ALGEBRAIC GOODWILLIE CALCULUS AND A COTRIPLE MODEL FOR THE REMAINDER ANDREW MAUER-OATS Abstract. We define an ä lgebraic" version of the Goodwillie tower, Palgn* *F(X), that de- pends only on the behavior of F on coproducts of X. When F is a functor t* *o connected spaces or grouplike H-spaces, the functor PalgnF is the base of a fibrati* *on |? *+1F| ! F ! PalgnF, whose fiber is the simplicial space associated to a cotriple ? built from* * the (n + 1)stcross effect of the functor F. When the connectivity of X is large enough (for * *example, when F is the identity functor and X is connected), the algebraic Goodwillie t* *ower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way o* *f studying the Goodwillie approximation to a functor F in many interesting cases. 1. Introduction Classically, computing the Goodwillie approximation PnF (X) for a homotopy fu* *nctor F (X) requires knowledge of F ( nX) for arbitarily high suspensions of X, as we* *ll as maps related to the map X ! F ( X). We present a new tower of approximations, which* * we call the algebraic Goodwillie tower, PnalgF (X), that depends only on F (X ^ -) on t* *he category of finite sets. The algebraic Goodwillie approximation PnalgF was created with the intent to * *be universal with respect to natural transformations of F to functors whose nth cross effect* * vanishes. However, it turns out that there is a subtle issue on ß0, so some hypotheses on* * F is needed. Our main theorem is that for good functors F , there exists a quasifibration (1.1) |? *+1n+1F| ! F ! PnalgF, where the fiber is built from a cotriple ? formed from the diagonal of the nthc* *ross effect. This result was inspired by the work of Johnson and McCarthy [8] in the entirel* *y algebraic context of chain complexes. In [8], the authors are able to define the analog o* *f Pnalgsimply by taking the cofiber of the left-hand map in (1.1). However, in an unstable topol* *ogical setting, that approach is not useful, and the proof is much more difficult. It is easy to show that when F commutes with realizations, the algebraic Good* *willie approximation PnalgF (X) agrees with the topological Goodwillie approximation P* *nF (X), so in this case (1.1)shows that the "remainderö f the n-excisive Goodwillie appro* *ximation is the ? n+1-homology of F . ___________ Date: December 5, 2002. 2000 Mathematics Subject Classification. 55P65. 1 ALGEBRAIC GOODWILLIE CALCULUS 2 1.1. Organization of the paper. We begin by summarizing our conventions in Sect* *ion 2. We then proceed to state the main theorem in Section 3, along with all of the b* *asic definitions necessary to understand it. The next section (x4) summarizes more technical ing* *redients of the proof. We then show in Section 5 that the approximation Pnalgpreserves the * *connectivity of natural transformations. To deal with another (important) technical issue, * *Section 6 sketches proofs that when the cross effect vanishes, a certain cube is actually* * Cartesian (rather than just having contractible total fiber). After these technicalities* *, we give an outline of the main theorem (x7). The proof of the main theorem is broken into * *two sections: Section 8 proves äC se I", and Section 9 proves äC se II". We include in Append* *ix A the technical proof that ? is in fact a cotriple. 1.2. Acknowledgements. This paper is based on my thesis. I would like to thank * *Randy McCarthy for his guidance, and Jim McClure for his helpful suggestions. 