Goodwillie calculus in the category of algebras over a chain complex operad
aa r X i v : . [ m a t h . A T ] J un Goodwillie calculus in the category of algebras over achain complex operad
MIRADAIN ATONTSA NGUEMOUniversité Catholique de Louvain, IRMP, Louvain La Neuve, 1348, BelgiumEmail: [email protected]
Abstract.
The goal of this paper is to furnish a literature on Goodwillie calculus for func-tors defined between categories which derive from chain complexes over a ground field k . We characterize homogeneous functors F : C −→ D where C , D = Ch (chain complexes), Ch + (non-negatively graded chain complexes) or Alg O (algebras over a chain complex op-erad O ). In the particular case when D = Alg O , our characterization requires k to be ofcharacteristics . We are then extending the results of Walter [Wal06] who studied in characteristics the chain complex cases and when O is the Lie operad. AMS Classification numbers. Primary: 18D50, 55P65 ; Secondary: 18G55, 55U35, 55U15
Key words.
Operads, algebras over an operad, model category, calculus of functors
Introduction
The chain complexes are over a ground field k . Let O be a fixed operad on Ch + . In thispaper, we give a characterization of homogeneous functors F : C −→ D , where C and D are either Alg O , Ch + or Ch.
This is an "algebraic" version of a couple of publications inFunctor Calculus. It starts with Goodwillie [Goo03] when C and D are the category ofpointed topological spaces or the associated category of spectra (S-modules of [EKMM97]).He proved that homogeneous functors F are completely described in term of symmetricsequences (of spectra) ∂ ∗ F, called "Derivatives". The Functor Calculus for continuousfunctors was extended by Kuhn, in [Kuh07], to the case the categories C and D respectthe following conditions:1. The categories C and D are simplicial or topological pointed model categories;2. C and D are proper: the pushout of a weak equivalence by a cofibration is a weakequivalence, and dually for the pullbacks.3. filtered homotopy colimit in D commutes with finite homotopy limits.By inspection, one see that under good conditions on the functors, the Kuhn (or Good-willie) Taylor tower construction can be defined when only the conditions . and . aresatisfied. In our cases the categories Alg O , Ch + and Ch satisfy the Kuhn’s requirements . and . We can therefore follow his lines to develop the approximation of functors. Onthe other hand, to get a characterization theorem for homogeneous functors similar toGoodwillie’s result, Kuhn needed condition 1. In our case even if the categories
Ch, Ch + and Alg O (by Hinich in [Hin97, § ain results We develop tools in the category Alg O to analyze the Taylor tower of homotopy functors.In fact we give an explicit model of the homotopy pullbacks and we deduce that any loop O -algebra has a trivial O -algebra structure when the ground field is in characteristics . On the other hand, we give an explicit model of homotopy pushouts. As a consequence, wededuce that the suspension of an O -algebra is equivalent to a free O -algebra. We applythese constructions to show that a homogeneous functor F is completely described bya symmetric sequences of unbounded chain complexes denoted ∂ ∗ F. In fact we analyzethe Taylor tower P n F of F and we show that under good conditions, there is a weakequivalence D n F ( X ) ≃ Ω ∞ ( ∂ ∗ F ⊗ h Σ n (Σ ∞ X ) ⊗ n ) . where D n F = hofib ( P n F −→ P n − F ) . In our construction, the pair (Σ ∞ , Ω ∞ ) has adifferent concept from the Kuhn’s construction. Namely, in Kuhn’s paper [Kuh07], givena simplicial category D , the pair Σ ∞ : D ⇄ Spectra ( D ) : Ω ∞ is an adjoint pair. As in [ BM05 ] , when D = Alg O , we will identify the category Spectra ( D ) with Ch.
We know that these two categories are related by a zig-zag of Quillen equiv-alences. This identification will imply a non canonical modification in the constructionof these functors. Roughly speaking, Σ ∞ : Alg O −→ Ch becomes the Quillen homology T Q ( − ) , and Ω ∞ : Ch −→ Alg O assigns to each chain complex a trivial O -algebra struc-ture. The pair (Σ ∞ , Ω ∞ ) we get is no more an adjoint pair. We give more detail aboutthese constructions in section 1.3.When C and D are either Ch or Ch + , or when O = Lie and the ground field is incharacteristics 0, these constructions and results appear in [Wal06].
Outline of the paper
In the first section 1, we briefly remind the preliminaries on the categories Alg O , Ch + and Ch.
In section 2 we make an explicit construction of homotopy pullbacks and homotopypushouts in Alg O . We remind in section 3 the Goodwillie approach in Functor calculus. Insection 4, we characterize homogeneous functors. Our method in that section is inspired bythe Goodwillie’s approach. Namely we prove that homogeneous functors are infinite loopspaces (as in [Goo03, Thm 2.1]). We use the "stabilization" of the cross effect and at theend of this process, we obtain a similar (with Goodwillie) characterization of homogeneousfunctors using the Quillen Homology TQ(-) viewed as Σ ∞ . We compute in section 5 thederivatives of some functors. In the last section 6 , we prove that there is a chain ruleproperty associated to the composition of two functors (composed on Ch ). Future Work
The work of this paper raises the question of extending the description of homogeneousfunctors to a classification of Taylor towers. This question was raised by Arone-Ching[AC11] and they investigated the module structure on the collection of derivatives ∂ ∗ F over a certain operad. They built a tower which fails to be the Taylor tower of F up toTate cohomology. Since Tate cohomologies vanish rationally ( see [Kuh04]), this questionis studied in our future work ([AN19]) when the ground field k is of characteristics . cknowledgements. The author is grateful to Pascal Lambrechts and Greg Arone fortheir suggestions and encouragement during this work.
All chain complexes are over a field k of any characteristics. The purpose of this section isto fix conventions and review basic properties which are background of our constructions.In this paper, we denote by Ch the category of Z - graded chain complexes over k . This category has a symmetric monoidal structure. The tensor product of chain complexes
V, W ∈ Ch is defined by: ( V ⊗ W ) n := ⊕ p + q = n V p ⊗ W q with the differential such that: ∀ x ⊗ y ∈ V p ⊗ W q , d ( x ⊗ y ) = d ( x ) ⊗ y + ( − p x ⊗ d ( y ) . The unit of the monoid − ⊗ − , which we denote abusively k , is the chain complex having k in degree and is trivial in all other degrees. The tensor product − ⊗ − has a rightadjoint hom ( − , − ) given by: hom ( V, W ) := ⊕ i ∈ Z hom i ( V, W ) where hom i ( V, W ) denotes the vector space of morphisms f : V ∗ −→ V ∗ + i of degree i. Similarly We denote by Ch + , the sub-category of Ch which consist of non negativelygraded chain complexes. Twisted chain complex
Let ( V, d V ) be a chain complex. A twisting homomorphism of degree − , d : V −→ V is amorphism of graded vector spaces of degree − which is added to the internal differential d V to produce a new differential d V + d : V −→ V on V. The equation ( d V + d ) = 0 isequivalent to the equation d V ( d ) + d = 0 , with d V ( d ) := d V d + dd V . Model category structure on Ch + The category Ch + is a proper closed model category (for instance see [GJ94, § We denote by
