Goodwillie Calculus in the category of algebras

over a spectral operad

Luıs Alexandre Pereira

October 25, 2017

The main purpose of this paper is to apply the theory developed in [26] tothe specific case of functors between categories of the form AlgO.

Section 1 introduces the basic necessary definitions and notations aboutsymmetric spectra SpΣ, model structures in that category, operads and algebrasover them, and model structures in the categories of algebras.

Section 2 then begins the task of understanding (finitary) homogeneous func-tors between algebra categories by understanding their stabilization. The re-sults here are Theorem 2.3 and its Corollary 2.9 showind that the stabilizationof AlgO is the module category ModO(1).

Section 3 completes the task of classifying (finitary) homogeneous functorsbetween algebra categories by classifying homogeneous functors between theirstabilizations, the module categories. The result here is Theorem 3.2, whichgeneralizes the well known characterization in the case of spectra.

Finally, section 4 concludes the paper by establishing our main results aboutGoodwillie calculus in the AlgO categories. In subsection 4.1 we finally estab-lish Theorem 4.3, saying that the Goodwillie tower for IdAlgO is indeed thehomotopy completion tower studied in [17]. Subsection 4.2 establishes Theorem4.8, which is probably more surprising. It roughly says that, as far as (finitary)n-excisive functors are concerned, Goodwillie calculus can not distinguish thecategory AlgO from AlgO≤n . Lastly, subsection 4.3 establishes Theorem 4.12,which shows the AlgO categories satisfy at least a weak analogue of the chainrule from [1].

1 Basic definitions

The majority of the material in this section is adapted from [15], which shouldbe consulted for details. We will only cover here the bare minimum necessaryfor the remainder of the paper.

1.1 Symmetric Spectra

Definition 1.1. The category SpΣ of symmetric spectra is the category suchthat

• objects X are sequences Xn ∈ SSetΣn∗ (i.e., Xn is a pointed simplicialset with a Σn action), together with structure maps (compatible with the

1

Σm × Σn action)Sm ∧Xn → Xm+n

satisfying appropriate compatibility conditions.

• maps f : X → Y are sequences of Σn maps fn : Xn → Yn compatible withthe structure maps in the obvious way.

Also, we denote by ∧ the standard monoidal structure on SpΣ with unit S,the canonical symmetric spectrum such that Sn = Sn, the n-sphere.

Definition 1.2 (Free symmetric spectra). Let H ⊂ Σm be a subgroup andA ∈ SSetH∗ .

The free spectrum FHm (A) generated by A is the symmetric spectrum withspaces

(FHm (A))n =

Σn ×Σn−m×H

(Sn−m ∧A) if n ≥ m

∗ if n < m

with the natural maps.Or, in other words, FHm is the left adjoint to the forgetful functor SpΣ →

SSetH∗ .

1.2 Stable model structures on SpΣ

Our interest on SpΣ is as a model category for spectra. The stable w.e.s used forthis are somewhat technical to define, as the actual definition resorts to defininginjective Ω-spectra first, hence we refer to [21] for the precise definition. Forour purposes it is enough to view these stable equivalences as being those mapsthat induce isomorphisms on the stable homotopy groups πsn

1.However, while we will only be interested in model structures on SpΣ which

use the stable w.e.s as the notion of w.e.s, there are multiple such model struc-tures, and it will be useful for us to be aware of several of them, which we listin the following definition.

Definition 1.3. We have the following stable model structures on SpΣ.

• level cofibrations stable model structure, where the cofibrations arelevelwise cofibrations of the underlying symmetric sequences.

• flat stable model structure, with generating cofibrations given byFHm ((δ∆k)+)→ FHm ((∆k)+) for m ≥ 0, H ⊂ Σm any subgroup.

• positive flat stable model structure, with generating cofibrationsgiven by FHm ((δ∆k)+)→ FHm ((∆k)+) for m ≥ 1, H ⊂ Σm any subgroup.

• stable model structure, with generating cofibrations given by

F ∗m((δ∆k)+)→ F ∗m((∆k)+)

for m ≥ 0, and where ∗ ⊂ Σm denotes the trivial subgroup.

1The catch here is that, when working with symmetric spectra, one can’t usually simplydefine πsn(X) = lim−→m

πn+mXm, unless X already satisfies some fibrancy type condition.

2

• positive stable model structure, with generating cofibrations given by

F ∗m((δ∆k)+)→ F ∗m((∆k)+)

for m ≥ 0, and where ∗ ⊂ Σm denotes the trivial subgroup.

Remark 1.4. It is worth noting the hierarchy of these model categories: thelevel cofibrations stable structure has the most cofibrations, followed by the flatstable model structure, followed by either the positive flat stable or the stablemodel structures (which refine the flat stable model structure in different ways),and followed finally by the positive stable model structure, which has the leastcofibrations of all.

Proposition 1.5. The five model structures listed above are all left proper cel-lular model categories.

Proof. We recall that left properness means that the pushout of a w.e. along acofibration is again a w.e.. Hence it suffices to prove this property for the modelcategory structure with the most cofibrations, namely the levelwise cofibrationmodel structure. This result is then Lemma 5.4.3 part (1) of [21].

Cofibrant generation of each of these model structures is proved in severalpapers: for the level cofibration model structure and the stable model structurethis is proved in [21]; for the flat stable model structure and the positive flatstable model structure this is proved in [32], and for the positive stable modelstructure this is proved in [24]2.

Cellularity of these model categories (see Definition A.1 of [20] for the def-inition) follows immediately from these being categories built out of simplicialsets.

Indeed, any set of objects A will be compact with respect to any set of levelinjections K by choosing a regular ordinal γ greater than the cardinality of allthe simplicies appearing in A, as then a map from a ∈ A into a relative K-cellcomplex will factor through the minimal complex containing the images of thesimplices, and this subcomplex will have less than γ cells.

Hence indeed the domains of I are compact with respect to I, and thedomains of J small relative to the cofibrations (by a similar but easier argument).Finally, it is clear that the cofibrations are categorical monomorphisms, sincethey are always levelwise monomorphisms.

1.3 Operads and algebras

Definition 1.6. An (spectral) operad O in SpΣ is a sequence of “spectra ofn-ary operations” O(n) ∈ SpΣ, for n ≥ 0, together with

• Σn actions on O(n),

2A little care is needed here because the results proven in [24] are for topologically basedspectra instead of for simplicial ones, so one needs to adapt the arguments present there.Alternatively, it is fairly straightforward to see that one has a positive level model structurein the simplicial case, and that this is a left proper cellular model category, hence the resultcan also be derived by applying the left Bousfield localization techniques of [18].

3

• multiplication maps

O(n) ∧ O(m1) ∧ · · · ∧ O(mn)→ O(m1 + · · ·+mn)

and unit mapS → O(1),

• together with associativity, unity and change of order of variables com-patibility conditions.

Definition 1.7. The category AlgO of algebras over the operad O is the cate-gory such that

• objects X are symmetric spectra plus algebra maps

O(n) ∧X∧n → X

satisfying appropriate compatibility conditions.

• maps f : X → Y are maps X → Y of the underlying symmetric spectrawhich are compatible with the algebra structure maps.

Definition 1.8 (Free algebras). The free O-algebra functor (usually denotedO) is the left adjoint to the forgetful functor AlgO → SpΣ.

It assigns to a symmetric spectrum X the canonical algebra O(X) whoseunderlying spectrum is

∨∞n=0(O(n) ∧X∧n)Σn .

We will also occasionally use the notation O X for this algebra, where ismeant to evoke the composition product of symmetric sequences.

Notations 1.9. Note that for any O-algebra X its structural multiplicationmaps can be packaged into a map

O(X)µ−→ X.

Note further that this is actually a map of O-algebras.

We will be using the following standard result.

Proposition 1.10. Let O → O′ be a map of operads. Then there is an adjunc-tion

AlgOO′O−//

AlgOforgetoo ,

where the functor O′ O − is defined by the natural coequalizer coeq(O′OX ⇒O′X).

Remark 1.11. As a particular case of the previous result, consider the obviousmap3 O(1) → O. Then each spectrum O(n) has a natural left action of Otogether with n different right actions of O(1). Note that these actions are allcompatible, and that the right actions are furthermore related by the symmetricgroup action. Altogether these make O(n) into a (O(1),O(1)on)-bimodule4.

3Notice that O(1) is itself a monoid, and can hence be viewed as an operad concentratedin degree 1.

4Here O(1)on denotes the wreath product ΣnnO(1)∧n = ∨σ∈ΣnO(1)∧n. Multiplication inthis ring spectrum is such that multiplying the σ component by the τ component lands in theστ component, and σ acts on the second O(1)∧n copy before multiplying those coordinatewise.

4

A direct calculation then shows that

O O(1)

X =∨n≥0

(O(n) ∧O(1)∧n

X∧n)Σn =∨n≥0

O(n) ∧O(1)on

X∧n.

Proposition 1.12. Let X be in AlgO, and consider the undercategory (AlgO)X/of O-algebras under X.

Then there exists an enveloping operad OX such that

(AlgO)X/ = AlgOX .

More specifically, one has

OX(n) = coeq(∐m≥0

O(n+m) ∧Σm (O X)∧m ⇒∐m≥0

O(n+m) ∧Σm X∧m)

with the two maps induced by the operad structure and the algebra structure.

Proof. This is just [15, Prop. 4.7] reinterpreted (and restricted from left modulesto algebras, i.e., left modules concentrated in degree 0.).

Indeed, what that proposition shows is that the forgetful functor (AlgO)X/has its left adjoint given by OX ·. The result then follows since the conditionsof Beck’s monadicity theorem are immediate.

1.4 Model structures on AlgO

The following is a particular case of the main result of [15] (as well as thecurrent author’s work5 in [27, Thm. 1.5]), combined with the simplicial structuredescribed in the follow-up paper [17, Sec. 6].

Theorem 1.13. Suppose SpΣ is endowed with either the positive stable modelstructure or the positive flat stable model structure.

Then, for any operad O, the projective model structure6 on AlgO exists.Furthermore, this is a simplicial model structure.

We will need a somewhat explicit description of the simplicial tensoring andcotensoring.

First, recall that in SpΣ those are given by

(K ⊗X)n = K+ ∧Xn

andMap(K,X)n = (Xn)K ,

where K ∈ SSet and X ∈ SpΣ.In other words, this these reflect pointwise the tensoring and cotensoring of

SSet∗ over SSet. Note that K ⊗X can also be described as F ∗0K ∧X.