2.Preliminaries The category of "spaces" is taken to be the category of compactly generated s* *paces with nondegenerate basepoint. The space 0 or * is the space with only one point. Whe* *n forming the function space Map (A, B), we always implicitly replace A by a cofibrant ap* *proximation (CW complex) and B by a fibrant approximation (which is no change with our defi* *nition of spaces). When forming the geometric realization, we first "thicken" each space * *([10, p. 308]) so the resulting functor has good homotopy behavior. When we say two spaces are* * equivalent, we mean they are weakly homotopy equivalent. We use the term fibration for quas* *ifibrations in the sense of Dold and Thom [4]. We briefly recall a few standard properties of functors. A functor is continu* *ous if the map Map (X, Y ) ! Map (F (X), F (Y )) sending f to F (f) is continuous. A functor i* *s a homotopy functor if it preserves weak homotopy equivalences. A functor is called reduced* * if F (0) ' 0. Definition 2.1. A homotopy functor F is said to satisfy the limit axiom if F co* *mmutes with filtered homotopy colimits of finite complexes. That is, if hocolimF (Xff) ' F * *(hocolimXff) for all filtered systems {Xff} of finite complexes. In this paper, we assume that all functors are continuous homotopy functors t* *hat satisfy the limit axiom. All functors are defined from pointed spaces to pointed space* *s unless otherwise stated. Cubical diagrams and homotopy fibers figure heavily into this work, so we rec* *all several definitions from [6]. Let T be a finite set. P(T ) is the poset of subsets of T* * (regarded as a category). P0(T ) is the poset of nonempty subsets of T . A T -cube is a functo* *r defined on P(T ). If X is a T -cubeQof pointed spaces, then its homotopy fiber, hofibX , i* *s the subspace the function space U T Map [0, 1]U, X (U)consisting of maps that are natural* * in U and have sent points with any coordinate 1 to the basepoint. A formal definition is* * given in [6, 1.1]. Alternatively, the homotopy fiber can be constructed by iterating the pro* *cess of taking fibers in a single direction. * * W We write n for the finite (unpointed) set {1, . .,.n}, and [n] for the finite* * space nS0 ~= {0, . .,.n}, with basepoint 0. ALGEBRAIC GOODWILLIE CALCULUS 3 3. The Main Theorem In this section, we briefly present all of the background necessary to unders* *tand the Main Theorem (3.10). In brief: cross effects, excisive functors, left Kan extensions* *, and Pnalg The cross effect measures the failure of a functor to take coproducts to prod* *ucts. Definition 3.1 (crn, ? n).iDefine the nthcross-effect cube, CR nF (X1, . .,.Xn)* *, to be the W j n-cube X with X (U) = F u62UXu and X (i : U ! V ) induced by the identity o* *f Xu if u 62 V and the map to the basepoint if u 2 V . Let crnF (X1, . .,.Xn) denote the homotopy fiber of the cube CR nF (X1, . .,.* *Xn), and let ? nF (X) = crnF (X, . .,.X) be the diagonal of the nthcross effect. As we will show in Appendix A, ? nis part of a cotriple. Example 3.2. The second cross effect of the functor F (X) = Q(X) is contractibl* *e, but the second cross effect of F (X) = Q(X ^ X) is cr2F (X, Y ) ' Q(X ^ Y ) x Q(Y ^ X). Before we define the algebraic Goodwillie tower, we will recall the classical* * definition of the topological Goodwillie tower. Definition 3.3 (Cartesian, co-Cartesian). An S-cube X is Cartesian if the map X* * (;) ! holimP0(S)X is a weak equivalence. An S-cube X is co-Cartesian if the map hocol* *imP1(S)X ! X (S) is a weak equivalence. An S-cube is strongly co-Cartesian if every sub-2-* *cube is co- Cartesian. Definition 3.4 (n-excisive). ([6, 3.1]) F is n-excisive if for every strongly * *co-Cartesian (n + 1)-cube X , the cube F X is Cartesian. In [7], Goodwillie constructs a universal n-excisive approximation PnF to a f* *unctor F . The approximations form a tower of functors equipped with natural transformatio* *ns of the following form: F _____//_FFPnF FF | FFF | FF""fflffl| Pn-1F Definition 3.5 (left Kan extension). Let LF denote the homotopy invariant left * *Kan ex- tension of a functor F along the