F inSet the category whose objects are finite sets and whose morphisms arebijections. We denote the category of all symmetric sequences in Ch + by [ F inSet, Ch + ] (in which morphisms are natural transformations). The composition M ◦ N, of the twosymmetric sequences M and N, is defined by: ( M ◦ N )( J ) := L J = ∐ j ∈ J ′ J j M ( J ′ ) ⊗ N j ∈ J ′ N ( J j ) . The coproduct here is taken over all unordered partitions, { J j } j ∈ J ′ , of J. The unit sym-metric sequence I is given by 3 ( J ) = k , if | J | = 1 , and I ( J ) = 0 otherwise; Definition 1.1 (Operads) . An operad in Ch + is a monoid over ([ F inset, Ch + ] , ◦ , I ) . Areduced operad is an operad O such that O (0) = 0 and O (1) = k . In this paper, we onlyconsider reduced operads in Ch + . Let O be a reduced operad. An O− algebra X is a chain complex together with structuremaps, for any n ≥ : m n : O ( n ) ⊗ Σ n X ⊗ n −→ X, satisfying the appropriate compatibilityconditions. Maps of O− algebras are given by chain complex morphisms f : X −→ X ′ which are degree and preserve the O− algebra structures of X and X ′ . The category of O -algebras is denoted Alg O . Model category structure on Alg O One use this adjunction O ( − ) : Ch + ⇄ Alg O : U , between the forgetful and the freefunctors, to define the projective model structure on Alg O (see [GJ94, Thm 4.4]). Namelyweak equivalences(resp. fibrations) of Alg O are equivalences (resp. fibrations) in the under-lined category Ch + . The cofibrations are morphisms having the right lifting property withrespect to acyclic fibrations. In particular, cofibrant O -algebras are retract of quasi-freealgebras. The notion of cooperad is dual to the notion of operad. The dual notion consists of con-sidering the opposite category (( Ch + ) op , ⊗ , I Ch ) . We define the dual composition product b ◦ of two symmetric sequences by replacing the coproduct in the definition ?? with aproduct. That is ( M b ◦ N )( J ) := Q J = ∐ j ∈ J ′ J j M ( J ′ ) ⊗ N j ∈ J ′ N ( J j ) . where the product is taken over all unordered partitions, { J j } j ∈ J ′ , of J. A cooperad in C is a triple ( Q, m c , η c ) , where Q is a symmetric sequence together withmaps m c : Q −→ Q b ◦ Q and η c : Q −→ I satisfying the co-associativity, the left and right co-unit condition.A cooperad Q is connected when e Q := ker ( η c ) is concentrated strictly in positivedegree.Since (finite) product and direct sum are equivalent in the underlying category Ch + , in the rest of this thesis, the dual composition product b ◦ will simply be denoted ◦ . Another dual analogy is the notion of the coalgebra over a cooperad. That is, any chaincomplex Y together with a structure map, ∀ n, m cn : Y −→ Q ( n ) ⊗ Σ n Y n satisfying the ap-propriate compatibility conditions. The maps of Q -coalgebras are degree 0 chain complexmorphisms f : Y −→ Y ′ which preserves the structures of Y and Y ′ . One denotes thecategory of Q -coalgebras by coAlg Q . odel category on coAlg Q We use this adjunction U : coAlg Q ⇄ Ch + : Q ( − ) between the, forgetful and the cofreefunctor, to define an injective model structure on coAlg Q (see [GJ94, Thm 4.7]). Namelyweak equivalences(resp. cofibrations) of coAlg Q are weak equivalences(resp. cofibrations)in the underlined category weak Ch + . The fibrations are morphisms having the left liftingproperty with respect to acyclic cofibrations J -tree Let J be a finite set. A J − tree is an abstract planar tree with one output edge on thebottom, and input edges on the top whose sources also called leaves are indexed by J. These input edges and the edge from the root are the external edges of the tree, andthe other edges are called internal edges. The vertices of internal edges are called internalvertices . Given an J − tree T, we denote by V ( T ) the set of its internal vertices, and E ( T ) the set of edges. The set of J − trees, denoted by β ( J ) , is equipped with a naturalgroupoid structure. Formally, an isomorphism of J − trees β : T ′ −→ T is defined bybijections β V : V ( T ′ ) −→ V ( T ) and β E : E ( T ′ ) −→ E ( T ) preserving the source andtarget of edges. In other word, β ( J ) is the groupoid of J − labeled trees and non-planarisomorphisms. Definition 1.2 (Free object) . Let M be a symmetric chain complex. The free object ,associated to M, and denote by F ( M ) consists of: chain complexes ( F ( M )( J ) , ∂ ) , forany finite set J, defined as F ( M )( J ) = ⊕ T ∈ β ( J ) T ( M ) / ≡ where T ( M ) = ⊗ v ∈ V ( T ) M ( J v ) , and the equivalence classes are made of non planar isomor-phisms of J − trees. The differential ∂ is induced naturally by the differentials of the chaincomplexes ( M ( J v ) , ∂ J v ) . A bijection J −→ J ′ gives an isomorphism F ( M )( J ) −→ F ( M )( J ′ ) by relabeling theleaves of the underlined trees. In this way F ( M ) becomes a symmetric sequence in chaincomplexes. Let O be an operad, R is a right O -module and L is a left O -module. Definition 1.3 (Two sided bar construction) . The two sided bar construction B ( R, O , L ) is the symmetric sequence of chain complexes given by: for any finite set J,B ( R, O , L )( J ) := ( R ◦ F ( s e O ) ◦ L ( J ) , ∂ + ∂ ) , with e O = kerε. The differential ∂ is induced in the natural way by the differentials of the chain complexes { ( R ( J ′ ) , d J ′ ) } J ′ ⊆ J , { ( O ( J ′ ) , d J ′ ) } J ′ ⊆ J , and { ( L ( J ′ ) , d J ′ ) } J ′ ⊆ J . The second differential ∂ = ∂ R + ∂ O + ∂ L of this complex is the derivation which integrates the structure morphisms: m R : R ◦ O −→ R, m L : O ◦ L −→ L, and m O : O ◦ O −→ O ( for explicit description,see [Fre04, § If L = R = I , then B ( R, O , L ) is the usual bar construction B ( O ) . ar construction with coefficients in O -algebras Given an O -algebra X, there is an associated left O -module ˆ X defined as follows: ˆ X (0) = X, ˆ X ( n ) = 0 if n ≥ and the left action O ◦ ˆ X −→ ˆ X is induced in the obvious way by the O -algebra structureof X. This defines an embedding functor b − : Alg O −→ O -mod of O -algebras to the categoryof left modules over O . We use this embedding and the bar construction with coefficientsin left and right O -modules described in Definition 1.3 to define the bar construction withcoefficients in O -algebras. Definition 1.4 (Bar construction on algebras) . Let X be an algebra over a reduced operad O . We define the bar construction on O with coefficient in X as the chain complex: B ( O , X ) := ⊕ n ( B ( I , O , ˆ X )( n ) , ∂ + ∂ ) , Let ( Q, Q m c −→ Q ◦ Q, Q η c −→ I ) be a connected cooperad in Ch + , and denote e Q := ker ( η c ) . The cobar construction of Q, denoted B c ( Q ) is the dual version of the bar construction(for operads). Namely, this is the quasi-free operad B c ( Q ) = ( F ( s − e Q ) , ∂ + ∂ ∗ ) , where ∂ is the internal differential of F ( s − e Q ) induced by that of Q, and ∂ ∗ is thedifferential defined by reversing all the arrows in the definition of ∂ on B ( O ) in Definition1.3 (when L = R = I ).In this definition, the cooperad Q needs to be connected to avoid the case where B c ( Q ) has elements in negative degree. We are now ready to state the next theorem(in characteristics 0) which gives a duality between the bar construction and the cobarconstruction. Theorem 1.5. [GJ94, Theorem 2.17] The functors B c and B form an adjoint pair betweenthe categories of connected cooperads and augmented operads. In addition, it is proved in [GK95, Theorem 3.2.16] that the unit Q −→ BB c ( Q ) andthe counit B c B ( O ) −→ O of this adjunction are quasi-isomorphisms. Cobar construction with coefficient in Q -coalgebras Let Q be a connected cooperad on chain complexes and Y a Q -coalgebra. One can followthe same procedure such as in the case of bar construction over algebras to define thecobar construction B c ( O , Y ) . In this sense we will get literally an object in the categoryof B c ( Q ) -algebra. We do not follow these steps here since we will need further as anapplication a cobar construction functor which send a B ( O ) -coalgebra (for a given reducedoperad O ) into an O -algebra.We consider to have from now a reduced operad O such that B c ( Q ) ≃ −→ O . This latermorphism induces a degree morphism s − e Q −→ O which gives a morphism θ : e Q −→ O of degree − . We use θ to define the composition6 : Q ( Y ) Q ( m cY ) / / Q ( Q ( Y )) θ (1 QY ) / / O ( Q ( Y )) The derivation d w : O ( Q ( Y )) −→ O ( Q ( Y )) of degree − associated to this w satisfies theequation of twisting homomorphism d ( w ) + d w .w = 0 on Q ( Y ) . This is equivalent to saythat ( O ( Q ( Y )) , d + d w ) is a quasi-free O -algebra. The morphism θ which is at the base ofthis construction will be called in the literature twisting cochain (see [GJ94, def 2.16]). Definition 1.6 (cobar construction on a Q -coalgebra) . Let Y be a Q -coalgebra. The cobarconstruction on Y, associated to the twisting cochain θ : e Q −→ O , and denoted B cθ ( Q, Y ) is the quasi-free O -algebra B cθ ( Q, Y ) = ( O ( Q ( Y )) , d + d w ) where d is the internal differential of O ( Q ( Y )) induced by the complexes O , Q and Y. When Q = B ( O ) , the map θ that we consider is given by the projection B ( O ) −→ s e O . In that specific case, we will always drop θ and simply write B c ( Q, Y ) to meanas θ is given by the projection B ( O ) −→ s e O . One form the cobar-bar adjoint pair B c ( B ( O ) , − ) : coAlg B ( O ) ⇄ Alg O : B ( O , − ) whose the unit and co-unit functors have the following property(in characteristics 0): Theorem 1.7 ([GJ94], Theorem 2.19) . Given an O -algebra X and a B ( O ) -coalgebra Y, the co-unit B c ( B ( O ) , B ( O , X )) −→ X and the unit Y −→ B ( O , B c ( B ( O ) , Y )) are weakequivalences. With the model structure defined on coAlg B ( O ) and Alg O , we can see that the cobar-bar adjunction is actually a Quillen pair, and Theorem 1.7 completes in proving that thisadjunction is a Quillen equivalence. Remark 1.8.