5In fact, the purpose of [27] was in part to correct a key technical problem with the proofin [15]. Namely, certain “Σn-cofibrancy” claims of “n-fold smash products” used in that proofturn out to be false, with [27] establishing correct (laxer) “Σn-cofibrancy” that still suffice forthe remainder of the proof to apply.

6We recall that in a category AlgC(C) of algebras over some monad C in C, the projectivemodel structure (when it exists) on AlgC(C) is the one where w.e.s/fibrations are the mapswhich are underlying w.e.s/fibrations in C.

5

In AlgO, the tensoring of a simplicial set K and a O-algebra X is then givenby the (algebraic) coequalizer

K ⊗alg X = O(K ⊗O(X))O(K⊗µ)

⇒O(τ)

O(K ⊗X)

where O(X)µ−→ X is the algebra structure map and K ⊗O(X)

τ−→ O(K ⊗X) isthe map

∞∨n=0

K ⊗ (O(n) ∧X∧n)Σn →∞∨n=0

(K×n ⊗O(n) ∧X∧n)Σn

which is induced at each level by the diagonal maps K → K×n.As for the cotensoring Mapalg(K,X), the underlying symmetric spectrum

is just Map(K,X), with the algebra structures maps

O(n) ∧Map(K,X)n →Map(K,X)

being the adjoints to the composite

K⊗O(n)∧Map(K,X)n → Kn⊗O(n)∧Map(K,X)n ∼= O(n)∧(K⊗Map(K,X))n → X

where the first map is induced by the diagonal K → Kn and the last one by thecounits K ⊗Map(K,X)→ X and algebra structure map O(n) ∧X∧n → X.

1.5 Spectra on model categories

As shown in [26] (good) homogeneous functors between (good) model cate-gories factor in an appropriate sense through the stabilization of those modelcategories, or in other words, spectra in those categories. We briefly recall herethe notion of spectra used there.

Assume C a pointed simplicial model category, so that it also has a tensoring∧ over SSet∗.

Definition 1.14. The category Sp(C)

• objects X are sequences Xn ∈ C, together with structure maps

S1 ∧Xn → X1+n

where S1 is the standard simplicial circle ∆1/∂∆1.

• maps f : X → Y are sequences of Σn maps fn : Xn → Yn compatible withthe structure maps in the obvious way.

It is shown in [20] that Sp(C) generally has a projective model structure.

Proposition 1.15 (Hovey). Suppose that C is a cofibrantly generated modelcategory with I and J the generating cofibrations and trivial cofibrations.

Then the projective model structure on Sp(C) exists and is cofibrantlygenerated with generating sets given by7

Iproj =⋃n

FnI, Jproj =⋃n

FnJ.

7Here Fn denotes the free spectrum generated by an object in C at level n. The constructionis entirely similar to that in 1.8, except one needs not care about Σn actions.

6

Definition 1.16. For C a cofibrantly generated model category the stablemodel structure on Sp(C) is the left Bousfield localization (should it exist) ofthe projective model structure with respect to the set of maps

Fn+1(S1 ∧Ai)→ FnAi)

where Ai ranges over the (cofibrant replacements of, when necessary) domainsand codomains of the maps in I.

Remark 1.17. For a detailed treatment of left Bousfield localizations, check[18, Chap. 3,4].

Notice that the existence of the stable model structure in Sp(AlgO) re-quires proof. Its existence does not follow for free from the result in [20], whichshows that that model structure always exists when C is left proper cellular,because AlgO is in fact almost never a left proper model category.

Notations 1.18. We denote by

Σ∞ : C Sp(C) : Ω∞

the standard Quillen adjoint functors.Note that Σ∞ is just what we previouly called F0.In fact, we will additionally denote by Ω∞−n the adjoint of Fn, which is

naturally also denotable Σ∞−n.Finally, by Σ∞−n and Ω∞−n we denote the appropriate left and right derived

functors8 of Σ∞−n and Ω∞−n.

It will be useful to know that the conclusion of the following result of [20]holds when C = SpΣ.

Theorem 1.19 (Hovey). Suppose C is a pointed simplicial almost finitely gen-erated model category. Suppose further that in C sequential colimits preservefinite products, and that Map(S1,−) preserves sequential colimits.

Then stable equivalences in Sp(C) are detected by the Ω∞−n functors.

Proof. This is just a particular case of [20, Thm. 4.9].

The hypothesis of the result above do hold when SpΣ is given the flat stablemodel structure.

Indeed, in that case SpΣ is almost finitely generated (see [20] Chapter 4for the definition), and in fact even finitely generated: searching the proofsgiven in [21] one sees that the generating cofibrations are maps FHm ((δ∆i)+)→FHm ((∆k)+), with compact domains and codomains, and that the generatingtrivial cofibrations are the maps FHm ((Λik)+) → FHm ((∆k)+) plus the simplicialmapping cylinders of the maps F ∗n+1((S1)+) → F ∗n((S0)+), which again havecompact domains and codomains.

The remaining two conditions, that sequential and finite products commute,and that Map(S1,−) preserves sequential colimits, follow immediately from thefact that all these constructions are levelwise at the simplicial set level, wherethe statements are known to be true.

8As in [26] we assume that a set functorial choice of cofibrant and fibrant replacements hasbeen made.

7

For the remaining model structures in SpΣ we listed it is not clear whetherthe hypotheses in Thm. 1.19 hold, but the conclusion of that result doesnonetheless hold.

Indeed, to see this it suffices to check that the w.e.s on Sp(SpΣ) are exactlythe same no matter what the base model category chosen on SpΣ is9.

Now note that clearly the various possible levelwise model structures onSp(SpΣ) are Quillen equivalent, and share the same weak equivalences. Butsince furthermore the notion of stable fibrant (i.e. local) objects in Sp(SpΣ)only changes with the base model structure on Sp(Σ) up to levelwise w.e.s, itindeed follows that all the stable model structrures on Sp(SpΣ) do have thesame w.e.s, as desired.

Remark 1.20. Note that for any of the choices of the underlying model struc-ture in SpΣ, the generating trivial cofibrations in Sp(SpΣ) can be chosen to havecofibrant domains and codomains. This follows from the localization theory in[18], specifically [18, Prop. 4.5.1], together with the fact that the generating cofi-brations in Sp(SpΣ) are directly seen to have cofibrant domains and codomains.

2 Model structure on Spectra of AlgO

As shown in [26] understanding homogeneous functors to/from a (good) modelcategory is closely related to understanding its stabilization. Hence, our goal inthis section will be to understand the (stable) model category Sp(AlgO).

Recall first of all that the underlying case model structures on AlgO arethose described in 1.4, i.e., the projective model structures built out of anyof the “positive” stable model structures in SpΣ. We shall have no need todistinguish between these two model structures in our results.

Second, recall that in order to stabilize a model category C, C must eitherbe a pointed model category, else one must first replace C by its category C∗ ofpointed objects and then stabilize that category. In our context, we notice thatAlgO is pointed precisely when O(0) = ∗ 10, hence we make that assumption forthe rest of this section. We also point out that, by 1.12, (AlgO)∗ is still alwaysan operadic algebra category, hence our results still cover that case.

Our main result will be that, Sp(AlgO) is just ModO(1)(SpΣ), up to a zig

zag of Quillen equivalences.However, some subtleties must be handled beforehand, like proving that the

model structure on Sp(AlgO) actually exists. The reason one can’t simply justuse the theory developed in [20] is that AlgO is rarely left proper.

Here’s a sketch for why: it is easy (by appealing to universal properties),to check that in AlgO it is OX(A)

∐X Y = OY (A). As the map X → OX(A)

is a cofibration whenever A is cofibrant, one sees that left properness wouldrequire X 7→ OX to be a homotopical construction. But now recall that OX(n)is constructed as a quotient of

∐m≥0O(n+m)∧ΣmX

∧m, which is not expectedto be a homotopical construction for general O, and hence neither is OX . It isnot too hard to craft specific examples where this can be seen explicitly.

9That these model structures always exist follows from Proposition 1.5 together with thefirst main theorem of [20].

10We shall also refer to this condition by saying that O is non unital.

8

In order to prove the existence of a model structure on Sp(AlgO) directly,we first notice that this category can also be regarded as AlgOsp(Sp(SpΣ)), thealgebras for a certain monad Osp on the category Sp(SpΣ). We hence needonly verify the hypotheses of [30, Lem. 2.3], a standard concerning existence ofmodel structures in algebra categories.

We point out that this is exactly what is done in [15], where the pushoutsnecessary to apply [30, Lem. 2.3] are studied by means of an adequate filtra-tion11, and we will make use of a similar filtration adapted to the monad Osp.

Proposition 2.1. Sp(AlgO) is the category AlgOsp(Sp(SpΣ)) of algebras forthe monad Osp given by

(OspX)n = O(Xn)

with the structure maps given by the composite12

S1 ∧ O(Xn)→ O(S1 ∧Xn+1)→ O(Xn+1).

Proof. The first task is to show that O is actually a monad (and indeed afunctor).

It is useful for this to consider the following auxiliary structures.

∧sp : Sp(SpΣ)× Sp(SpΣ)→ Sp(SpΣ)

(X ∧sp Y )n = Xn ∧ Yn,

∧ : SpΣ × Sp(SpΣ)→ Sp(SpΣ)

(X ∧ Y )n = X ∧ Yn,

where the structure maps for X∧spY are the natural composite S1∧Xn∧Yn →(S1)∧2 ∧ Xn ∧ Yn ∼= S1 ∧ Xn ∧ S1 ∧ Yn → Xn+1 ∧ Yn+1, with the first mapinduced by the diagonal map, and the structure maps for X ∧ Y given bynatural composite S1 ∧X ∧ Yn ∼= X ∧ S1 ∧ Yn → S1 ∧ Yn+1.

It is then clear that ∧sp is a non unital symmetric monoidal structure13 onSp(SpΣ).

Furthermore, ∧ : SpΣ×Sp(SpΣ)→ Sp(SpΣ) behaves unitaly with respect tothe unit of SpΣ and associatively with respect to both the monoidal structure14

∧ on SpΣ and the monoidal structure ∧sp on SpΣ.Since furthermore each of these operations preserves colimits in each variable,

it is formal to check that non unital operads in SpΣ induce monads on Sp(SpΣ),the monad associated to O being what we just called Osp.