inclusion of finite pointed sets into pointed * *spaces. Letting S denote the category of finite pointed sets and T denote the category of poin* *ted spaces, the realization of the following simplicial space can taken to be the definitio* *n of LF(-): ` (3.1) [n] 7! F (C0) ^ (MapS(C0, C1) x . .x.Map T(Cn, -))+. (C0,...,Cn) The coproduct is taken over all (C0, . .,.Cn) 2 Sxn. The left Kan extension LF (Y ) comes equipped with a map to F (Y ) by composi* *ng the maps in each dimension. This map is an equivalence when Y is a finite set [9, x* *X.3, Cor. 3]. ALGEBRAIC GOODWILLIE CALCULUS 4 When working with the left Kan extension, we will frequently want to shift th* *e functor so that we can examine its value on coproducts of spaces other than S0. To do this* *, we write FX (-) for the functor F (X ^ -). We primarily understand LF as "F made to commute with realizations", in the s* *ense of the following lemma. Lemma 3.6. Let Y.be a simplicial set. Then LFX (|Y.|) ' |FX (Y.)| = |F (X ^ Y.)|. Q Proof.Using (3.1), this follows from the observation that Map (S, Y ) ~= s2SY * *commutes with realizations when S is a finite set. We are now in a position to define the algebraic Goodwillie approximation. Definition 3.7 (PnalgF ). The algebraic Goodwillie approximation PnalgF (X) is * *defined to be the functor given by applying Pn to the left Kan extension of F shifted over* * X; that is, PnalgF (X) = Pn(LFX )(S0). The natural transformation F (X) ! PnalgF (X) arises from the evaluating the * *map LFX ! Pn(LFX ) at S0. There remains a technical hypothesis on F , related to the "ß*-Kan condition"* *, needed to apply the main theorem. We give the hypotheses here, but defer discussion of ho* *w they are used until Section 4.3. Hypothesis 3.8 (Connected Values). F has connected values (on coproducts of X) * *if the functor LFX is always connected. Hypothesis 3.9 (Group Values). In the following definition, let T denote the ca* *tegory of pointed spaces. Let G denote the category of grouplike H-spaces, and let U : G * *! T be the forgetful functor. F has group values (on coproducts of X) if there exists a functor F 0so that * *the following diagram commutes: ?G? F0X(-)~~~ ~~~ U|| ~~ fflffl| TLFX(-)//_T In this case, we will conflate LFX (-) with its lift to groups. We are now able to state our theorem relating the cross effects of a functor * *and the algebraic Goodwillie approximation. Theorem 3.10 (Main Theorem). Let F be a reduced homotopy functor from pointed s* *paces to pointed spaces. If F has either connected values (Hypothesis 3.8) or group * *values (Hy- pothesis 3.9) on coproducts of X, then the following is a quasifibration: (3.2) |? *+1n+1F (X)| ! F (X) ! PnalgF (X), and furthermore the right hand map is surjective on ß0. ALGEBRAIC GOODWILLIE CALCULUS 5 4.Technical Conditions In this section, we address many technical aspects needed to make our machine* *ry work. 4.1. Limit axiom. It is of primary importance to know that the functors we will* * be working with satisfy the limit axiom. Lemma 4.1. For any functor F , the functor LF satisfies the limit axiom (2.1). Proof.The functor Map (K, -) satisfies the limit axiom for any compact K. Exam* *ining (3.1), we see that this implies that LF satisfies the limit axiom. 4.2. Eilenberg-Zilber. The Eilenberg-Zilber theorem for bisimplicial sets provi* *des a source for connectivity estimates. In this paper, we use homotopy invariant realizatio* *n of simplicial spaces, so the same result holds for the realization in two dimensions and the * *realization of the diagonal of a bisimplicial space. Lemma 4.2 (Eilenberg-Zilber). ([2, B.1]) If X is a bisimplicial set, then Tot(X* *) '-!diag(X) is a weak equivalence of simplicial sets. Corollary 4.3 (Eilenberg-Zilber connectivity estimate). Let X and Y be functor* *s from ( op)xN to spaces, and let p = (p1, . .,.pN ) 2 ( op)xN denote an index for the* *se multisim- plicial spaces. Let f : X ! Y . Suppose that f(p) is w-connected for all indi* *ces p. If in addition f(p) is an equivalence when any pi< ni, then |f| is ( ni+ w)-connected. 