Fresse proved in [Fre04, Prop 3.1.12] that the counit B c B ( O ) −→ O is aquasi-isomorphism when the ground field k is of any characteristics. Thus one can applythis fact in [Fre09, Thm 4.2.4] to deduce that the co-unit B c ( B ( O ) , B ( O , X )) −→ X is acofibrant resolution of X if k is a field of any characteristics. Ω ∞ and Σ ∞ Let X be an O -algebra. We define the functor τ O ◦ O − : Alg O −→ Alg O as follows: τ O ◦ O X := colim Alg O ( τ O ◦ O ◦ ( X ) ⇒ τ O ◦ ( X )) , where τ O is the operad: τ O (1) = k , and τ O ( t ) = 0 if t = 1; The first map of thiscolimit is produced by the multiplication τ O ◦ O −→ τ O and the second map is givenby the algebra structure map O ( X ) −→ X. Strictly speaking, the algebra τ O ◦ O X has atrivial O -algebra structure, thus we will define the abelianization functor as its compositewith the forgetful functor ( − ) ab : Alg O τ O ◦ O − −→ Alg O U −→ Ch + I ֒ → Ch I is the inclusion fucntor defined by I ( V ) t := (cid:26) V t if t ≥ if t < The abelianization functor has a right Quillen adjoint functor: Ω ∞ : Ch red −→ Ch + ( − ) triv −→ Alg O where for any chain complex C ∗ , red ( C ∗ ) t := (cid:26) C t if t > ker ( d ) if t = 0 and ( − ) triv is the functor which assigns to any non negative chain complex the trivial O -algebra structure.The functor ( − ) ab does not preserves quasi-isomorphisms in general, apart from pre-serving quasi-isomorphisms between quasi-free algebras (since they are cofibrant objectsin Alg O ). Its associated derive functor is called in the literature Quillen homology. Definition 1.9 (Quillen homology) . If X is an O -algebra, then the Quillen homology
T Q ( X ) of X is the O -algebra τ O h ◦ O X. Again since the algebra structure on
T Q ( X ) is trivial, we will abuse notation andconsider it as an object in Ch + . We will give in the next lines an explicit model of thefunctor
T Q ( − ) which we will need to define Σ ∞ . Let X be an algebra over O . One associate to X the symmetric sequence ˆ X definedas ˆ X ( k ) = X, if k = 0 and ˆ X ( k ) = 0 , if k = 0 . One can see that ˆ X is a left O -modulewith the structure map induced naturally by the algebra structure map. In this sense thecategory Alg O embeds in Lt O ( the category of left O -modules) as a full subcategory ofleft modules concentrated at degree , via the functor ˆ − : Alg O −→ Lt O , X ˆ X. According to Theorem 1.7, B c ( B ( O ) , B ( O , X )) is a cofibrant replacement of X, there-fore T Q ( X ) ≃ U B ( O , X ) , where U : Coalg B ( O ) −→ Ch + is the forgetful functor. Underthis last quasi-isomorphism, we will consider the functor U B ( O , − ) as our explicit modelfor the functor T Q ( − ) and we will denote by Σ ∞ the composition:Alg O Σ ∞ B ( O , − ) / / Coalg B ( O ) U / / Ch + (cid:31) (cid:127) I / / Ch Before concluding these constructions, we make the following remark:We consider the following two adjunctionscoAlg B ( O ) U / / Ch + I / / B ( O , ( − ) triv ) o o Ch red o o , where the top functors are each left adjoint and the bottom functors are each right adjoint.We then observe that the associate comonad is IU B ( O , ( − ) triv ) red ∼ = Σ ∞ Ω ∞ . Thereforeeven if the functor Ω ∞ is not adjoint to Σ ∞ , we can say that T = Σ ∞ Ω ∞ : Ch −→ Ch isa comonad and is a "homotopy good model" of the comonad ( − ) ab ( − ) triv . We extend construction of the functors Σ ∞ and Ω ∞ to other categories as follows:- Σ ∞ := I : Ch + −→ Ch ; - Σ ∞ = Id : Ch −→ Ch ; - Ω ∞ = red − : Ch −→ Ch + ; - Ω ∞ = Id : Ch −→ Ch. Homotopy limits and colimits in Alg O The purpose of this section is to remind a brief notion of homotopy limits and colimits, andgive their explicit description in Alg O in terms of holims and hocolims in chain complexes.Let C and D be any of the categories Alg O , coAlg B ( O ) and Ch + . These categories arecomplete and cocomplete. The authors of [DHKS04] proved, in a general argument forcomplete and cocomplete model categories, that holims and hocolims always exists in C (see [DHKS04, 19.2]). More explicitly, given a small category J, and an J -diagram D in C , they replace D through a functor D D vf (resp. D D vc ) which associate a socalled "virtually fibrant replacement" (resp "virtually cofibrant replacement") D vf (resp. D vc ) such that there is a map D ≃ −→ D vf (resp. D vc ≃ −→ D ) natural in D. According to this vocabulary we can now set the definition of holims and hocolims:
Definition 2.1.
Given an J -diagram D in C ,holim C ( D ) := lim C ( D ) vf and hocolim C ( D ) := colim C ( D ) vc . O We assume in this section that the ground field k is of characteristics 0. The homotopylimit in Alg O is calculated using observations in the underlined category Ch + . Given adiagram D : X g −→ Z f ←− Y in Ch + , if either f or g is a surjection (to mean fibration),then holim Ch + ( D ) ≃ lim Ch + ( D ) . This comes out easily when we apply the homology longexact sequence theorem to the two parallel fibrations of the pullback square associated to D. Our methodology to define an explicit homotopy limit in Alg O is to replace the mapsof the O -algebra diagram by explicit surjections. Construction of path objects in Alg O Let I = ( ∧ ( t, dt ) , d ) be the free differential graded commutative algebra generated by theelement t in degree and dt in degree − , with differential d given by d ( t ) = dt and d ( dt ) = 0 . It is useful to notice that an element α of I has the form α = P ( t ) + Q ( t ) dt with P, Q ∈ k [ t ] . There are natural commutative algebra maps s : k −→ I and p , p : I −→ k definedas: ∀ ( α = P ( t ) + Q ( t ) dt ∈ I ) and k ∈ k ,p ( α ) := P (0) , p ( α ) := P (1) and s ( k ) = ks is a quasi-isomorphism and p s = p s = 1 k . For any O -algebra X, there is a natural O -algebra structure on I ⊗ X (see [Liv99, § a ∈ O ( n ) , α i ⊗ x i ∈ I ⊗ X, for ≤ i ≤ n,m ( a ⊗ α ⊗ x ⊗ ... ⊗ α n ⊗ x n ) := ± α ...α n ⊗ m X ( a ⊗ x ⊗ ... ⊗ x n ⊗ ); One then get the factorization in O -algebras (unbounded algebras) X s ⊗ X / / I ⊗ X p ⊗ X / / p ⊗ X / / X which yield to the diagram in Alg O : s X / / red ( I ⊗ X ) p X / / p X / / X One can prove that p X and p X are trivial surjections. Definition 2.2 (path object) . A path object associated to an O -algebra X is the O -algebra X I := red ( I ⊗ X ) together with the O -algebra morphisms p X , p X and s X . Construction of homotopy pullbacks in Alg O Let us consider the commutative diagram in Alg O : Y f % % Y ( ( ( s Z f,Y ) ' ' Z s Z ' ' Z I × Z Y π (cid:15) (cid:15) π ≃ / / Y f (cid:15) (cid:15) Z I p Z ≃ / / Z where the square in the middle is a pullback. From the left triangle, we build the followingfactorization of f : Y ( s Z f,Y ) ≃ / / / / f = p Z s Z f " " ❊❊❊❊❊❊❊❊❊❊❊ Z I × Z Y p Z π (cid:15) (cid:15) Z We use this factorization to replace f in a diagram D : X g −→ Z f ←− Y by the fibration p Z π . Proposition 2.3.
Given an O− algebra diagram D : X g −→ Z f ←− Y, a homotopy pullbackof D is the O− algebra P D = X × Z Z I × Z Y, namely P D = lim Alg O ( X g −→ Z p Z π ←− Z I × Z Y ) . Proof.
1. The morphism p Z π is a surjection, it then follows from our comment in theintroduction of this section that in Ch + , we have U P D ≃ holim Ch ( U X g −→ U Z p Z π ←− U Z I × Z Y ) ≃ holim Ch U D
As a consequence we deduce that the functor P − preserves weak equivalence ofdiagrams in Alg O . Therefore we retain that P D ≃ −→ P D vf .
2. We now prove that lim
Alg O ( D vf ) ≃ −→ P D vf . Lets consider D vf : X ′ g ′ −→ Z ′ f ′ ←− Y ′ and the cube in Ch + : P D vf U Z ′ I × Z ′ Y ′ U X ′ U Z ′ U lim
Alg O ( D vf ) U Y ′ U X ′ U Z ′ h h h ( s Z ′ f ; Y ′ ) h = = where we have applied the forgetful functor U : Alg O −→ Ch + to the original naturalcube in Alg O . The square obtained from the homotopy fibers of the morphisms h , h , h and h has the following characteristics: hof ibre ( h ) (1) ≃ / / / / hof ibre ( h ) hof ibre ( h ) O O (3) ≃ / / hof ibre ( h ) (2) ≃ O O where(1) is a weak equivalence as the top square of the cube is a homotopy pullback;(2) is a weak equivalence because UZ ′ and U ( s Z f, Y ) are weak equivalences, andthese imply that the right hand square is a homotopy pullback;(3) is a weak equivalence since the bottom square is a homotopy pullback.We can conclude that hof ibre ( h ) −→ hof ibre ( h ) is a weak equivalence and there-fore that lim Alg O ( D vf ) ≃ −→ P D vf . We then conclude in conclusion that P D ≃ −→ P D vf ≃ ←− lim Alg O ( D vf ) In general this construction is extended in the obvious manner in order to define higherdimensional limits in Alg O . Lemma 2.4. If X is an O -algebra, then the map Φ : ( red s − X ) triv −→ Ω Xs − x (0 , dt ⊗ x, is a weak equivalence in Alg O . Proof.