It now remains to see that the categories Sp(AlgO) and AlgOsp(Sp(SpΣ))are indeed the same.

11Also, check [27, Sec. 5.2] for a slightly different approach.12Here the first map is the map we called τ in 1.4.13This monoidal structure may look strange and unfamiliar at first. There is a very good

reason for this, namely the fact that ∧sp is always homotopically trivial, i.e., X∧spY is alwaysnullhomotpic. This fact plays a crucial role in the proof of the main result of this section.

14We purposefully abuse notation here in using the symbol ∧ to denote two different oper-ations. We believe this should not cause confusion.

9

First, we show the objects are the same. It is immediately clear that Ospalgebras X are made out of O-algebras Xn at each level. So it really onlyremains to see that having a map Osp(X)→ X is equivalent to having maps15

S1 ∧O Xn → Xn+1 of algebras. To see this first rewrite the structure maps ofspectra in adjoint form16. But it is now clear that both conditions are just thecommutativity of the following diagram

O(Xn)

// O(Map(S1, Xn+1)) // Map(S1,O(Xn+1))

Xn

// Map(S1, Xn+1),

the only difference being that the first condition concerns the squares obtainedby ommiting O(Map(S1, Xn+1)) and the second those obtained by ommitingMap(S1,O(Xn+1)).

It remains only to see that maps in the two categories are the same, but thisis clear: compatibility with spectra structure maps gives the same condition inboth cases, and compatibility with the Osp algebra structures is the same ascompatibility with the O-algebra structures at all the levels.

Lemma 2.2. The class of maps in Sp(SpΣ) which are both levelwise monomor-phisms stable equivalences is closed under pushouts, transfinite compositions andretracts.

Proof. Recall that weak equivalences in Sp(SpΣ) are detected as equivalencesat the hocolim ΩnhXn+k level.

Now consider a pushout of a such a map. Levelwise all these pushoutsare actually homotopy pushouts (since the level cofibration model structure onSpΣ is left proper), and since SpΣ is a stable, they are actually also levelwisehomotopy pullbacks. But then applying Ωn turns such (levelwise) squares intohomotopy pullbacks. But then it is obvious that the original pushout square isa homotopy pushout after applying any of the Ω∞−n functor.

A similar easier argument deals with the case of transfinite compositions,and the statement for retracts is obvious.

Theorem 2.3. The (monadic) projective model structures on

Sp(AlgO) ∼= AlgOsp(Sp(SpΣ))

based on either the positive stable or the positive flat stable model structures onSpΣ exist (and are cofibrantly generated).

Proof. We verify the conditions of [30, Lem. 2.3].It is immediate that Osp commutes with filtered direct colimits, and the

smallness conditions are obviously satisfied by adapting the argument made inthe proof of Proposition 1.5.

15Here we use ∧O to denote the pointed simplicial tensoring of AlgO.16I.e., Xn →Map(S1, Xn+1) rather than S1 ∧Xn → Xn+1.

10

It hence remains to check that for J a set of generating trivial cofibrationsin Sp(SpΣ,Σ) then Osp(J)reg, the closure of Osp(J) under pushouts andtransfinite compositions, consists of w.e.s..

Suppose now that X → Y is in J , and consider any pushout diagram inSp(AlgO) of the form

Osp(X)

// A

Osp(Y ) // B.

(2.4)

By Lemma 2.2, we will be done provided we can show that the map A→ Bis a level monomorphism and an underlying stable equivalence in Sp(SpΣ).

The strategy for this is to filter the map A→ B (analogously to [15] or [27,Sec. 5.2]). Explicitly, we write B as colim(A0 → A1 → A2 → . . . ), where thesuccessive An are built as pushouts

Osp,A(t) ∧Σt Qtt−1

// At−1

Osp,A(t) ∧Σt Y

∧spt // At.

(2.5)

A little care is needed in explaining the notation here. We do not quitedefine terms Osp,A(t) in this context since the colimits used for the analogousdefinition in [15] would make no sense, as they would involve both objects inSpΣ and in Sp(SpΣ) simultaneously17. Rather, we define expressions Osp,A(t)∧Z1 ∧sp · · · ∧sp Zt as a whole by the coequalizers18

Osp,A(t)∧ZT = coeq(∐p≥0

O(p+t)∧Σp (OA)∧p∧ZT ⇒∐p≥0

O(p+t)∧ΣpA∧p∧ZT ),

(2.6)with the equalized maps corresponding to the algebra structure of A and thestructure maps of the operad19.

Additionally, as in [15], the symbol Qtt−1 is meant to suggest the “union”(i.e. colimit) of the terms in the t-cube (X → Y )∧spt with the Y ∧spt terminalvertex removed20. Finally, the subscripts ∧Σt are meant to denote that onetakes coinvariants with respect to the obvious (diagonal) Σt action.

The existence of the relevant maps in 2.5, along with the existence of therelevant maps At → B, is then formally analogous to the treatment in [15] (or[27, Sec. 5.2]), merely replacing the monad O by the monad Osp.

It follows immediately that B is indeed the colimit of the At, since thesecolimits are levelwise on the (outer) spectra coordinates, and our filtration justrestricts to the one in [15](or [27, Sec. 5.2]) for each level.

Hence by Lemma 2.2 it suffices to show that the maps

Osp,A(t) ∧Σt Qtt−1 → Osp,A(t) ∧Σt Y

∧t

17This is a reflection of the fact that the monoidal structure ∧sp is non unital.18We use the shorthand ZT = Z1 ∧sp · · · ∧sp Zt.19Though note that here we only ever need to use the first p partial composition products.20Though, again, one needs to view the expression Osp,A(t) ∧Qtt−1 defined as a whole.

11

are all level monomorphisms and stable equivalences.To see this, we note first that, since we are in the case O(0) = ∗, the equalizer

in 2.6 splits into two summands: that with p = 0, which is just O(t)∧Z∧T , andthat with p ≥ 1. We deal with these summands separately.

For the p = 0 summand, the t = 1 term is just O(1) ∧X → O(1) ∧ Y , andsince Remark 1.20 ensures X, Y are levelwise cofibrant, this map is a levelwisemonomorphism and, further, the levelwise smash products are homotopicallymeaningful, so that

Ω∞−n(O(1) ∧ Z) =hocolimk

(Ωk(O(1) ∧ Zk+n)

)∼

∼hocolimk(O(1) ∧ ΩkZk+n) ∼∼O(1) ∧ hocolimk(ΩkZk+n) = O(1) ∧ Ω∞−nZ

(2.7)

and by Thm. 1.19 one concludes O(1) ∧X → O(1) ∧ Y is also a stable equiva-lence.

Still in the case of the p = 0 summand, the t > 1 terms are O(t)∧Σt Qtt−1 →

O(t) ∧Σt Yt. Combining Remark 1.20, which implies X → Y is a levelwise

positive cofibration, with [27, Thm. 1.3] and (part (a) of) [27, Thm. 1.4], onesees that those terms are indeed levelwise monomorphisms. To see that these arealso stable equivalences, we claim that in fact both O(t)∧Σt Q

tt−1 and O(t)∧Σt

Y t are actually null homotopic spectra. Indeed, since stable equivalences aredetected by the Ω∞−n functors, this will follow if we show that the structuremaps of these spectra are null homotopic. The structure maps of O(t) ∧Σt Y

t

are built by first considering the maps

S1 ∧ (Yn)t → St ∧ (Yn)t ∼= (S1 ∧ Yn)t → (Yn+1)t

and then applying the functor O(t)∧Σt −. The map above is clearly null homo-topic if one forgets the Σt action (as it factors through a higher suspension), andit remains null homotopic after applying the functor O(t) ∧Σt − since Remark1.20 and [27, Thm. 1.3] imply Y ∧t is S Σ-inj Σn-proj cofibrant21 so that [27,Thm. 1.4] guarantees that O(t)∧Σt − is a homotopical construction. The exactsame analysis works for O(t) ∧Σt Q

tt−1.

It remains to deal with the p ≥ 1 summand. Showing that the desired mapsare levelwise monomorphisms follows by repeating the arguments for the p = 0case. To see that they are also stable equivalences, we again reduce to showingthat both the domains and codomains are null homotopic. Indeed, we claim thestructure maps for (Osp,A(t) ∧Σt Y

∧t)p≥1 factor as

S1∧ (OAn(t)∧Σt Ytn)p≥1 → S2∧ (OAn(t)∧Σt Y

tn)p≥1 → (OAn+1

(t)∧Σt Y∧tn+1)p≥1.

This can be seen from the following commutative diagram between the defining

21The reader is encouraged to think of this as a “lax” form of (projective) Σt-cofibrancy.

12

equalizers (displayed vertically)

Σ

( ∐p≥1

O(p+ t) ∧Σp×Σt

(OAn)∧p ∧ (Yn)∧t

)////

Σ

( ∐p≥1

O(p+ t) ∧Σp×Σt

(An)∧p ∧ (Yn)∧t

)

∐p≥1

O(p+ t) ∧Σp×Σt

Σ(OAn)∧p ∧ Σ(Yn)∧t////

∐p≥1

O(p+ t) ∧Σp×Σt

Σ(An)∧p ∧ Σ(Yn)∧t

∐p≥1

O(p+ t) ∧Σp×Σt

(OAn+1)∧p ∧ (Yn+1)∧t////∐p≥1

O(p+ t) ∧Σp×Σt

(An+1)∧p ∧ (Yn+1)∧t

(2.8)

Here the top vertical maps are induced by the diagonal maps S1 → S2, andthe bottom vertical ones are the induced by the previously described spectrastructure maps. Note that the group actions are trivial on these sphere coordi-nates. That the diagram commutes follows from the fact that the bottom mapsare themselves built using diagonal maps. As in the p = 0, t > 1 it follows thatthe (Osp,A(t)∧Σt Y

t)p≥1, (Osp,A(t)∧ΣtQtt−1)p≥1 spectra summands are null ho-

motopic (note however that no form of Σt-cofibrancy is required in this case),concluding the proof.

Corollary 2.9. The induced adjunctions Sp(ModO[1]) Sp(AlgO) are Quillenequivalences.

Proof. We need to show that, for A ∈ Sp(SpΣ) cofibrant and D ∈ Sp(AlgO)fibrant, a map OspA→ D is a w.e. iff the adjoint map A→ D is. However, the

second of these maps factors through the first as Aµ−→ OA→ D, where µ is the

unit of the free-forget adjunction. Hence it suffices to show that µ is a stableequivalence for any cofibrant A.