4.3. The ß*-Kan condition. For us to be able to say anything useful about LF , * *we need to know that the Kan extension of the fiber of a map is the fiber of the Kan ex* *tensions. Definition 4.4 (ß*-Kan functor). A functor F is called a ß*-Kan functor if give* *n a map p : Y ! Z with a section, the simplicial spaces F (Y.) and F (Z.) satisfy the ß* **-Kan condition, and Lß0F (p) is a fibration of simplicial sets. This is useful in practice due to the following theorem of Bousfield and Frie* *dlander, re- stated for simplicial spaces. Theorem 4.5. ([2, Theorem B.4]) Let A, B, X, and Y be simplicial spaces, and s* *uppose that the cube A ____//_X | | | | fflffl|fflffl| B ____//_Y has the property that evaluation at every [n] 2 op produces a Cartesian cube. * *If X and Y satisfy the ß*-Kan condition and if ß0X ! ß0Y is a fibration of simplicial sets* *, then after realization the cube is still Cartesian. In this work we restrict ourselves to functors satisfying the hypotheses of c* *onnected values (3.8) or group values (3.9) so that the simplicial spaces involved always satis* *fy the ß*-Kan condition. The ß*-Kan condition is a technical fibrancy condition introduced i* *n [2, xB.3] that we do not repeat here. ALGEBRAIC GOODWILLIE CALCULUS 6 Corollary 4.6. If F is a ß*-Kan functor, then given a map p : X ! Y with a sect* *ion, the spaces L hofibF (p) and hofibLF (p) are equivalent. In view of Lemma 3.6, this* * is implies |hofibF (p.)| ' hofibF (|p.|). Proof.By definition, a ß*-Kan functor causes F (X.) ! F (Y.) to satisfy the hyp* *otheses on the right hand vertical map in Theorem 4.5, so letting B = * and A = hofibF (p.* *) produces the desired result. Lemma 4.7. If either F has connected values (3.8) on coproducts of X or F has g* *roup values (3.9) on coproducts of X, then FX is a ß*-Kan functor (4.4). Proof.In these cases, LFX always satisfies the ß*-Kan condition ([2, p. 120]).* * Also, for any surjective map p, the function ß0FX (p.) is a fibration of simplicial sets,* * since surjective maps of simplicial groups are fibrations. Hence Theorem 4.5 applied with B = * * *shows that |hofibF (p.)| ' hofibF (|p.|), as desired. Remark 4.8. Theorem 4.5 also implies that if each space Xiis connected then | X* *i | ' |Xi|. The cross effect of the Kan extension of a ß*-Kan functor can be computed fro* *m finite sets, as the following lemma shows. Lemma 4.9. If F is a ß*-Kan functor, then crn(LF )(Y 1, . .,.Y n) ' Ln(crnF )(Y 1, . .,.Y n), where Ln indicates the Kan extension is taken in each of the n variables of crn* *F . The statement above can be abbreviated to ? n(LF )(Y ) ' L(? nF )(Y ). * * W Proof.The space crn(LF )(Y 1, . .,.Y n) is the homotopy fiber of a cube involvi* *ng LF u2UY u for U {1, . .,.n}. We will write this part of our argument assuming U = {1, .* * .,.n} for simplicity. By Lemma 3.6, this is the realization of the simplicial space F (diag(Y.1_ . ._.Y.n)) = diagF (Y.1_ . ._.Y.n). Then the Eilenberg-Zilber theorem (4.2) shows that the diagonal has the same ho* *motopy type as the (multidimensional) realization of the n-dimensional simplicial space F (* *Y.1_ . ._.Y.n). All of the maps in CR n(LF ) have sections, so the hypothesis that F is a ß*-* *Kan functor means that we can compute fibers before taking realizations, as desired for the* * lemma. 