We first prove that Φ is a map of O -algebras. Namely let x , ..., x n ∈ X, and a ∈ O ( n ) , ( n ≥ , then m Ω X ( a ⊗ Φ( s − x ) ⊗ ... ⊗ Φ( s − x n )) = (0 , dt n ⊗ m X ( a ⊗ x ⊗ ... ⊗ x n ) , ( since dt n = 0 )This computation proves that Φ is a map of O -algebras as the O -algebra structure on ( red s − X ) triv is trivial. It is obvious that the map Φ commutes with differentials of thetwo complexes.Now we prove by hand that H ∗ (Φ) is injective and surjective. Let us take11 = a + Σ l ≥ t l a l + Σ k ≥ t k dtb k ∈ X I such that (0 , x, ∈ Ω X ∩ Kerd where for each l and k, a l , b k ∈ X ;(0 , x, ∈ Ω X ⇐⇒ p X ( x ) = 0 = p X ( x ) ⇐⇒ a = 0 = Σ l ≥ a l One can also see that dx = 0 ⇐⇒∀ l ≥ , da l = 0 and Σ l ≥ lt l − dta l = Σ k ≥ t k dtdb k This last equality implies that ∀ l ≥ , a l = l db l − and thus Σ l ≥ l db l − = 0 One then get: x = Σ l ≥ l t l db l − + Σ l ≥ t l − dtb l − = Σ l ≥ l ( t l db l − + lt l − dtb l − )= d ( Σ l ≥ l t l b l − )= d ( Σ l ≥ l t l b l − − t Σ l ≥ l b l − + t Σ l ≥ l b l − )= d ( Σ l ≥ l t l b l − − t Σ l ≥ l b l − ) + d ( t Σ l ≥ l b l − ) One can see that Σ l ≥ l t l b l − − t Σ l ≥ l b l − ∈ Ω X and that d ( t Σ l ≥ l b l − ) = dt ⊗ Σ l ≥ l b l − , therefore [ x ] = [ dt ⊗ Σ l ≥ l b l − ] = H ∗ (Φ)([ s − Σ l ≥ l b l − ]) This implies that H ∗ (Φ) is surjective.To prove that H ∗ (Φ) is injective, let’s take [ s − x ] ∈ ( red s − X ) triv such that H ∗ (Φ)([ s − x ]) =0 . This implies that dtx = dx, for a given x ∈ Ω X. As before we set x = Σ l ≥ t l a l + Σ k ≥ t k dtb k , with Σ l ≥ a l = 0 . An easy comparison on the degree of the polynomials proves that dtx = Σ l ≥ lt l − dta l + Σ l ≥ t l dtda l − Σ k ≥ t k dtdb k ⇐⇒ x = a − db and ∀ l ≥ , a l = 1 l db l − = ⇒ x = − Σ l ≥ l db l − − db = d ( − Σ l ≥ l b l − ) this means that [ s − x ] = 0 and proves that H ∗ (Φ) is injective. Lemma 2.5. If Y is an O -algebra such that Y ≃ Ω X then Y ≃ Ω ∞ U Y.
Proof.
From Lemma 2.4, we deduce that Y ≃ Ω ∞ s − X. When we apply the forgetfulfunctor U, we get the quasi-isomorphism in chain complexes U Y ≃ U Ω ∞ s − X. We applyagain the functor Ω ∞ and get the O -algebra weak equivalences Ω ∞ U Y ≃ Ω ∞ U Ω ∞ s − X ∼ = Ω ∞ s − X ≃ Y. In this sense, strictly speaking we will just say that any loop space in Alg O has atrivial O -algebra structure. 12 .2 Homotopy pushouts in Alg O We assume in this section that the ground field k is of characteristic 0 in order to de-scribe an explicit model for homotopy pushouts. In the particular case of describing thesuspension of O -algebras, the ground field k can be of any characteristics. Construction of a cylinder of a quasi-free O -algebra We give in this part the construction of a cylinder of a quasi-free O -algebra in the sameline that the definition for differential graded Lie algebras in [Tan83, II.5.], and for closedDGL’s in [BFMT16, § ( O ( V ) , d ) be a quasi-free O -algebra, and let V ′ be a copy of V. We define :- O ( V ) b ⊗I := ( O ( V ⊕ V ′ ⊕ sV ′ ) , D ) , where: ( sv ′ ) n = v ′ n − , Dv ′ = 0 , Dsv ′ = v ′ ,Dv = dv. - λ : ( O ( V ) , d ) −→ O ( V ) b ⊗I the canonical injection;- p : O ( V ) b ⊗I −→ ( O ( V ) , d ) is the O -algebra morphism given by: p ( v ) = v ; p ( v ′ ) = p ( sv ′ ) = 0; p is a quasi-isomorphism since O ( V ′ ⊕ sV ′ ) is acyclic.- i : O ( V ) b ⊗I −→ O ( V ) b ⊗I is the degree +1 O -algebra derivation given by: i ( v ) = sv ′ ; i ( sv ′ ) = i ( v ′ ) = 0; - The O -algebra derivation of degree , θ = Di + iD verifies θD = Dθ, θ ( v ′ ) = θ ( sv ′ ) =0 . We have the induced automorphism of O -algebras e θ = Σ n ≥ θ n n ! ( with inverse e − θ ).The automorphism e θ is well defined for the following reason: let v ∈ V n . We write downexplicitly the differential d of ( O ( V ) , d ) by d = d + d + ..., where d k v ∈ O ( k ) ⊗ Σ k V ⊗ k , forany given k. Computation gives that θ ( v ) = θi ( d v + d v + ... ) ∈ O ( V Given a O -algebra diagram D : X Z f / / g o o Y , a homotopy pushoutof D is given by C D = X c ∐ Z c Z c b ⊗I ∐ Z c Y c . Namely C D = colim Alg O ( X c Z c π i / / g c o o Z c b ⊗I ∐ Z c Y c ) . Proof. This is analogue as the proof of Proposition 2.3. We simply replace holims byhocolims and Z I by Z b ⊗I . 1. Let D i : X i Z i f i / / g i o o Y i , i ∈ { , } , be two O -algebra diagrams so that D ≃ −→ D . One make the following computations: Σ ∞ C D (1) ≃ colim Ch ( U B ( O , X ) B ( O ,g ) ←− U B ( O , Z ) UB ( O ,π i ) −→ U B ( O , Z c b ⊗I ∐ Z c Y c )) (2) ≃ hocolim Ch ( U B ( O , X ) B ( O ,g ) ←− U B ( O , Z ) UB ( O ,π i ) −→ U B ( O , Z c b ⊗I ∐ Z c Y c )) ≃ hocolim Ch ( U B ( O , X ) ←− U B ( O , Z ) −→ U B ( O , Z c b ⊗I ∐ Z c Y c )) ≃ Σ ∞ C D where- (1) is obtained by applying the left adjoint functor ( − ) ab (which is equivalentin this case to Σ ∞ ) to the diagram C D ; - (2) is obtained by replacing colim with hocolim since U B ( O , π i ) is an injec-tion ( cofibration).One then obtain C D ≃ C D , and deduce that the functor C − preserves weak equiva-lences of diagrams of the form • ←− • −→ • in Alg O . One deduce from this propertythat C D vc ≃ −→ C D , where D vc ≃ −→ D is a virtually cofibrant replacement of D. 2. Now we prove that C D vc ≃ −→ colim Alg O ( D vc ) . We consider D vc : X ←− Z −→ Y, and we form the diagram: 14 YX colim Alg O D vc Z c Z c b ⊗I ∐ Z c Y c X c C D vc h h ≃ h ≃≃ h The square obtained from the homotopy fibers of the horizontal morphisms h , h , h and h is described as follows: hocof ibre ( h ) (1) ≃ / / / / hocof ibre ( h ) hocof ibre ( h ) (2) ≃ O O (3) ≃ / / hocof ibre ( h ) (4) O O where(1) is a weak equivalence as the top square of the cube is a homotopy pushout;(2) is a weak equivalence because the back face of the cube is trivially a homotopypushout;(3) is a weak equivalence. In fact the functor Σ ∞ ( − ) applied to the diagram at thebottom (of cofibrant algebras) gives a homotopy pushout diagram in Ch. Onethen deduce that Σ ∞ hocof ibre ( h ) ≃ −→ Σ ∞ hocof ibre ( h ) and equivalently we get hocof ibre ( h ) ≃ −→ hocof ibre ( h ) By this we conclude that (4) is a weak equivalence, and therefore that C D vc ≃ −→ colim Alg O ( D vc ) . Remark 2.7. If D : X Z f / / g o o Y is a diagram of quasi-free O -algebras, then wedon’t need the cofibrant replacement functor ( − ) c in the construction, and we have C D = X ∐ Z Z b ⊗I ∐ Z Y. Lemma 2.8. Let ( O ( V ) , d ) be a quasi-free algebra with the notation for the differential: d = d + d + ... . Then Σ( O ( V ) , d ) ≃ ( O ( sV ′ ) , D ) , where D ( sv ′ ) := − sd v ′ and V ′ is acopy of ( V, d ) . roof. We set for short Z = ( O , d ); In Proposition 2.6, we have proved that (0 ∐ Z Z b ⊗I ∐ Z , D ) ≃ Σ Z. Since ( e θ ) ab ( v ) = v ′ + sd v ′ , we deduce that in (0 ∐ Z Z b ⊗I ∐ Z ab , [ Dsv ′ ] = [ v ′ ]= [ v ′ + sd v − sd v ′ ]= [ − sd v ′ ] Now we consider the morphism of O -algebras ψ : ( O ( sV ′ ) , D ) −→ (0 ∐ Z Z b ⊗I ∐ Z , D ) given by ψ ( sv ′ ) = [ sv ′ ] . This is a well defined chain complex morphism since [ Dψ ( sv ′ )] = ψ ( D ( sv ′ )) and inaddition B ( O , ψ ) ≃ ψ ab is a quasi-isomorphism. We deduce that ψ is a quasi-isomorphism. Remark 2.9. The result of Lemma 2.8 holds in general when the ground field k is of anycharacteristics. In fact, we have the following pushout diagram O ( V ) / / (cid:15) (cid:15) O ( V ⊕ sV ) ≃ (cid:15) (cid:15) / / O ( sV ) This is also a homotopy pushout diagram, thus we deduce that Σ O ( V ) ≃ O ( sV ) . Corollary 2.10. We assume that the ground field