To see this, a retraction argument shows that one can reduce to the case ofA being transfinitely built out as A = colimβ<κAβ where each successor ordinalmap Aβ → Aβ+1 is a pushout as in (2.4). It then suffices to inductively showthat in the diagram

Osp(Xβ)

// Aβ

// OAβ

Osp(Yβ) // Aβ+1

// OAβ+1.

(2.10)

the right side “pushout corner map” Aβ+1

∐AβOAβ → OAβ+1 is both a

monomorphism and a stable equivalence (so that Aβ → OAβ is a trivial cofi-bration in the transfinite directed diagram category (Sp(SpΣ))κ). But this nowclearly follows from the analysis in the proof of Theorem 2.3, which shows thatthe p = 0, t = 1 part of the filtration of OAβ → OAβ+1 is just the pushout ofAβ → Aβ+1, and all the other parts of the filtration are null-homotopic.

13

Remark 2.11. We point out that Theorem 2.3 could have itself been provenby regarding Sp(AlgO) as the algebras for a certain monad over22 Sp(ModO[1]).

This would entail studying colimits of the form

O O(1) (X)

// A

O O(1) (Y ) // B,

where X and Y are O(1)-modules, by producing a filtration analogous to theone described in Diagram 2.5. Indeed, the only difference between the nor-mal construction and this one is that ∧ symbols get replaced by relative ∧O(1)

symbols.

One would then obtain a sequence of (O(1),O(1)on)-bimodules OO(1)A (n),

forming a “relative enveloping operad”. We point out, however, that OO(1)A (n)

is just OA(n) again23. Indeed, it is straightforward to see that the OA(n) arethemselves (O(1),O(1)on)-bimodules (either by direct analysis of the formula,or by using the canonical map of operads O → OA), and one sees that thesesequences must match since they are both codifying left adjoints to forgetfulfunctors.

Remark 2.12. Notice that in the commutative diagram of Quillen adjunctions(with vertical adjunctions induced by the map of operads O → O(1))

AlgO//

Sp(AlgO)oo

ModO(1)

//

OO

Sp(ModO(1))oo

OO

both the lower and right adjunctions are Quillen equivalences. Indeed, thatthe lower adjunction is a Quillen equivalence follows from Theorem 5.1 of [20],since ModO(1) was stable to start with. The right adjunction is also the rightadjunction in the diagram of adjunctions

Sp(ModO(1)) Sp(AlgO) Sp(ModO(1))

induced by the maps of operads O(1)→ O → O(1). Since these maps composeto the identity in O(1), so do the adjunctions, so that the right adjunction willbe a Quillen equivalence if the first is, and this we showed in Corollary 2.9.

It then follows that we can think of the adjunction

AlgOO(1)O−// ModO(1)forgetoo

as the stabilizing adjunction for AlgO, with O(1) O − playing the role of Σ∞

and forget the role of Ω∞, so that we will occasinally refer to these functors bythose names.

22Note also that the theory from [20] does ensure that a stable model structure onSp(ModO[1]) exists, since ModO[1] is left proper.

23Note that this is not quite obvious from the defining formulae.

14

We notice that both of these functors also go by other names in the literature.Σ∞ is often called the indecomposables functor, since it is obtained from analgebra X by killing elements that can be written as higher n-ary operations (i.e.n ≥ 2). It is also costumary, at least for some operads, to denote its derivedfunctor by topological Andre-Quillen homology. Ω∞ is often called thetrivial extension, since its image consists of those algebras where the highern-ary operations identically vanish.

Finally, notice that we do not at this point yet know if the model structurein AlgO we constructed fits the paradigm described on Subsection 1.5.

Proposition 2.13. The model structure defined by Theorem 2.3 coincides withthe left Bousfield localized model structure as required by Definition 1.16.

Proof. We first show that the two model structures have the same cofibrations.Both sets of generating cofibrations are constructed based on the generatingcofibrations I of SpΣ. For the model structure of Theorem 2.3 the generatingcofibrations are then Osp(

⋃n FnI), while for the model structure in Definition

1.16 they are⋃n FnOI. It is straightforward to check that these sets match

(indeed, this just repeats the analysis in Proposition 2.1).Now repeating the analysis above it also follows immediately that the gen-

erating trivial cofibrations for the projective model structure on Sp(AlgO) aretrivial cofibrations for the model structure from Theorem 2.3, so that one con-cludes that the latter model structure is a left Bousfield localization of the for-mer. Since left Bousfield localizations are completely determined by the classesof local objects 24, it now suffices to check that those are the same in both modelstructures.

By Proposition 3.2 of [20]25, the local objects as defined using Definition1.16 are levelwise fibrant spectra X∗ ∈ Sp(AlgO) such that the structure mapsXn → Map(S1, Xn+1) are weak equivalences. Since clearly these are also thefibrant objects for the model structure of Theorem 2.3, the proof is complete.

3 Homogeneous functors from ModA to ModB

Our goal in this section is to classify finitary homogeneous functors from ModAto ModB . We start by recalling the analogous result for finitary homogeneousfunctors from SpΣ to SpΣ (this can be found, for instance , as Corollary 2.5 in[22]).

Theorem 3.1 (Goodwillie). Let F : SpΣ → SpΣ be a finitary homogeneousfunctor. Then there exists a spectrum δnF with a Σn action such that F ishomotopic to the functor X 7→ (δnF ∧X∧n)hΣn .

The purpose of section is to show the following natural generalization.

24Note that this is not the same as saying that a model structure is determined by thecofibrations and the cofibrant objects.

25Note that though that the statement of that Proposition requires left properness, theremarks immediately after the proof point out that that condition is unnecessary when thedomains of the generating cofibrations are themselves cofibrant, as is the case for AlgO.

15

Theorem 3.2. Let F : ModA → ModB be a finitary homogeneous simplicialfunctor. Then there exists a (B,A∧n)-bimodule δnF with a Σn action inter-changing the A-module structures (or, in other words, a (B,Aon)-bimodule) suchthat such that F is homotopic to the functor X 7→ (δnF ∧A∧n X∧n)hΣn .

Here we assume the functor is already simplicial both for simplicity andbecause that suffices for the purpose we have in mind.

Naturally the bimodule δnF in the theorem has to be F (A ∧ S0), the valueat the free module over the sphere spectrum. The hardest task is to prove thatin general this object can actually be given the desired bimodule structure in astrict sense.

Recalling that a (strict) A-module structure on X is just a map of (strict)ring spectra A → End(X) it is clear that this problem would be immediatellysolved if one were dealing with spectral functors rather than mere simplicialones. Further, since a heuristic argument shows that sprectral functors andlinear simplicial functors share many homotopical properties, the case for n = 1would essentially be solved were one to show that the homotopy theories of suchfunctors are equivalent, and this is the essential goal of the next subsection.

Further, we note that the case n > 1 does not really present a bigger chal-lenge. Indeed, though n-homogeneous functors can not, of course, be madespectral, they correspond to symmetric n-multilinear functors (when the targetcategory is stable), and hence the n > 1 will also be solved if one generalizes then = 1 result to show that the homotopy theories of multilinear multisimplicialsymmetric functors and multispectral symmetric functors are equivalent.

3.1 Linear simplicial functors are spectral functors

In this subsection we shall be using several basic notions of enriched categorytheory, enriched representable functors and weighted enriched colimits, alongwith some basic results about those. We recomend [29] as a general referencefor these.

Remark 3.3. Let SpΣ be given the flat stable model structure, and considerthe induced projective model structure on ModA, for A any ring spectrum.

Then it follows from SpΣ being a monoidal category that ModA is a SpΣ

model category with mapping spectra

SpA(X,Y ) = eq(Sp(A ∧X,Y ) ⇒ Sp(X,Y ))

and the obvious (underlying) spectral tensoring and cotensoring.Notice that ModA is also a simplicial model category by reduction of sc-

truture, with the simplicial mapping spaces induced given by SSetA(X,Y ) =Ω∞SpA(X,Y ), and the obvious (underlying) simplicial tensoring and cotensor-ing.

We will have use for the following technical result concerning cofibrant/fibrant replacements in these categories.

Proposition 3.4. ModA with the model structure described above has simplicialcofibrant and fibrant replacement functors. Furthermore, it also has spectralfibrant replacement functors.

16

Proof. The statement about simplicial cofibrant and fibrant replacements is wellknown (see, for example, Theorem 13.5.2 in [29]), as one needs only to performthe enriched version of the Quillen small object argument, which then worksbecuse all simplicial sets are cofibrant.

The claim about the existence of a spectral fibrant replacement functor ismore delicate precisely because not all spectra are cofibrant, but a careful anal-ysis of the enriched small object argument in that case still provides a fibrantreplacement.

Indeed, note that a general generating trivial cofibration for ModA has theform A ∧ Z → A ∧W , for Z → W a generating trivial cofibration of SpΣ, andthat the enriched argument requires, at the stage Xβ for one to build Xβ+1 asthe pushout ∨

gen. triv. cof. Z→WSpΣ(A ∧ Z,Xβ) ∧A ∧ Z

// Xβ

∨gen. triv. cof. Z→W

SpΣ(A ∧ Z,Xβ) ∧A ∧W // Xβ+1.

One now sees that the map SpΣ(A ∧ Z,Xβ) ∧ A ∧ Z → SpΣ(A ∧ Z,Xβ) ∧A ∧ W is still a trivial cofibration in the levelwise model structure (though,crucially, possibly not in ModA), so it still follows that the map X → X∞ is anequivalence, and it is formal to check that X∞ is fibrant, proving the result.

Proposition 3.5. Let C ⊂ModA be a small subcategory.Consider the categories FunSSet(C,ModB) and FunSpΣ(C,ModB) of, re-

spectively, enriched simplicial and enriched spectral functors from C to ModB.Then the projective model structures on both of these functor categories exist.Furthermore, these model categories are simplicial and cofibrantly generated,

and FunSpΣ(C,ModB) is also a spectral model category.