4.4. Goodwillie calculus: classification of homogeneous functors. The functor * *Pn gives the universal n-excisive approximation to a functor. The functor Dn gives* * the homo- geneous n-excisive part of a functor; it is part of a fibration sequence Dn ! Pn ! Pn-1. Goodwillie shows that there is actually a functorial delooping of the derivativ* *e, so this fibration sequence can be delooped once: Theorem 4.10. ([7, Lemma 2.2]) If F is a reduced, analytic functor from spaces * *to spaces, then the map PnF ! Pn-1F is part of a fibration sequence PnF ! Pn-1F ! -1DnF, where -1DnF is a homogeneous n-excisive functor. ALGEBRAIC GOODWILLIE CALCULUS 7 Definition 4.11 (Derivative of F ). The nthderivative of F (at *), denoted @(n)* *F (*), is the following spectrum with n action, which we will denote Y. The space Yk in the * *spectrum is k(n-1)crnF (Sk, . .,.Sk). The structure map Yk ! Yk+1 arises from suspendi* *ng inside and looping outside each variable of crn. When F satisfies the limit axiom (2.1), we can express DnF (X) using the deri* *vative: Theorem 4.12. ([7, x5]) If F is an analytic functor from spaces to spaces that * *satisfies the limit axiom (2.1), then the functor DnF is given by (n) ^n (4.1) DnF (X) ' 1 @ F (*) ^h n X . where smashing over h n denotes taking homotopy orbits. 5. Pnalgpreserves connectivity In this section, we establish a property of fundamental importance when worki* *ng with Pnalg: the n-additive approximation preserves the connectivity of natural tran* *sformations that satisfy some basic good properties. Actually, we prove the slightly strong* *er result that before evaluation at S0, the functor PnL(-)X preserves connectivity. Theorem 5.1. Let F and G be reduced ß*-Kan functors (4.4). If j : F ! G is a na* *tural transformation that is w-connected, then the natural transformation Pn(Lj) is w* *-connected. Once this theorem is established, we have the following immediate corollary. Corollary 5.2. Let F and G be reduced ß*-Kan functors. If j : F ! G is a nat* *ural transformation that is w-connected, then the natural transformation Pnalg(j) is* * w-connected. Lemma 5.3. Let j : F ! G be a natural transformation on ß*-Kan functors. If j * *is w- connected, then the induced map of derivative spectra @(n)LF (*) ! @(n)LG(*) is* * w-connected. Proof.Using the Eilenberg-Zilber connectivity estimate (4.3), for all n 1, th* *e map |? nF (Sk.)| ! |? nG(Sk.)| is (nk + w)-connected. The derivative spectrum @(n)LF (*) then has* * as its kth space the space k(n-1)?nLF (Sk.), which by Lemma 4.9, is equivalent to k(n-1)* *|? nF (Sk.)|. On these spaces the map induced by j is (k + w)-connected, exactly as required * *to produce a w-connected map @(n)LF (*) ! @(n)LG(*). Corollary 5.4. The derivative spectrum @(n)LF (*) is connective. Proof.The map F ! 0 is always 0-connected. Corollary 5.5. If j : F ! G is a w-connected map of ß*-Kan functors, then Dn(Lj* *) is w-connected. Proof.Recall from Equation 4.1 that for any functor H, (n) ^n Dn(LH)(Y ) = 1 @ LH(*) ^h n (Y ) . Taking homotopy orbits and smashing with a fixed space preserves connectivity, * *so this is really a question about the connectivity of the map @(n)LF (*) ! @(n)LG(*). The* * required connectivity was established in Lemma 5.3. ALGEBRAIC GOODWILLIE CALCULUS 8 Corollary 5.6. If F is a reduced ß*-Kan functor then the natural transformation* * Pn+1LF ! PnLF is surjective in ß0. Proof.Theorem 4.10 (Goodwillie's delooping of Dn) shows that the fibration Dn+1LF ! Pn+1LF ! PnLF deloops to a fibration (5.1) Pn+1LF ! PnLF ! -1Dn+1LF. The delooping of Dn+1LF consists of smashing with the suspension of @(n+1)LF (** *) and taking homotopy orbits. By Corollary 5.4, the spectrum @(n+1)LF (*) is connect* *ive, so its suspension is 0-connected. From the long exact sequence on ß*, this implies Pn+* *1LF ! PnLF is surjective on ß0. Proof of Theorem 5.1.As above, the spectrum @(n+1)LF (*) in Equation (5.1)is co* *nnective, so the map on the delooping of Dn+1 induced by j is (w + 1)-connected. The map * *on the fibers (Pn+1(Lj)) is therefore w-connected. 