k is of any characteristics. Given an O -algebra Z, then Σ Z is the free O -algebra ( O ( sU B ( O , Z )) , d ) , where d is the internaldifferential induced by the differential of ( B ( O , Z ) , d ) . Proof. We make the following computation Σ Z ≃ Σ B c ( B ( O ) , B ( O , X )) ( Using Thm 1.7 and Remark 1.8 ) ≃ O ( sU B ( O , Z )) ( Using Proposition 2.8 and Remark 2.9 ) Finally, we remind the following relation between holims and hocolims. Lemma 2.11. In Alg O , filtered homotopy colimits, that are colimits of filtered diagrams,commute with finite homotopy limits.Proof. This follows from the fact that this property is true in Ch + , and that the forgetfulfunctor U : Alg O −→ Ch + commutes with finite limits and filtered colimits.16 Goodwillie approach in functor calculus We assume in that section that the ground field k is of any characteristics. Let C and D be any of the model categories Alg O , Ch + or Ch. We remind in this section the theoryof functor calculus, for functors F : C −→ D . We follow the lines of [Kuh07, § § F ( X ) ⊗ K −→ F ( X ⊗ K ) , where K ∈ sSets, and X ∈ C . In fact, we will see (in Lemma 4.13) that homotopy functors F : Ch −→ Ch have a natural assembly map (at least at the level of the homotopycategory HoCh ). However, we will require our functors to be homotopy(preserve weakequivalences), which is a weaker version of being continuous, since continuous implieshomotopy. Definition 3.1 (Homotopy functor) . Let C and D be any of the model categories Alg O ,Ch + , or Ch and F : C −→ D be a functor.1. The functor F is reduced if F (0) ≃ F is a homotopy functor if it preserves weak equivalences.3. F is finitary if it preserves filtered homotopy colimits. Definition 3.2 ( n -excisive functor) . Let C and D be any of the model categories Alg O ,Ch + or Ch and F : C −→ D be a homotopy functor.1. An n -cube in C is a functor X : P ( n ) −→ C , where P ( n ) is the poset of subsets of n := { , ..., n } . 2. The functor F : C −→ D is called n -excisive if whenever X is a strongly coCartesian n + 1 -cube in C , F ( X ) is a cartesian cube in D . Definition 3.3 ([Kuh07], 4.6) . Let C be any of the model categories Alg O , Ch + or Ch. Let X ∈ C and T be a finite set. We define the joint X ∗ T, of X and T, to be the homotopycofiber of the folding map X ∗ T = hocof ( ∐ T X ▽ −→ X ) Example 3.4. Using Proposition 2.6, we make the following computation: for X ∈ Alg O , - X ∗ X ∗ ∅ = B c ( B ( O ) , B ( O , X )); - X ∗ cB c ( B ( O ) , B ( O , X )); - X ∗ B c ( B ( O ) , B ( O , X )) . Let C and D be any of the model categories Alg O , Ch + or Ch and let F : C −→ D bea homotopy and reduced functor. For X ∈ C , define the n -cube χ n ( X ) : P ( n ) −→ C by χ n ( X ) : T X ∗ T. This is a strongly coCartesian n -cube (see [Wal06, lemma 7.1.4]), the fact is that homotopycolimits commute with themselves. One set T n − F ( X ) := holim T ∈P ( n ) −{∅} F ( χ n ( X )( T )) F is n -excisive, then the natural map t n − F : F ( X ) = F ( χ n ( X )( ∅ )) −→ T n − F ( X ) isa weak equivalence. Write T in − F defined inductively by T i +1 n − F := T n − ( T in − F ) and P n − F := hocolim ( F t n − F −→ T n − F T n − ( t n − F ) −→ T n − ( T n − F ) T n − ( t n − F ) −→ ... ) Example 3.5. T F ( X ) = holim ( F ( X ∗ −→ F ( X ∗ ←− F ( X ∗ if F is reducedthen F ( X ∗ ≃ and we deduce that T F ( X ) ≃ Ω F (Σ X ); Therefore inductively we get P F ( X ) ≃ hocolim p →∞ Ω p F Σ p Definition 3.6 (homogeneous functors) . Let F : C −→ D be a homotopy and reducedfunctor. F is called n -homogeneous if- F is n -excisive and- P n − F ≃ . When D = Alg O , we make the following remark: Remark 3.7. Let C = Alg O , Ch + or Ch. If a functor F : C −→ Alg O is n -homogeneous,then for any X ∈ C , F ( X ) has a trivial O -algebra structure. In fact Goodwillie [Goo03,Lemma 2.2] proves in a completely general argument that there is a homotopy pullbackdiagram P n F / / (cid:15) (cid:15) P n − F (cid:15) (cid:15) / / R n F , where R n F : C −→ Alg O is n -homogeneous. Thus if F is n -homogeneous, then F ≃ P n F ≃ Ω R n F. Therefore, when the ground field is of characteristics 0, we can rewrite F as F ≃ Ω ∞ U F, where U : Alg O −→ Ch + is the forgetful functor (see Lemma 2.5). Since F = T n − F, the functor P n − F is equipped with a map F −→ P n − F. In additionthe inclusion of categories P ( n ) −→ P ( n + 1) induces a map T n F −→ T n − F whichextends formally to give a map q n F : P n F −→ P n − F which is a fibration (see [Goo03,Page 664]). By inspection this map is again a fibration in Alg O and in Ch + , since themaps T n F −→ T n − F will always be a surjection, and filtered colimits of surjections isagain a surjection. Theorem 3.8. [Goo03, 1.13] A homotopy functor F : C −→ D determines a tower offunctors { P n F : C −→ D} n , where P n F are n -excisif, q n F : P n F −→ P n − F are fibrations,the functors D n F = hofibre ( q n F ) are n -homogeneous. Remark 3.9. A straight consequence of Lemma 2.11 for this section is that the functor P n , which is basically a homotopy colimit, commutes with finite homotopy limits. In particular P n preserves fiber sequences. Characterization of homogeneous functors In this section, we characterize homogeneous functors with the cross effect. Before gettingto this result, we will make a couple of constructions and provide intermediate results.The characterization itself appears in Corollary 4.12 at the end of this section.There are two ways to define the cross effect associated to a functor. One can define itas a homotopy fiber (hofib) and we can also define it as a total homotopy fiber (thofib).These definitions are reported here bellow. Definition 4.1 (Cross-effects) . Let C and D be any of the model categories Alg O , Ch + or Ch and let F : C −→ D be a homotopy and reduced functor. We define cr n F : C × n −→ D , the n th cross-effect of F, to be the functor of n variables given by cr n F ( X , ..., X n ) = hofib { F ( ∐ i ∈ n X i ) −→ holim T ∈P ( n ) F ( ∐ i ∈ n − T X i ) } This is equivalent to define the n th cross-effect of F as: cr n F ( X , ..., X n ) = thofib ( T ⊇ n F ( ∐ i ∈ n − T X i )) . In the particular cases where C = Ch + or Ch and D = Ch, we can also describe thecross effect of a functor F using the total homotopy cofiber (thocofib) of a certain cube.This dual construction, also called the "co-cross-effect", was considered by McCarthy[McC01, . ] in studying dual calculus, and the equivalence between the cross-effect andco-cross-effect was proved by Ching [Chi10, Lemma . ] for functors with values in spectra.Let W , ..., W n ∈ C , we associate the n -cube X in C defined as follows:- T ⊆ n, X ( T ) := ⊕ i ∈ T W j ; - For T ( n and j ∈ n \ T, the map X ( T ) −→ X ( T ∪ { j } ) (in the cube) is induced bythe inclusion ⊕ i ∈ T W i −→ ( ⊕ i ∈ T W i ) ⊕ W j x ( x, Definition 4.2 (Co-cross-effects) . Let C = Ch + or Ch and F : C −→ Ch be a homotopyfunctor. The n th co-cross effect of F is the functor cr n F : C × n −→ Ch which computesthe homotopy total fiber of F ( X ) . That is: cr n F ( W , ..., W n ) := hocofib { hocolim T ( n F ( ⊕ i ∈ T W i ) −→ F ( W ⊕ ... ⊕ W n ) } . Lemma 4.3. Let C = Ch + or Ch and F : C −→ Ch be a homotopy functor. Then the n th cross-effect of F is equivalent to the n th co-cross-effect of F. That is: cr n F ( W , ..., W n ) ≃ −→ cr n F ( W , ..., W n ) Proof. Since Ch is a stable category and that in C products and coproducts are