Proof. Letting I and J denote the sets of generating cofibrations and generatingcofibrations of ModB , one obtains the natural candidates for the generating cofi-brations for the functor categories FunSSet(C,ModB) and FunSpΣ(C,ModB),the sets

∐c∈Ob(C) SSet(c,−)∧I, SSet(c,−)∧J and SpΣ(c,−)∧I, SpΣ(c,−)∧J

To see this defines a cofibrantly generated model category one needs to checkthe conditions of Theorem 2.1.19 of [19]. The only non obvious condition is 4,the fact J − cell ⊂ W ∩ (I − cof) or, more specifically, proving J − cell ⊂ W .But this follows from the fact that the maps in SSet(c,−)∧J and SpΣ(c,−)∧Jare levelwise monomorphisms and stable equivalences (since so is the smash ofa stable trivial cofibration with any spectrum).

That these categories are simplicial is formal, with mapping spaces definedby

SSet(F,G) = eq(∏c∈C

SSet(F (c), G(c)) ⇒∏

c,c′∈CSSet(F (c) ∧ SSet(c, c′), G(c′)))

with each map induced by adjointess using either the structure of either F orG as simplicial functors. It is also formal that one has simplicial tensoring andcotensorings, which are just pointwise, and it is then obvious that the modelstructure is simplicial (by verifying that condition using the cotensoring).

17

The proof that FunSpΣ(C,ModB) is also a spectral model category is en-tirely analogous.

Proposition 3.6. There is a simplicial Quillen adjuntction

Spf : FunSSet(C,ModB) FunSpΣ(C,ModB) : fgt

Proof. The right adjoint is the natural “restriction” of spectral functors to sim-plicial ones obtained by applying Ω∞ to the maps

SpΣC(c, c′)

G(c,c′)−−−−→ SpΣModB (G(c), G(c′))

that compose a spectral functor G.The left adjoint is defined freely by its value on the representable functors

SSet(c,−) ∧ X, which one easily verifies are necessarily sent to SpΣ(c,−) ∧X. Since any simplicial functor is canonically an enriched weighted colimit ofrepresentables, namely

F = colimWCop×ModB (SSet(c,−) ∧X),

where the weight W : C × Dop is W (c, x) = SSet(X,F (c)), one obtains theexplicit formula

SpfF = colimWCop×ModB (SpΣ(c,−) ∧X).

It is worth noting that the model categories we just defined are not the oneswe are ultimately interested in, both because Goodwillie calculus deals onlywith homotopy functors and because we want to restrict to those topologicalfunctors which happen to be 1-homogeneous.

In order to do this, we need to localize the model structures we just defined,and being able to do this is the main purpose of the following result.

Lemma 3.7. The model structures on FunSSet(C,ModB) and FunSpΣ(C,ModB)are left proper cellular.

Proof. Reall that it was shown in the proof of 3.5 cofibrations in these categoriesare in particular pointwise monomorphic natural transformations.

Hence left properness follows immediately from the fact that in ModB thepushouts of weak equivalences by monomorphisms are weak equivalences.

As for cellularity, the proof is essentially a repeat of the proof of 1.5, butwith a minor wrinkle, which we explain now. Suppose given a natural transfor-mation F

τ−→ G, where G is a cellular functor. Since the generating cofibrationsare pointwise monomorphisms, the argument from the proof of 1.5 applies toshow that there is a small enough26 subcellular functor G of G such that the τ(c)

factor uniquely as Fτ(c)−−→ G

i(c)−−→ G. The wrinkle to verify is that these τ(c)assemble to an enriched natural transformation. This amounts to a straightfor-ward diagram chase of diagrams of enriched mapping spaces, though one needsto point out that the i(c) are monomorphisms in the enriched sense, rather thanjust categorically.

26The cardinal of cells being bounded by the union of the cardinals of all the simplices inthe F (c).

18

Lemma 3.7 means that, by the main Theorem of [18], one is free to localizethese model structures, and hence to obtain appropriate model categories ofhomotopical, excisive, etc, functors. Doing this requires some technical care,however, since our source category C is a small subcategory of ModA. Hence,for instance, demanding a functor to be homotopical is not something one wouldexpect meaningful if C is too small, so that w.e. objects can not actually belinked by (a zig zag of) weak equivalences in C.

Ensuring these type of problems do not occur is the goal of the followingdefinition.

Definition 3.8. Let C ⊂ModA be a small full simplicial/spectral subcategory.We say C is Goodwillie closed, if ∗ ∈ C, C is closed under finite hocolim-

its, tensoring with finite spectra, a chosen spectral fibrant replacement functor(as is ensured to exist by Proposition 3.4), and a chosen simplicial cofibrantreplacement functor.

Notice that any small C has a small Goodwillie closure C, since the closureconditions do not increase the cardinal of any (infinite) C.

Definition 3.9. The homotopical model structures FunhSSet(C,ModB) andFunhSpΣ(C,ModB) are the left Bousfield localizations with respect to the maps

SSet(c,−)→ SSet(c′,−) induced by weak equivalences in C.Notice that the fibrant objects for these model structures are just the level-

wise fibrant homotopical functors.

Remark 3.10. Notice that since the left adjoint clearly sends fibrant objectsto fibrant objects, the adjunction from 3.6 descends to a Quillen adjunctionbetween the homotopical model structures

FunhSSet(C,ModB) FunhSpΣ(C,ModB).

Definition 3.11. The linear model structure

Funh,linSSet(C,ModB)

is the further localization with respect to the maps

∗ → SSet(∗,−) ∧X,

and, for each hococoartesian square

A //

B

C // D

in C, the maps SSet(B,−)∧X∐hSSet(D,−)∧X SSet(C,−)∧X → SSet(A,−)∧X.

(Here in both cases X ranges over the domains/codomains X of the generatingcofibrations of ModB .)

Remark 3.12. Notice that, by Proposition 3.2 of [20], which says that w.e.s inMobB can be detected by the mapping spaces out of such X, it follows that thefibrant objects of Funh,linSSet(C,ModB) are precisely the levelwise fibrant, homo-topy functors which are homotopically pointed27 and sending pushout squaresto pullback squares.

27I.e., sending ∗ to a contractible object.

19

Lemma 3.13. Suppose C is Goodwillie closed.Then for F any homotopical spectral functor, the reduced topological functor

fgtF is pointed and 1-excisive.

Proof. To see that fgtF is pointed, note that ∗ is the only object in either Cor ModB where the identity map and the null self map coincide. But sinceany functor preserves identity maps and spectral functors preserve null maps itfollows that fgtF is pointed.

To check that such a functor is 1-excisive it suffices, by [26], to check that thenatural map28 F → ΩFΣ is a weak equivalence (here we are free to assume thatF is pointwise bifibrant, as necessary). Since the target category is stable thisis equivalent to showing that the adjoint map ΣF → FΣ is a weak equivalence.We show this by proving the existence of left and right inverses up to homotopy.To do this, let Σ−1 denote F1S

0 (recall that Σ and Ω are defined based on F0S1).

Notice that there is a natural w.e. Σ−1Σε−→ S0.

To prove the existence of a left inverse is suffices to do so for Σ−1ΣF →Σ−1FΣ, and this inverse provided by the diagram

Σ−1ΣF → Σ−1FΣ→ FΣ−1Σ→ F

where the second map follows from F being a spectral functor, and the last mapis induced by ε. The full composite is then a weak equivalence since it too isinduced by ε.

The other side is analogous. To show a left inverse it suffices to do so forΣFΣ−1 → FΣΣ−1. This inverse is provided by the diagram

ΣΣ−1F → ΣFΣ−1 → FΣΣ−1 → F

where the first map again follows by F being a spectral functor, the last mapfrom ε, and again the full composite is a weak equivalence since it is induced byε.

Lemma 3.14. Suppose C is Goodwillie closed.Then the fibrant replacement of SSet(c,−)∧X in Funh,linSSet(C,ModB), where

X is stable cofibrant, is, up to levelwise fibrant replacement, given by

fgt(SpΣ((c)c, (−)f ) ∧X),

where (−)f denotes a (spectral) functorial fibrant replacement functor, and (−)c

denotes any cofibrant replacement functor.

Proof. Since levelwise fibrant replacement does not affect the other fibrancyconditions in Funh,linSSet(C,ModB) we will largely omit it so as to simplify nota-tion.

First we note that SSet(c,−) ∧ X a priori fails all fibrancy conditions, asit is neither homotopical, homotopically pointed, or 1-excisive. We deal withthese conditions in succession.

First, we claim the “homotopification” of SSet(c,−)∧X is SSet((c)c, (−)f )∧X. That the canonical map SSet(c,−) ∧ X → SSet((c)c,−) ∧ X is w.e. in

28Here we drop fgt for simplicity of notation.

20

the homotopical model structure is immediate since this is precisely one of themaps being localized. Hence one needs only deal with the case of c a cofibrantobject, which we assume in the remainder of the proof. Now notice that thereis a canonical natural transformation SSet(c,−) ∧ X → SSet(c, (−)f ) ∧ X,induced by the natural transformation id → (−)f , and hence, since clearlySSet(c, (−)f ) ∧X is homotopic, the claim will follow if we show that it is theuniversal homotopy functor with a map from SSet(c,−)∧X, where universalityis read in the homotopy category of functors. Hence let SSet(c,−) ∧ X →F be a natural transformation with F a homotopy functor. Existence of thefactorization then follows immediatly from the fact that the map on the rightin the natural diagram

SSet(c,−) ∧X

// F

∼

SSet(c, (−)f ) ∧X // F (−)f

is a levelwise equivalence. Similarly, uniqueness follows from the bottom leftmap in

SSet(c,−) ∧X

// SSet(c, (−)f ) ∧X //

∼

F

∼

SSet(c, (−)f ) ∧X ∼ // SSet(c, ((−)f )f ) ∧X // F (−)f

being a levelwise equivalence. Indeed, the claim is that the right upper mapis determined by the upper composite. But for this is suffices for it to bedetermined by the lower composite, and this follows from knowing that theindicated maps are levelwise equivalences.

We next show that the homotopically pointed localization of SSet(c, (−)f )∧X is SSet∗(c, (−)f )∧X, where we denote by SSet∗(−,−) the enrichement overpointed simplicial sets SSet∗ of the simplical pointed category C. This may seemslightly confusing since SSet(−,−) and SSet∗(−,−) have the same underlyingsimplicial set, but the crucial point here is that the ∧ operations are differentdepending on which context one is working in, so that SSet(c, (−)f )∧X becomesreinterpreted as (SSet∗(c, (−)f ))+ ∧ X in the pointed simplicial context. Butnow notice that the natural pushout diagram

X ' (SSet∗(∗, (−)f ))+ ∧X

// (SSet∗(c, (−)f ))+ ∧X

∗ // SSet∗(c, (−)f ) ∧X

is in fact a levelwise homotopy pushout (this follows from the properties of thelevel cofibration model structure on spectra, since the top map is an injection),and hence it is now clear that

Map(SSet(c, (−)f ) ∧X,F ) 'Map(SSet∗(c, (−)f ) ∧X,F )

for any F a homotopically pointed functor.

