203.

[Kl95] J.R. Klein, On the derivative of the stable homotopy of mapping spaces, Bielefeld preprint (1995).

[Ly98] M. Lydakis, Simplicial functors and stable homotopy theory, Bielefeld preprint (1998).

[Sch01] S. Schwede, Stable homotopy of algebraic theories, Topology 40 (2001), 1–41.

[Wa85] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419.

[Wa96] F. Waldhausen, On the construction of the Kan loop group, Doc. Math. 1 (1996), 121– 126.

Department of Mathematics, Wayne State University, Detroit, USA

   E-mail address: klein@math.wayne.edu

Department of Mathematics, University of Oslo, Oslo, Norway

   E-mail address: rognes@math.uio.no