isomorphic,we simply mimic Ching’s proof.To understand homogeneous functors, Goodwillie[Goo03] pointed the following propo-sition for functors with values in spectra. We reformulate it in our algebraic context thoughthe proof follows literally [ [Goo03], proposition 3.4] and [[Goo92], proposition 2.2].19 roposition 4.4. Let C and D be any of the model categories Alg O , Ch + or Ch. If H : C −→ D is an n − excisive and reduced functor such that cr n H ≃ , then H is ( n − − excisive.Proof. (i) One define the n -cube X = S ∗ ( X , ..., X n ) , for objects X , ..., X n in C , asfollows: ∀ T ⊆ [ n ] , X ( T ) := ∐ i ∈ T X i and X ( ∅ ) = 0 . The maps in the cube X areinclusions. We associate to this cube X the n -cube S ( X , ..., X n ) which has thesame objects with X , but where the inclusions are reversed to the projections. Let U : D −→ Ch be the forgetful functor when D = Alg O and be the identity functorwhen D = Ch + . We make the following computations: U thofib H ( X ) ∼ = thofib U H ( X ) = thofib U H ( S ∗ ( X , ..., X n ))= Ω n thofib U H ( S ( X , ..., X n ))= Ω n cr n ( U H )( X , ..., X n )= Ω n U cr n H ( X , ..., X n ) ≃ , One will then conclude from these that thofib H ( X ) ≃ ( or equivalently that H ( X ) is cartesian) for all strongly coCartesian cubes X in which X ( ∅ ) = 0 , since any suchcube X is naturally equivalent to S ∗ ( X ( { } ) , ..., X ( { n } )) (see [Goo92, proposition2.2]).(ii) Let ∀ T ⊆ [ n ] , and a, b ∈ [ n ] . Given an arbitrary strongly coCartesian n-cube X in C , put X ′ ( T ) = hocolim (0 ←− X ( ∅ ) −→ X ( T )) . We have the following commutative diagram X ( ∅ ) / / (cid:15) (cid:15) X ( T ) (cid:15) (cid:15) / / X ( T ∪ { a } ) (cid:15) (cid:15) / / X ′ ( T ) / / X ′ ( T ∪ { a } ) where the largest square is a homotopy pushout along with the most left square. Itthen follows that the most right square is also a homotopy pushout and thereforethat the following square is a homotopy pushout: X ( T ) / / (cid:15) (cid:15) X ( T ∪ { a } ) (cid:15) (cid:15) X ′ ( T ) / / X ′ ( T ∪ { a } ) and therefore it follows that X ′ ( T ) / / (cid:15) (cid:15) X ′ ( T ∪ { a } ) (cid:15) (cid:15) X ′ ( T ∪ { b } ) / / X ′ ( T ∪ { a, b } ) is a homotopy pushout diagram. This proves that the n-cube X ′ is strongly coCarte-sian and that the map X −→ X ′ is a strongly cocartesian n +1 -cube. H is n -excisive,thus H ( X ) −→ H ( X ′ ) is cartesian. In addition since X ′ ( ∅ ) = 0 , we deduce from ( i ) that H ( X ′ ) is cartesian and conclude that H ( X ) is also cartesian.20e get the following consequence: Corollary 4.5. Let F and G be two n − homogeneous functors C −→ D , where C and D are any of the model categories Alg O , Ch + or Ch, and a natural transformation F J −→ G. If cr n ( J ) : cr n F −→ cr n G is an equivalence, then so is J. Proof. Let H = hofib ( F J −→ G ) . H is n − homogeneous and then n − excisive. By hypoth-esis cr n H ∼ = hofib ( cr n F cr n J −→ cr n G ) ≃ . The functor H gathers then the hypothesis ofProposition 4.4, thus H is n − − excisive . Hence we get H ≃ P n − H = hofib ( P n − F −→ P n − G ) = 0 q q One deduce from the long exact sequence obtained from the homotopy fiber sequence of J that J is a weak equivalence. Definition 4.6. Let C and D be any of the model categories Alg O , Ch + or Ch, and F : C −→ D be a homotopy and reduced functor.1. The functor L n F : C n −→ D is obtained from cr n F by L n F ( X , ..., X n ) ≃ hocolim p i →∞ Ω p + ... + p n cr n F (Σ p X , ..., Σ p n X n ) In the case that D = Alg O , this filtered homotopy colimit can be seen as a homotopycolimit in the underlying category of chain complexes.2. The functor △ n F : C −→ D is obtained from L n F by: △ n F = ( L n F ) ◦ △ where △ : C −→ C × n is the diagonal map. The symmetric group Σ n acts on △ n F by permuting its n entries of the cross effect cr n F. 3. The functor b △ n F ( X ) : C −→ Ch is obtained from △ n F by dropping the functor red . Namely, b △ n F ( X ) := hocolim p i →∞ s − p − ... − p n cr n ( U F )(Σ p X, ..., Σ p n X ) where U : Alg O −→ Ch is the forgetful functor and this colimit is taken in thecategory Ch. The symmetric group Σ n acts on b △ n F ( X ) by permuting its n entriesof the cross effect cr n U F. Remark 4.7. The functor L n F of Definition 4.6 can also be seen as the stabilization ofthe cross effect, that is the functor obtained by applying the first Taylor approximationfunctor P to each variable position of the multi-variable functor cr n F. For instance,1. L F = P F (see Example 3.5);2. L F ( X, Y ) = P ( Y P ( X cr ( X, Y ))); 3. and so on. 21e assume from now, when it is not specified, that the ground field k is of characteristic0. Lemma 4.8. Let C be any of the model categories Alg O , Ch + or Ch, and F : C −→ Alg O be a homotopy and reduced functor. Then for any X ∈ C , there is a weak equivalence of O -algebras △ n F ( X ) ≃ ( red b △ n F ( X )) triv . Proof. If U : Alg O −→ Ch denotes the forgetful functor, we make the following compu-tation: U △ n F ( X ) ≃ hocolim Chp i →∞ [ red s − p − ... − p n cr n U F (Σ p X, ..., Σ p n X )] ≃ red hocolim Chp i →∞ [ s − p − ... − p n cr n U F (Σ p X, ..., Σ p n X )] This last equivalence is justified by the fact that the functor red commutes with filteredcolimits. Now by applying the functor ( − ) triv , we get the weak equivalence of O -algebras ( U △ n F ( X )) triv ≃ ( red b △ n F ( X )) triv . In addition since the functor △ n F is n -homogeneous, we know from Remark 3.7 that △ n F ( X ) ≃ ( U △ n F ( X )) triv , therefore the result follows.We are now ready to state the next theorem which was inspired by [Kuh07, Thm 5.12]for functors with values in stable model categories. Theorem 4.9. Let C and D be any of the categories Alg O , Ch + and Ch, and F : C −→ D be a homotopy and reduced functor. Then there is a weak equivalence D n F ( X ) ≃ Ω ∞ ( b △ n F ( X ) h Σ n ) . where ( − ) h Σ n denotes the homotopy orbits. When D = Ch + or Ch then this result holdswhen the ground field k is of any characteristics. To prove this, we need the following lemma. Lemma 4.10. Let C be either Alg O , Ch + or Ch, and F : C −→ D be a homotopy andreduced functor. Then we have a weak equivalence P n ( L n F ◦ △ ) ≃ L n ( P n F ) ◦ △ Proof. One make the following observation: T n ( L n F ◦ △ )( X ) := holim T ∈P ( n +1) hocolim p i →∞ Ω p + ... + p n cr n F (Σ p ( X ∗ T ) , ..., Σ p n ( X ∗ T )) (1) ≃ holim T ∈P ( n +1) hocolim p i →∞ Ω p + ... + p n cr n F ((Σ p X ) ∗ T, ..., (Σ p n X ) ∗ T )= holim T ∈P ( n +1) hocolim p i →∞ Ω p + ... + p n thofib ( A ⊇ n F ( ∐ n − A ((Σ p j X ) ∗ T )) (2) ≃ holim T ∈P ( n +1) hocolim p i →∞ Ω p + ... + p n thofib ( A ⊇ n F (( ∐ n − A Σ p j X ) ∗ T ) (3) ≃ hocolim p i →∞ Ω p + ... + p n thofib ( A ⊇ n T n F ( ∐ n − A Σ p j X ))= hocolim p i →∞ Ω p + ... + p n cr n ( T n F )(Σ p X, ..., Σ p n X )= L n ( T n F ) ◦ △ ( X ) where 221) is due to the isomorphism Σ p j ( X ∗ T ) ∼ = (Σ p j X ) ∗ T, for each j ; (2) is due to the isomorphism ∐ n − T (Σ p j X ∗ T ) ∼ = ( ∐ n − T Σ p j X ) ∗ T, for each T ⊆ n ; (3) is because finite holims commute with filtered colimits (see Lemma 2.11), and holimscommute with loops Ω and total fibers.One also deduce from this observation steps that the following square is commutative L n F ◦ △ = / / t n L n F ◦△ (cid:15) (cid:15) L n F ◦ △ L n t n F ◦△ (cid:15) (cid:15) T n ( L n F ◦ △ ) ≃ / / L n ( T n F ) ◦ △ Thus we can deduce by induction