21

Finally, it remains to show that the “1-excisification” of SSet∗(c, (−)f )∧Xis fgt(SpΣ(c, (−)f )). Recall that, as proven in [26], and following Goodwillie,the linearization of a pointed (pointwise fibrant) simplicial functor F can becomputed by hocolimk(Ωk F Σk). Consider the natural morphism29

hocolimk(Ωk(SSet∗(c, (−)f )∧X)fΣk)→ hocolimk(Ωk(SpΣ(c, (−)f )∧X)fΣk).

By Lemma 3.13, it suffices to show that this map is a weak equivalence. Butup to equivalence this map can be rewritten as

hocolimk(ΩkΣ∞SSet∗(c, (−)f )Σk)∧X → hocolimk(ΩkSpΣ∗ (c, (−)f )Σk)∧X,

and it hence suffices to show that hocolimk(Ωk Σ∞SSet∗(c, (−)f ) Σk) →hocolimk(Ωk SpΣ

∗ (c, (−)f ) Σk) is a weak equivalence, but this follows fromthe fact that a spectrum K can be described as hocolim(Σ∞−k(ΣkK)), finishingthe proof.

Theorem 3.15. The adjunction

Funh,linSSet(ModfinA ,ModB) FunhSpΣ(ModfinA ,ModB)

is a Quillen equivalence.

Proof. That this is indeed a Quillen adjunction is just a restatement of Lemma3.13.

Next we check that the (homotopy) unit of the adjunction is a w.e.. By thesmall object argument for the formation of cofibrant replacements it suffices toprove the statment for functors built cellularly from the generating cofibrationsSSet(c,−) ∧X → SSet(c,−) ∧ Y . Obviously the left adjoint functor preserveshomotopy colimits, but note further that so does the right adjoint fgt, sincecolimits of natural transformations are pointwise in the target. It hence sufficesto show that the (homotopy) unit is a w.e. for functors of the form SSet(c,−)∧X. But this is just Lemma 3.14 together with the fact that the homotopicalreplacement of SpΣ(c,−) ∧X is SpΣ((c)c, (−)f ) ∧X, as is clear from the proofof that Lemma.

Finally, it suffices to check that the (right derived functor of the) right adjointis conservative with respect to weak equivalences. But this is clear since in bothcategories weak equivalences between fibrant functors are given by pointwiseweak equivalences, and clearly fgt preserves those.

The previous theorem naturally generalizes to the case of multilinear func-tors. We briefly indicate the changes that need to be made to the previousdiscussion to obtain that result.

Definition 3.16. Let C, D be categories. A symmetric n-multifunctor from Cto D is a functor Cn F−→ D together with natural isomorphisms

F∼−→ F σ,

where σ denotes the natural action of a permutation on Cn, together with theobvious compatibility conditions.

29Here we omit the reduction fgt of spectral functors to simplicial functors for simplicityof notation.

22

Theorem 3.17. Suppose C is Goodwillie closed.Then there is a Quillen equivalence

SymMFunh,multlinSSet (ModfinA ,ModB) SymMFunhSpΣ(ModfinA ,ModB).

Proof. First notice that in the category MFunSSet(C,D) of multifunctors with-out symmetry constraints, the role of representable functors is played by thefunctors of the form SSet(c1,−) ∧ · · · ∧ SSet(cn,−) ∧X, for (c1, . . . , cn) ∈ Cn.There is an obvious action of Σn on such functors (permuting the ci), and henceone sees that the analog of 3.5 showing the existence of a cofibrantly gener-ated projective model structure follows by considering generating cofibrationsI ∧ SSet(c1,−) ∧ · · · ∧ SSet(cn,−) ∧Σn Σn and generating trivial cofibrationsJ ∧ SSet(c1,−) ∧ · · · ∧ SSet(cn,−) ∧Σn Σn. The case of multispectral functorsfollows analogously.

The proof of the analog of Lemma 3.7, showing these are left proper cellularmodel structures, is entirely analogous, and the sets of localizing maps to obtainmultihomotopical and multilinear model structures are induced from the n = 1case by30

Smulth = Sh ∧ SSet(c2,−) ∧ · · · ∧ SSet(c2,−) ∧Σn Σn

Smultlin = Slin ∧ SSet(c2,−) ∧ · · · ∧ SSet(c2,−) ∧Σn Σn.

The analog of Proposition 3.6, the existence of an adjunction is perhaps alittle less clear, since it may not be obvious how to canonically express a sym-metric functor as a colimit of appropriate representables. Rather than provingsuch a colimit expression, we notice instead that since such a technique doeswork for the non symmetric functor categories MFun, and since the right ad-joint in that case is obviously compatible with the Σn action on Cn, then so isthe left adjoint, and hence that left adjoint naturally sends symmetric functorsto symmetric functors, providing the desired adjuntion.

Finally, to finish the proof that this adjunction is indeed a Quillen equiv-alence one needs only the appropriate analogs of Lemmas 3.13 and 3.14, theproofs of which require no noteworthy alterations.

3.2 Characterization of n-homogeneous finitary functors

In this section we finish the proof of Theorem 3.2, after a couple of additionalLemmas.

Here is a sketch of the main idea, ignoring some technicalities. As mentionedbefore, it suffices to prove the associated result saying that any symmetric Gmultilinear functor has the form

X1, . . . , Xn 7→ δnG ∧A∧n X1 ∧ · · · ∧Xn.

By a cofibrant replacement argument, one can essentially assume that theobjects in the source category are cell complexes. Lemma 3.18 then says thatany cell complex is the filtered hocolimit of its finite sucomplexes, meaning thatG can be completely recovered from its values on the category of finite complexes

30Here we are abusing notation by noting that Sh and Slin are built out of functors of theform XSSet(c,−), so that the notion of “permuting the ci” makes sense.

23

C0. But then letting C be a Goodwillie closure of C0, Theorem 3.17 allows oneto replace that restriction by a spectral functor G, so that G(A, . . . , A) thenhas a genuine (B,An) module structure, and Lemma 3.19 finally provides thedesired map G(A, . . . , A) ∧An X1 ∧ · · · ∧ Xn → G(X1, . . . , Xn), and it is theneasy to finish the proof.

We now prove the aforementioned Lemmas.

Lemma 3.18. Let X be any cellular object of ModA (based on any of thegenerating sets of cofibrations in Definition 1.3 other than those for the levelcofibration model structure).

Let further Xfin denote the category of finitesubcomplexes of X together withtheir inclusions.

Then one has a canonical equivalence

hocolimC∈XfinC∼−→ X.

Proof. We first show that colimC∈XfinC'−→ X is an isomorphism. Since this

colimit is filtered (as the union of finite subcomplexes is clearly a finite subcom-plex), this is just equivalent to showing that any simplex of (any spectral level)of X belongs to some finite subcomplex, which we further claim can be chosenminimal. Call such a simplex s.

Now recall that X is presented as a transfinite colimit of monomorphisms

∗ = X0 → X1 → X2 → . . . X

with Xβ+1 obtained from Xβ by attaching cells31 A∧Fn∆m along their “bound-ary” A ∧ Fnδ∆m.

Notice that there must be a minimal β+ 1 for which s ∈ Xβ+1, and a singlespecific cell es being attached to Xβ for which s ∈ es (abusing notation). Theclaim now follows by induction on β + 1. Indeed, the “boundary” of es has theform A∧Fnδ∆m, and a map out this boundary is determined completely by theimage of finitely many simplices y0, . . . , ym and, by induction, those images arecontained in minimal finite subcomplexes C1, . . . , Cm, and it is then clear thatC1 ∪ · · · ∪Cm ∪es is the minimal finite subcomplex containing s. Notice thatfrom such such a complex being minimal it then follows that the intersection offinite subcomplexes is still a finite subcomplex.

We have now proven that colimC∈XfinC'−→ X, so that the hocolim result

will immediately follow from knowing that the identitiy diagram Xfin →ModAis projectively cofibrant. This ammounts to showing that for any C the canonicalmap colimC′ CC

′ → C is a cofibration. But this is clear since colimC′ CC′ is

ensured to be a subcomplex of C due to the intersection of finite subcomplexesbeing a finite subcomplex.

Lemma 3.19. Let G : Cn → ModB be a multispectral functor, where C ⊂ModA is a full subcategory closed under the free module construction on objectsand containing A, the free module on the sphere spectrum S. Then there is anatural transformation

G(A, . . . , A) ∧An X1 ∧ · · · ∧Xn → G(X1, . . . , Xn)

31Note that though we present the argument for the generating cofibrations based on thestable model structure on SpΣ, the argument applies to the other model structures with nomajor alterations.

24

induced by the “identity” map at (A, · · · , A).Furthermore, this natural transformation is compatible with Σn actions when

F is a symmetric functor.

Proof. We first notice that G(A, · · · , A) does indeed have n commuting rightmodule A structures, which are induced by the composites

A→ SpΣA(A,A)

Gi−−→ SpΣB(G(A, . . . , A), G(A, . . . , A))

where the first map describes the right A-module structure of A in ModA andthe second map corresponds to spectrality of F in its i-th variable. For theremainder of the proof we deal only with the case n = 1 so as not to overbearthe notation, but we note that the case n > 1 presents no further difficulties.

Now consider the diagram

G(A) ∧A ∧X ⇒ G(A) ∧X → F (A ∧X)→ F (X),

where the first two maps correspond to the left A-module structure on X andthe right A-module structure on G(A), the middle map is adjoint to X →SpΣ

A(A,A ∧ X) → SpΣB(G(A), G(A ∧ X)), and the final map is obtained by

applying G to the structure multiplication map A ∧X → X, which we note isa map in ModA. Since F (A) ∧A X is the coequalizer of the first two maps wewill be finished by showing that the top and bottom compositions coindice.

Consider first the composition corresponding to the left module structure onX. By adjointness this corresponds to a composite

A ∧X → X → SpΣA(A,A ∧X)→ SpΣ

B(G(A), G(A ∧X))→ SpΣB(G(A), G(X)),

and by functoriality of F with respect to A ∧ X → X this is the same as acomposite

A ∧X → X → SpΣA(A,A ∧X)→ SpΣ

A(A,X)→ SpΣB(G(A), G(X)).

On the other hand, the composition corresponding to the right module struc-ture on G(A) corresponds by adjointness to a composite

A ∧X → SpΣA(A,A) ∧X → SpΣ

B(G(A), G(A)) ∧X → SpΣB(G(A), G(A) ∧X)

→ SpΣB(G(A), G(A ∧X))→ SpΣ

B(G(A), G(X)),(3.20)

which is rewritten using naturatily of F with respect to X → SpΣA(A,A∧X) as

A ∧X →SpΣA(A,A) ∧X → SpΣ

A(A,A ∧X)→→SpΣ

B(G(A), G(A ∧X))→ SpΣB(G(A), G(X)),

(3.21)

which can be further rewritten using naturality with respect to the multiplica-tion map A ∧X → X as

A ∧X → SpΣA(A,A) ∧X → SpΣ

A(A,A ∧X)→ SpΣA(A,X)→ SpΣ

B(G(A), G(X)).(3.22)

Looking at both composites one sees that they both factor as a map A∧X →SpΣ

A(A,X), and unwinding definitions one sees that these two maps are preciselythe maps that must match for X to be an A-module, finishing the proof.

25

Proof of Theorem 3.2. It suffices to show that any symmetricGmultilinear mul-tisimplicial functor is equivalent to one of the form

X1, . . . , Xn 7→ δnG ∧A∧n X1 ∧ · · · ∧Xn.

First one replaces G by G Qn, where Q denotes the Quillen small objectargument cofibrant replacement functor. Since Q(f) for f : X → Y is always acellular map, one has induced funtors32

hocolim((QX)fin)nG→ colim((QX)fin)nG→ G Qn,

where (QX)fin denotes the category of finite subcomplexes of QX. It thenfollows by Lemma 3.18 that the full composite is a weak equivalence for all X,since it is a weak equivalence on finite complexes and both functors commutewith filtered colimits.

Furthermore, it is clear that one can replace the restriction of G to thecategory of finite complexes by any other equivalent functor G while still havinga zig zag of weak equivalences between G and hocolim((QX)fin)nG.

Now let C be the smallest Goodwillie closed category that contains (a skele-ton of) all finite complexes, and that is further closed under forming the freealgebra over its objects. Then Theorem 3.17 provides an equivalent spectralfunctor G, and Lemma 3.19 then provides a map of functors G(A, . . . , A) ∧AnX1∧· · ·∧Xn → G(X1, . . . , Xn). While it is not necessarily obvious whether thisnatural transformation is a weak equivalence over the whole of C, since it is soon (A, · · · , A) it is also so for any tuple with coordinates of the form A∧Fnδ∆m

(as this is a suspension of A and both sides are linear functors), and hence alsoon any finite complexes.

It hence finally follows that F is equivalent to hocolim((QX)fin)nG, and sincethis last one is clearly (the left derived functor of)

X1, . . . , Xn 7→ δnG ∧A∧n X1 ∧ · · · ∧Xn,

the proof is concluded.

4 Goodwillie Calculus in the AlgO categories

In this section we present our main results concerning Goodwillie Calculus as itrelates to the AlgO categories. We point out that we will throughout deal onlywith the case where those categories are pointed or, equivalently, O(0) = ∗.

In subsection 4.1 we finally assemble the results proven so far to characterizethe Goodwillie tower of the identity in AlgO as being the homotopy completiontower studied in [17] and associated to the truncated operads O≤n.

In subsection 4.2 we show that, when studying n-excisive functors eitherto or from AlgO one can equivalently study n-excisive functors to or fromAlgO≤n . Combining this with subsection 4.1 this effectively says that the cat-egory AlgO≤n can be recovered from the category AlgO purely in Goodwillie

32Here we use for hocolim((QX)fin)nG the models B(∗, (QX)fin, G) 'N(−/(QX)fin) ⊗(QX)fin G, as described in Theorem 6.6.1 of [29], and were we make

a (simplicial) pointwise cofibrant replacement of G if necessary.

26

calculus theoretic terms. One might then wonder whether an arbitrary homo-topical category C admits an analogous “truncation” C≤n, a question the authorwould like to examine in future work.

Finally, subsection 4.3 shows that for finitary functors between categories ofthe form AlgO one does have at least a weak version of the chain rule as provedfor spaces and spectra (and conjectured in general) in [1].

4.1 The Goodwillie tower of IdAlgO

In this subsection we characterize the Goodwillie tower of IdAlgO . First weintroduce some notation.

Definition 4.1. Let O be an operad such that O(0) = ∗. Then the truncationoperad O≤n is the operad33 whose spaces are

O≤n(m) =

O(m) if m ≤ n∗ if n < m

Note that there is a canonical map of operads O → O≤n.

Definition 4.2. Notice that putting together Theorem 3.2, the results of [26]and Remark 2.12 it follows that any n-homogeneous finitary functor F : AlgO →AlgO has the form34

X 7→ Ω∞O (δnF ∧O(1)n (Σ∞OX)n)hΣn

where δnF is a (O(1),O(1)on)-bimodule, which we call the n-derivative of F .

The following is the main theorem of this section.

Theorem 4.3. The Goodwillie tower of the identity for AlgO is given by the(left derived) truncation functors O≤n O −.

Furthermore, the n-derivative is O(n) itself with its canonical (O(1),O(1)on)-bimodule structure.

We first show that the proposed Goodwillie tower is indeed composed ofn-excisive functors. This will follow by combining the results of [27] with themain result of the previous section.

Lemma 4.4. The left derived functors of O≤n O − : AlgO → AlgO are n-excisive.

Proof. In order to avoid the need to introduce cofibrant replacements every-where we instead deal only with the restrictions of these functors to the cofibrantobjects in ModO. Further, note that by [27, Thm. 1.1] the functor O≤n O −sends trivial cofibrations between cofibrant objects to w.e.s so that, by KenBrown’s Lemma, it is in fact a homotopic functor when restricted to cofibrantobjects.

The proof follows by induction on n. The case n = 1 follows since O≤1 O−is just Ω∞Σ∞.

33We notice that the fact that this forms an operad depends on the fact that O(0) = ∗.34Here we use the notation of Remark 2.12.

27

Letting On denote the O-bimodule with values On(n) = O(n) and On(m) =∗ for n 6= m, by [27, Thm. 1.9] one has a hofiber sequence of functors

On O − → O≤n O − → O≤(n−1) O −, (4.5)

where the first functor is On O X = O(n) ∧Σn (O(1) ∧O X)∧n, hence an n-homogeneous functor. To finish the argument, note that since holims in algebracategories are underlying, it suffices to prove that O≤nO is n-excisive as afunctor landing in spectra, and the result now follows by induction and thetwo-out-of-three property for n-excisive functors landing in a stable category.

Proof of Theorem 4.3. In order to avoid the need to introduce cofibrant replace-ments everywhere we instead deal only with the restrictions of these functors tothe cofibrant objects in ModO.

First note that since holimits in AlgO are underlying one is free to justprove that the On O − form the Goodwillie tower of the identity when viewedas functors landing in SpΣ. Thanks to Lemma 4.4 it remains only to show thatthe natural map Id→ O≤n O − induces an isomorphism on the homogeneouslayers 1 through n.

Now consider the natural left Quillen adjoints

ModO(1)

OO(1)−−−−−−−→ AlgOO(1)O−−−−−−−→ModO(1)

which, as noticed in Remark 2.12, compose to the identity. Now, since OO(1)−is a left Quillen functor it preserves homotopy colimits and hence precompositionwith it commutes with the process of forming Goodwillie towers and layers. Onthe other hand, by [26] and Remark 2.12 any such homogeneous functors factorsthrough O(1) O−, and it hence now follows that one can just as well verifythe theorem after precomposing with O O(1) −.

We have hence reduced the result to the claim that∐nk=1O(n) ∧Σn X

n isthe n-th excisive approximation to

∐∞k=1O(n) ∧Σn X

n. This is a well knownfact about analytic functors, finishing the proof.

Remark 4.6. Notice that as a particular case of Theorem 4.3 it follows thatfor any truncated operad O≤n the category of algebras AlgO≤n has the propertythat its identity functor is n-excisive.

Remark 4.7. Theorem 4.3 asserts that the n-stage of the Goodwillie tower forIdAlgO is the monad associated to the adjunction

AlgO AlgO≤n ,

hence describing it as a left Quillen functor, followed by IdO≤n , followed bya right Quillen functor, and from this perspective the n-excisiveness of thiscomposite is then a consequence of the n-excisiveness of IdAlgO≤n .

Now suppose that A is a (O′,O)-bimodule (in symmetric sequences), andconsider the associated functor

AlgOFA=AO−−−−−−−−→ AlgO′ .

28

It is then not hard to see that the proof of Theorem 4.3 can be adapted toshow that the Goodwillie tower for FA is given by the functor FA≤n , where A≤ndenotes the obvious truncated (O′,O)-bimodule.

Notice that we then have a factorization35

AlgOFA≤n //

O≤nO−

AlgO′

AlgO≤nFA≤n // AlgO′≤n

fgt

OO

associating to the n-excisive functor FA≤n : AlgO → AlgO′ an n-excisive functorFA≤n : AlgO≤n → AlgO′≤n between algebras over the truncated operads. Per-

haps more surprising is the fact that any finitary n-excisive functor admits asimilar factorization. That is the content of the next subsection.

4.2 n-excisive finitary functors can be “truncated”

The objective of this subsection is to prove the following result.

Theorem 4.8. Let AlgOF−→ AlgO′ be a finitary n-excisive functor.

Then there is a finitary n-excisive AlgO≤nF−→ AlgO′≤n such that one has a

factorization (up to homotopy)

AlgOF //

O≤nOQ

AlgO′

AlgO≤nF // AlgO′≤n .

fgt

OO

(here Q denotes a fixed cofibrant replacement functor)Furthermore, any two such F are equivalent.

The existence part of Theorem 4.8 is a fairly straightforward consequenceof Theorem 4.3 and the following Lemma, which is a fairly direct adaptation ofProposition 3.1 of [1], while the uniquess part will use the additional Lemma4.10.