on the iterations from this square that P n ( L n F ◦ △ ) ≃ ( L n P n F ) ◦ △ . Proof of theorem 4.9. Let F : C −→ D be a homotopy and reduced functor. Let J bethe composition in Ch :(( cr n U D n F ) ◦ △ ( X )) h Σ n −→ ( U D n F ( ∐ n X )) h Σ n −→ U D n F ( X ) where the first map is the projection, the second map is induced by the folding map ∐ n X ∇ −→ X, and if D = Alg O , then U : Alg O −→ Ch is the forgetful functor, and U is simply theidentity functor when D = Ch ; Since we want to prove that J is a quasi-isomorphism, wewill simply show that cr n J is a quasi-isomorphism and conclude using Corollary 4.5.For the sake of simplicity we set L ( X ) = cr n ( U D n F )( X, ..., X ) h Σ n .cr n L ( X , ..., X n ) = thof ib ( L ◦ S ( X , ..., X n ))= thof ib ( n − T L ( ∐ T X i ))= thof ib ( n − T cr n D n F ( ∐ T X i , ..., ∐ T X i ) h Σ n )= thof ib ( χ ) h Σ n , where χ : n − T cr n ( U D n F )( ∐ T X i , ..., ∐ T X i ) . Since cr n ( U D n F ) is multilinear, we deducethe weak equivalence (natural in T ) χ ( n − T ) ≃ −→ Y π : n → T cr n ( U D n F )( X π (1) , ..., X π ( n ) ) . (1) Let’s consider the map π : n → n and consider the cube Y π defined by: Y π ( n − T ) = ( cr n ( U D n F )( X π (1) , ..., X π ( n ) ) if π ( n ) ⊆ T otherwiseThe morphism (1) is equivalent to χ ( n − T ) ≃ −→ Q π : n → n Y π ( n − T ) . - If π is not a permutation and then not surjective, we can find an element s / ∈ π ( n ) . All the maps Y π ( n − T ) −→ Y π ( n − T ∪ { s } ) are isomorphisms, so Y π is cartesian.- If π is a permutation, thof ib ( Y π ) ∼ = Y π ( n ) = cr n U D n F ( X π (1) , ..., X π ( n ) ) . herefore thof ib ( χ ) ≃ −→ Q π ∈ Σ n cr n ( U D n F )( X π (1) , ..., X π ( n ) ) . Thus thof ib ( χ ) h Σ n ≃ −→ ( Y π ∈ Σ n cr n ( U D n F )( X π (1) , ..., X π ( n ) )) h Σ n ≃ −→ cr n ( U D n F )( X , ..., X n ) . Now that we have showed that J is a quasi-isomorphism, let us consider the chain map α using Lemma 4.10: α : L n ( U F ) ◦ △ p n ( L n UF ◦△ ) −→ P n ( L n ( U F ) ◦ △ ) ≃ −→ ( L n ( U P n F )) ◦ △ which is a weak equivalence since L n ( U F ) ◦ △ is n -excisive. Putting all these together weform the following diagram: ( L n ( U F ) ◦ △ ) h Σ n =( △ n ( UF )( X )) h Σ n α ≃ / / α ≃ / / α ≃ / / (( L n ( U P n F )) ◦ △ ) h Σ n ( cr n ( U P n F ) ◦ △ ) h Σ n p ...p cr n P n F ≃ O O ( cr n ( U D n F ) ◦ △ ) h Σ n ≃ o o J ≃ / / U D n F Note that ( △ ( U F )( X )) h Σ n ≃ red ( b △ ( U F ( X ))) h Σ n . One then deduce the following special-izations for D : - When D = Ch + , we have : red ( b △ F ( X ) h Σ n ) ≃ D n F ( X ) - When D = Alg O , we apply the functor Ω ∞ ( − ) to the above diagram and since Ω ∞ ( U D n F ) ≃ D n F (by Remark 3.7), we get the weak equivalence Ω ∞ ( b △ F ( X ) h Σ n ) ≃ D n F ( X ) - When D = Ch, the diagram itself gives the proof. Corollary 4.11. Let D = Alg O , Ch + or Ch, and F : Alg O −→ D be a homotopy andreduced functor. Then there is a weak equivalence D n F ( X ) ≃ Ω ∞ H (Σ ∞ X ) where H : Ch + −→ Ch k is the n -homogeneous functor given by: H ( V ) := b △ n ( F O ( − ))( V ) h Σ n . In particular when D = Ch + or Ch then this result holds when the ground field k is ofany characteristics.Proof. The functor H is n -homogeneous since it is the n -th stabilization of the cross effectof F O ( − ) . Let X be an algebra over the operad O , and F : Alg O −→ D be a homotopy andreduced functor. We observe that b △ n F ( X ) ≃ Ω n ( b △ n F )(Σ X ) ( since L n U F is n-multilinear ) ≃ Ω n ( b △ n F )( O ( s Σ ∞ X )) ( since Σ X ≃ O ( s Σ ∞ X ) from Corollary . ∼ = Ω n b △ n ( F O ( − ))( s Σ ∞ X ) ( since O ( − ) commutes with coproducts ) ≃ b △ n ( F O ( − ))(Σ ∞ X ) ( since L n ( U F O ( − )) is n-multilinear ) One deduce from this observation that ( b △ n F ( X )) h Σ n ≃ b △ n ( F O ( − ))(Σ ∞ X ) h Σ n . UsingTheorem . , we obtain the quasi-isomorphism24 n F ( X ) ≃ Ω ∞ ( b △ n ( F O ( − ))(Σ ∞ X ) h Σ n ) . Corollary 4.12. Let C and D be any of the categories Alg O , Ch + or Ch, and F : C −→ D be a homotopy and reduced functor. If either F is finitary or if X is finite we have thefollowing quasi isomorphisms:1. If C = Alg O , then there is a weak equivalence D n F ( X ) ≃ Ω ∞ ( b △ n F ( O ( k )) ⊗ (Σ ∞ X ) ⊗ n ) h Σ n ; 2. If C = Ch + or Ch, then there is weak equivalence D n F ( X ) ≃ Ω ∞ ( b △ n F ( k ) ⊗ (Σ ∞ X ) ⊗ n ) h Σ n . In particular when D = Ch + or Ch then this result holds when the ground field k is ofany characteristics. To prove this result which classifies homogeneous functors in our algebraic point ofview, we will need the following lemma. Lemma 4.13. Let L r : ( Ch + ) × r −→ Ch be a r -reduced-multilinear functor. Then forany chain complexes V , ..., V r and finite chain complexes W , ..., W r , there is a zig-zag ofquasi-isomorphisms W ⊗ ... ⊗ W r ⊗ L r ( V , ..., V r ) ≃ L r ( W ⊗ V , ..., W r ⊗ V r ) . Proof. 1. We first consider the case r = 1 and we want to construct a zig-zag of quasi-isomorphisms W ⊗ L ( V ) ≃ L ( W ⊗ V ) , for a given chain complex V and a finiteone W. Let us consider the following commutative diagram L ( sV ⊕ V ) (cid:15) (cid:15) ≃ o o ≃ / / (cid:15) (cid:15) L ( sV ) ⊕ s − L ( sV ) (cid:15) (cid:15) L ( sV ) L ( sV ) = o o = / / L ( sV ) L (0) O O ≃ o o = / / O O O O O O A homotopy limit functor applied on each column gives the zig-zag of quasi-isomorphisms L ( V ) ≃ ←− • ≃ −→ s − L ( sV ) where the homotopy limit result of the first column in due to the fact that thefunctor L is linear. This later zig-zag can also be re-written as: sL ( V ) ≃ ←− • ≃ −→ L ( sV ) and thus equivalent to: k u ⊗ L ( V ) ≃ ←− • ≃ −→ L ( k u ⊗ V ) , u of degree . One deduce inductively from thisconstruction that , ∀ n ≥ , we have ( k u ) n ⊗ L ( V ) ≃ ←− • ≃ −→ L (( k u ) n ⊗ V ) andtherefore , given any homogeneous element u of arbitrary degree, we have a zig-zagof quasi-isomorphisms: α u : k u ⊗ L ( V ) ≃ ←− • ≃ −→ L ( k u ⊗ V ) . If W = ( k u ⊕ k v, d ) is a chain complex with 2 generators, we set α u + α v to be thecomposition α u + α v : W ⊗ L ( V ) • ≃ o o ≃ / / L ( k u ⊗ V ) ⊕ L ( k v ⊗ V ) ≃ / / L ( W ⊗ V ) , where the last quasi-isomorphism is due to the fact that L is linear. We generalizethis construction inductively on the number of generators to any arbitrary finitechain complex W. 2. In the case that r = 2 , let L ,V be the linear functor V L ( V , V ); One have: W ⊗ W ⊗ L ( V , V ) ∼ = −→ W ⊗ ( W ⊗ L ,V ( V )) ≃ ←− • ≃ −→ W ⊗ L ,V ( W ⊗ V ) ∼ = −→ W ⊗ L ,W ⊗ V ( V ) ≃ ←− • ≃ −→ L ,W ⊗ V ( W ⊗ V ) = L ( W ⊗ V , W ⊗ V ) Again, we generalize this argument inductively to any arbitrary r. Proof of corollary 4.12. We give the proof using the result of Theorem 4.9 and Corollary4.11. Namely, let F : C −→ D be a homotopy and reduced functor with C and D be any ofthe categories Alg O or Ch + . Then D n F ( X ) ≃ (Ω ∞ ) red H (Σ ∞ X ) , where H : Ch + −→ Ch is a special case of a n -multilinear functor ( V , ..., V n ) H ( V , ..., V n ) . 1. When C = D = Alg O , H ( V ) = b △ n ( F O ( − ))( V ) h Σ n = L n ( F O ( − ))( V, ..., V ) h Σ n . If Σ ∞ X is finite, then using Lemma 4.13, we have a Σ n -equivariant zig-zag of quasi-isomorphisms: (Σ ∞ X ) ⊗ n ⊗ b △ n ( F O ( − ))( k ) ≃ ←− • ≃ −→ b △ n ( F O ( − ))(Σ ∞ X ) and therefore we deduce the quasi-isomorphism (Σ ∞ X ) ⊗ n ⊗ h Σ n b △ n ( F O ( − ))( k ) ≃ ←− • ≃ −→ b △ n ( F O ( − ))(Σ ∞ X ) h Σ n . In addition, if F is finitary then L n ( F O ( − )) is finitary on each variable. In thiscase for any arbitrary algebra X, we rewrite Σ ∞ X as a filtered colimit of its finitesubcomplexes and then apply again Lemma 4.13 to these finite subcomplexes as aboveand recover a quasi-isomorphism (Σ ∞ X ) ⊗ n ⊗ h Σ n b △ n ( F O ( − ))( k ) ≃ ←− • ≃ −→ b △ n ( F O ( − ))(Σ ∞ X ) h Σ n . 