Lemma 4.9. Let F,G be pointed simplicial homotopy functors between cate-gories of the form AlgO (allowing for different operads for the source and targetcategories). Assume F and G are composable. Then

1. The natural map Pn(FG)→ Pn((PnF )G) is an equivalence.

2. If F is finitary, then the natural map Pn(FG)→ Pn(F (PnG)) is an equiv-alence.

Proof. The proof is a straightforward adaptation of that of Proposition 3.1 of[1], and we hence indicate only the main differences.

35Here we abuse notation by also viewing A≤n as a (O′≤n,O≤n)-bimodule, and by hence

also denoting by FA≤n the functor associated to that (O′≤n,O≤n)-bimodule.

29

First, and as was usual in the proofs of [26], note that one is free to restrictoneself to cofibrant objects and to assume that F and G take bifibrant values.

The proof of part (1) is then essentially unchanged, with the map

Pn(Pn(F )G)vn−→ Pn(FG)

built using the enrichment of these functors over pointed simplicial sets, and it’skey properties being retained.

For part (2), the claim that the proof from [5] works when the middle cate-gory is spectra is replaced with the claim that that proof works whenever thatmiddle category is ModA, with the use of Proposition 6.10 in [14] replaced byits generalization in Proposition 3.7 of [26] . The remainder of the proof equallyfollows, as after reducing to the case where F is n-homogeneous one can alsowrite FF ′Σ∞O , where Σ∞O denotes the O(1) O − functor, and since Σ∞O sharesall the relevant properties of Σ∞, the result follows.

Lemma 4.10. Let C denote the composite of (derived functors)

AlgO≤nfgt−−→ AlgO

O≤nOQ−−−−−−→ AlgO≤n .

(here Q denotes a cofibrant replacement functor)Then the canonical counit map C → IdAlgOn becomes a weak equivalence

after applying Pn.

Proof. Notice that since fgt is conservative (i.e. it reflects w.e.s) and it com-mutes with forming Pn, one may has well prove the result after postcompositionwith fgt.

But now consider the canonical composite

fgt Q→ fgt O≤n O Q fgt→ fgt.

This full composite is a weak equivalence, and it hence suffices to checkthat the first map is a weak equivalence after applying Pn. But this follows byLemma 4.9, since the first map is induced by the unit IdO → fgt O≤n O −,which is just the n-excisive approximation of IdO by Theorem 4.3.

Proof of Theorem 4.8. Set F ′ to be the composite

AlgO≤nftgO≤n−−−−−→ AlgO

F−→ AlgO′O′≤nO′Q−−−−−−→ AlgO′≤n

and let F = Pn(F ′).Since fgtO′≤n commutes with filtered hocolims and with holims, and since

O≤n O Q commutes with hocolimits, one has (a zig zag of) an equivalence

fgtO′≤n Pn(F ′) O≤n O Q ∼ Pn(fgtO′≤n F′ O≤n O Q).

But this is then just Pn(Pn(IdAlgO ) F Pn(IdAlgO )), and Lemma 4.9 thenapplies to show that, since F was assumed finitary and n-excisive, this functoris just equivalent to F itself, proving the existence of factorization.

For uniqueness, the claim is then that any such F is weak equivalent toPn(O′≤nO′QF fgtO≤n), and our hypothesis says that this is weak equivalent

30

to Pn(C F C), where C is the composite appearing in Lemma 4.10. One thenfinishes the proof by applying Lemmas 4.9 and 4.10.

Remark 4.11. Theorem 4.8 roughly proves an equivalence of homotopy cate-gories

Ho(Funfin,≤n(AlgO, AlgO′)) ' Ho(Funfin,≤n(AlgO≤n , AlgO′≤n)).

(here the notation Funfin,≤n is meant to denote n-excisive finitary functors)We note however that this should underlie an actual Quillen equivalence of

model categories Funfin,≤n(AlgO, AlgO′) and Funfin,≤n(AlgO≤n , AlgO′≤n).

Indeed, ignoring excisiveness for the moment, the O≤n O −, fgtO type ad-junctions should indude a combined pre and post-composition adjunction be-tween the functor categories, and this adjunction descends to n-excisive functors,which one should be able to view as a localization of the “model category of allfunctors”. The proof of Theorem 4.8 is then essentially checking that the unitand counit in this hypothethical Quillen adjunction are weak equivalences.

Unfortunately it seems hard to construct appropriate Funfin,≤n model cat-egories. For instance, the techniques of Section 3.1 essentially break down dueto AlgO generally not being left proper36, and one hence does not have directaccess to the localization machinery of [18], and so we are at current reduced tothe weaker form of the result presented here.

4.3 Proto chain rule

Throughout this section we assume that each level O(n) is flat cofibrant forn ≥ 1 after forgetting the Σn actions. Note that in this case O − is anhomotopy functor when restricted to positive flat cofibrant spectra and it sendspositive flat cofibrant spectra to positive flat cofibrant spectra (both claimsfollow from37 [27, Thm.1.1] when O = I and f2 = ∗ → O). These propertieshence ensure that the functors appearing in Theorem 4.12 are homotopicallymeaningful.

Our goal in this subsection is to provide some evidence that a result analo-gous to the Chain Rule proved in [1] should also hold for functors between theAlgO categories.

Firstly, recall that in that paper it was conjectured that for C a category inwhich one can do Goodwillie Calculus one should expect that the derivatives38

δ∗(IdC) form in some sense an operad, and one can hence view Theorem 4.3 asevidence of this, since as a symmetric sequence the derivatives δ∗IdAlgO are justO itself. Unfortunately we have no intrisic construction of an operad structureon the δ∗IdAlgO with which to compare the operad structure on O, but we canoffer at least some heuristic in that sense. Namely, given a homotopy functorF : AlgO → AlgoO′ consider the composite

SpΣ OQ−−−→ AlgOF−→ AlgO′

fgt−−→ SpΣ.

36In the rare case where AlgO is indeed left proper, the main cases of this being when O isa monoid (i.e. concentrated in degree 1) or an enveloping operad of the form ComA, we dohowever expect such a treatment to be viable.

37Note that in [27] positive flat cofibrancy is refered to as positive S cofibrancy.38We note though that also part of this conjecture is that there is a sensible way to generally

define such objects in the first place.

31

As was argued in the proof of Theorem 4.3 one can read the derivatives of Fas the derivatives of this composite39. Applying this when F = IdAlgO , andusing the already known chain rule for functors in SpΣ, one gets a hypotheticaloperad structure map

δ∗(IdAlgO ) δ∗(IdAlgO ) ' δ∗(O −) δ∗(O −)→ δ∗(O −) ' δ∗(IdAlgO )

with the middle map40 induced by the natural map of functors fgtOfgtO →fgt O.

Taking this idea further one can also construct “module structures” over Ofor the derivatives of other functors to or from AlgO. For instance, in the caseof F a functor to AlgO one can write

δ∗(IdAlgO )δ∗(F ) ' δ∗(O−)δ∗(fgtF O′−)→ δ∗(fgtF O′−) ' δ∗(F ),

where the middle map is induced by the natural transformation fgtOfgtF →fgt F . The case of functors from AlgO is similar.

One has the following result, which according to the previous remarks canbe viewed as a weak version of the chain rule. We note that this is very closelyin form (and proof) related to Theorem 16.1 of [1].

Theorem 4.12. Consider finitary homotopy functors

AlgO′F−→ AlgO

G−→ AlgO′′

Then the canonical maps

1. |PnB•(fgt G O, fgt O, fgt F O′)|∼−→ Pn(fgt G F O′)

2. |DnB•(fgt G O, fgt O, fgt F O′)|∼−→ Dn(fgt G F O′)

are equivalences.Here the geometric realizations are taken in the homotopical sense (i.e. they

are performed after making a Reedy cofibrant replacement).

Proof. The overall strategy of the proof mimics that of Theorem 16.1 of [1], andwe hence focus on the points at which specific properties of AlgO need to beused.

Notice that part (2) will follow immediatelly once we know (1), as homotopyfibers commute with geometric realization.

We first deal with part (1). The first step is to notice that, by 4.9, themap G → PnG induces levelwise equivalences between the relevant augmentedsimplicial objects, so it suffices to prove the result assuming G is n− excisive.

Consider now a hofiber sequence of functors G′ → G → G′′, so that onewants to check that the result will follow for G if one knows it for both G′ andG′′. Notice that one then obtains an associated levelwise hofiber sequence ofaugmented simplicial objects. But since both weak equivalences and (homo-topically meaningful) geometric realizations in AlgO are computed in SpΣ, the

39Strickly speaking the argument used in that proof precoomposed with the maps

ModO(1)

OO(1)Q−−−−−−→ AlgO and AlgO′fgt−−→ ModO′(1), so now we are actually disregarding

the (O′(1),O(1)∧n)-bimodule structures.40Tecnichally speaking this map actually requires “inverting” the equivalence

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result follows immediately from noting that the hofiber sequence of augmentedsimplicial objects is also an underlying hocofiber sequence.

We have now hence reduced to proving the claim in the case that G is a fini-tary n-homogeneous functor. At this point notice that Pn actually commuteswith the geometric realization (since geometric realizations commute with bothfiltered hocolimits and the punctured cube holimits), so it sufficed to checkthat in this case one has an equivalence as in (1) with the Pn removed. Re-call that G has the form X 7→ trivO′′(δnG ∧hΣn TAQO(X)∧n), where δnG isa (O′′(1),O(1)on)-bimodule. This means that in the terms of the the simpli-cial augmented object one has a fgt trivO′′ on the leftmost side, and sincethis composite is just forgetting from O′′(1)-modules to SpΣ, one sees thatthe homotopy coinvariants hΣn commute with the whole construction, and canhence be removed, so that one can deal instead with a functor of the formX 7→ trivO′′(δnG ∧ TAQO(X)∧n). But in this case the augmented simplicialobject being discussed is naturally the diagonal of a n multisimplical object41,so that one can replace the geometric realization by a n-multigeometric real-ization. Since this amounts to moving the δnG ∧ − out of the whole thing, wehave reduced to the case G = TAQ. But TAQ is a left Quillen functor, hencecommuting with the (homotopical) realization, and one can hence take G asthe identity. The result now follows by Theorem 1.8 of [16], saying that anyO-algebra is canonically the (homotopical) realization of its bar construction.

41By considering the multilinear functor associated to G, X1, · · · , Xn 7→ δnG ∧TAQ(X1, · · · , Xn).

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