2. In all other cases of the categories C and D , we refer again to Theorem 4.9 andCorollary 4.11 to chose the appropriate H and follows an analogue road map as in . efinition 4.14 (Goodwillie derivatives) . Let D be either Alg O or Ch + . 1. If F : Alg O −→ D is a homotopy and reduced functor, then b △ n F ( O ( k )) is called the n th derivative (or n th Goodwillie derivative) of F and is denoted ∂ n F. 2. If F : Ch + −→ D is a homotopy and reduced functor, then b △ n F ( k ) is called the n th derivative (or n th Goodwillie derivative) of F and is denoted ∂ n F. This definition holds when D = Ch + or Ch and the ground field k is of any characteristics. In this section, we show how one can compute the Goodwillie derivatives for a couple offunctors. Example 5.1. The computation below shows that the Goodwillie derivatives of the identityfunctor Id : Alg O −→ Alg O is given by: ∂ ∗ Id ≃ O .∂ n Id ≃ hocolim p i →∞ s − p − ... − p n cr n I ( O (Σ p k ) , ..., O (Σ p n k ))= hocolim p i →∞ s − p − ... − p n thof ib ( n − T 7→ O ( ⊕ i ∈ T Σ p i k )) ∼ = hocolim p i →∞ s − p − ... − p n O ( ⊕ i ∈ n Σ p i k )= hocolim p i →∞ s − p − ... − p n M r ≥ ( O ( r ) ⊗ Σ r ( ⊕ i ∈ n s p i k ) ⊗ r )= O ( n ) ⊗ Σ n ( k ) ⊗ n ∼ = O ( n ) . We use an analogue computation to obtain ∂ ∗ Σ ∞ Ω ∞ ≃ B ( O ) . The next example is transformed into a lemma. Let V be a finite non negatively gradedchain complex. By finite, we mean of finite dimension in each degree and bounded above.We define the functor N k Hom Ch + ( V ⊗ N k △ • , − ) : Ch + −→ Ch + where, N : sAb −→ Ch + is the normalization functor and k Hom Ch + ( V ⊗ N k △ • , W ) denotes the free simplicial k -vector space generated by the simplicial set Hom Ch + ( V ⊗ N k △ • , W ); Lemma 5.2. Let V ∈ Ch + . Then we have the quasi-isomorphism (in Ch ) ∂ n N k Hom Ch + ( V ⊗ N k △ • , − ) ≃ hom ( V, k ) ⊗ n Before we give the proof of this quasi-isomorphism, we remind the following fact whichseem to be a classical construction: Let p ∈ N , A be a simplicial k -vector space and considerthe following notations:- We write A [ p ] to mean the simplicial k -vector space given levelwise by A [ p ] n := A n ⊗ k s p . - If ( X • , ∗ ) is a pointed simplicial set then e k X • := k X • / k ∗ [ p ] is a p -connected Kan complex (as any simplicial abelian group), thus the Hurewiczmap A [ p ] h −→ e k A [ p ] , which is in fact the unit of the adjoint pair k ( − ) : sV ect k ⇄ sSet : U, is p -connected. The Hurewicz theorem stated on this current form appears in [GJ99, Thm3.7] for abelian groups.In addition, considering the natural projection l : e k A [ p ] −→ A [ p ] , ⊕ i x i Σ i x i . since the composite A [ p ] h −→ e k A [ p ] l −→ A [ p ] is the identity on A [ p ] \{ } , we deducethat the map l is also p -connected. Therefore the map Ω p e k A [ p ] Ω p ( l ) −→ Ω p A [ p ] is p -connectedand the map hocolim p →∞ Ω p e k A [ p ] −→ hocolim p →∞ Ω p A [ p ] is a weak equivalence of simplicial abelian groups. Now using the fact the the functor N is a left and right Quillen functor of the Dold Kan correspondence we deduce thequasi-isomorphism hocolim p →∞ Ω p N e k A [ p ] −→ hocolim p →∞ Ω p N A [ p ] (2) Proof of Lemma 5.2. We use Lemma 4.3 to obtain the quasi-isomorphism: cr n ( N k Hom Ch + ( V ⊗ N k △ • , − ))( W , ..., W n ) ≃ thocofib ( n ⊇ T N k Hom Ch + ( V ⊗ N k △ • , ⊕ i ∈ T W i )) On the other hand the functors N : sAb −→ Ch + and k ( − ) : sSet −→ sAb are leftQuillen functors, we therefore have the equivalencesthocofib ( N k Hom Ch + ( V ⊗ N k △ • , ⊕ i ∈ T W i )) ≃ N k thocofib ( Hom Ch + ( V ⊗ N k △ • , ⊕ i ∈ T W i )) ≃ N k thocofib ( ⊕ i ∈ T Hom Ch + ( V ⊗ N k △ • , W i )) Since the maps in the n -cube of pointed simplicial sets: T i ∈ T Hom Ch + ( V ⊗ N k △ • , W i ) are inclusions, the total homotopy colimit is the strict total cofiber (tcofib), and computa-tion shows (inductively) thattcofib ( ⊕ i ∈ T Hom Ch + ( V ⊗ N k △ • , W i )) ∼ = N k ( Hom Ch + ( V ⊗ N k △ • , W ) ∧ ... ∧ Hom Ch + ( V ⊗ N k △ • , W n )) ∼ = N ( e k Hom Ch + ( V ⊗ N k △ • , W ) ⊗ ... ⊗ e k Hom Ch + ( V ⊗ N k △ • , W n )) We then conclude the quasi-isomorphism: cr n ( f Ch + ( V, − ))( W , ..., W n ) ≃ N e k Hom Ch + ( V ⊗ N k △ • , W ) ⊗ ... ⊗ N e k Hom Ch + ( V ⊗ N k △ • , W n ) (3) If V is bounded below degree k, we have Hom Ch + ( V ⊗ N k △ • , s p + k k ) ∼ = Hom Ch + ( N k △ • , hom ( V, s p + k k )) ∼ = Hom Ch + ( N k △ • , hom ( V, k ) ⊗ s p + k k ) (1) ←− ≃ Hom Ch + ( N k △ • , hom ( V, k ) ⊗ s k k )[ p ] where the weak equivalence (1) is given by the weak equivalence of simplicial vector spaces om Ch + ( N k △ • , hom ( V, k ) ⊗ s k k ) ⊗ Hom Ch + ( N k △ • , s p k ) ≃ Hom Ch + ( N k △ • , hom ( V, k ) ⊗ s p + k k ) defined in [SS03, (2.8), p 295].Now, when we replace A in the map (2) with Hom Ch + ( N k △ • , hom ( V, k ) ⊗ s k k ) andcompose it with Ω k ( − ) , we get the quasi-isomorphismshocolim p →∞ Ω p N e k Hom Ch + ( N k △ • , hom ( V, k ) ⊗ s p k ) ≃ hocolim p →∞ Ω p + k N e k Hom Ch + ( N k △ • , hom ( V, k ) ⊗ s k k )[ p ] ≃ hocolim p →∞ Ω p + k N Hom Ch + ( N k △ • , hom ( V, k ) ⊗ s k k )[ p ] ≃ hocolim p →∞ Ω p + k hom ( V, k ) ⊗ s p + k ≃ hom ( V, k ) Using this above equivalence, we consider the specific case W i = s p i k in Equation (3) andapply the functor hocolim p i →∞ to the left-hand and right-hand side of this same equation, weget the quasi-isomorphism ∂ n N k Hom Ch + ( V ⊗ N k △ • , − ) ≃ hom ( V, k ) ⊗ n . This section intends to describe the chain rule property that have the Goodwillie deriva-tives when we compose two functors. Theorem 6.1. [Chi10, Thm 1.15] Let F, G : Ch −→ Ch be homotopy functors, andsuppose that F is finitary, then ∂ ∗ ( F G ) ≃ ∂ ∗ ( F ) ◦ ∂ ∗ ( G ) Corollary 6.2. Let C or D be any of the model categories Alg O , Ch, or Ch + , and F, G be homotopy and reduced functors: C G −→ Ch F −→ D , and suppose that F is finitary, then ∂ ∗ ( F G ) ≃ ∂ ∗ ( F ) ◦ ∂ ∗ ( G ) Proof. We prove this result only when C = Alg O since the other cases are obtained froma slight adaptation and following the same idea. If U : Alg O −→ Ch denotes the forgetfulfunctor, then we make the following computation: ∂ n ( F G ) ≃ b △ n ( F G )( O ( k ))= hocolim p i →∞ s − p − ... − p n cr n ( F G )( O (Σ p k ) , ..., O (Σ p n k )) ∼ = hocolim p i →∞ s − p − ... − p n cr n ( U F G )( O (Σ p k ) , ..., O (Σ p n k )) ∼ = hocolim p i →∞ s − p − ... − p n cr n ( U F G O ( − ))(Σ p k , ..., Σ p n k ) ≃ hocolim p i →∞ s − p − ... − p n cr n ( U F G O ( − ) red )(Σ p k , ..., Σ p n k ) ≃ ∂ n ( U F G O ( − ) red ) The two functors Ch G O ( − ) red −→ Ch UF −→ Ch are homotopy functors and U F preservesfiltered homotopy colimits. Therefore, we use Theorem 6.1 to claim that ∂ ∗ ( F G ) ≃ ∂ ∗ ( U F ) ◦ ∂ ∗ ( G O ( − ) red ) ≃ ∂ ∗ ( F ) ◦ ∂ ∗ ( G ) . eferences [AC11] Greg Arone and Michael Ching. Operads and chain rules for the calculus offunctors. Astérisque , (338):vi+158, 2011.[AN19] Miradain Atontsa Nguemo. Goodwillie calculus: Characterisation of polyno-mial functors. In preparation , 2019.[BFMT16] Urtzi. 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