6. Fiber contractible implies Cartesian In this section, we will sketch proofs of the critical but mainly technical f* *act that in the cases we consider, the cross effect vanishing is equivalent to the cross effect* * cubes being Cartesian. We generally want to use the fact that the cross effect is contracti* *ble to conclude that the initial space in the cross-effect cube is equivalent to the (homotopy)* * inverse limit of the rest of the spaces. Unfortunately, this is not always true; the problem* * is that the homotopy fiber does not detect failure to be surjective on ß0. Here we should * *that our hypotheses on F are sufficient that this does not happen. Lemma 6.1. Let F be a functor satisfying Hypothesis 3.8 (connected values) on c* *oproducts of X. If ? nF (X) ' 0, then the cube CR nF (X, . .,.X) defining ? nF (X) is Car* *tesian. Proof sketch.The first step is to consider the pullback diagram p q X -!Z- Y when the spaces X, Y , and Z are connected, and p and q have sections. In this * *case, one can directly show that ß0 of the homotopy inverse limit is 0. In general, decompose the desired inverse limit into iterated pullbacks of th* *e form in the first step. All of the maps have sections because of the very special form* * of the cube CR nF (X, . .,.X). This shows that the map from the initial space to the homotopy inverse limit * *is (trivially) surjective in ß0; then Cartesian-ness follows because the total fiber is contra* *ctible. Lemma 6.2. Let F be a functor satisfying Hypothesis 3.9 (group values) on copro* *ducts of X. If ? nF (X) ' 0, then the cube CR nF (X) defining ? nF (X) is Cartesian. ALGEBRAIC GOODWILLIE CALCULUS 9 Proof sketch.This strategy here is to decompose F into a fibration involving F0* *, the con- nected component of the identity, and ß0F , a functor to discrete groups. F0 ! F ! ß0F. The statement for F0 follows from Lemma 6.1. The statement for ß0F is Lemma 6.3* *, below. Then a short argument shows no problem arises on ß0 in CR nF (X). The case of discrete groups is key, so we give a complete proof for this case. Lemma 6.3. Let X be an n-cube (n 1) of discrete groups. If X has compatible s* *ections to all structure maps (for example, X = CR nF (X)), then the map X (;) ! lim X (U) U2P0(n) is surjective. Proof.We need to show that the map above is surjective. This is equivalent to s* *howing that there exists an x; 2 X (;) mapping to each coherent system of elements xU 2 X (* *U), with U 6= ;. X is a cube of groups, and hence all of the structure maps are group homomorp* *hisms. This allows us to subtract an arbitrary w 2 X (;) from x;, and subtract the images I* *mU(w) of w in X (U) from each xU, to show the question is equivalent to the existence of an x* *;- w 2 X (;) mapping to each coherent system of elements xU - ImU(w). Given a coherent system of elements xU in an n-cube, let w be the image of x{* *n} in X (;) using the section map X ({n}) ! X (;). Define zU = xU - ImU(w), noting th* *at when {n} V , we have ImV (w) = xW , so zW = 0. By the preceding paragraph, the su* *rjectivity that we are trying to establish is equivalent to the existence of a z; mapping * *to each coherent collection zU. If n = 1, then limU2P0(1)X (U) = X({1}), so the section map X ({1}) ! X (;) p* *roduces a z; mapping to z{1}, as desired. If n > 1, then we proceed by induction, assuming the lemma is true for smalle* *r n. Taking the fiber of X in the direction of {n}, we have an (n - 1)-cube Y(U) := fib(X (U) ! X (U [ {n})). The cube Y satisfies the hypothesis of the lemma because taking fibers preserve* *s compatible sections. Notice that for {n} 6 U, the element zU passes to the fiber, since * *it maps to zU[{n}= 0. Now Y is an (n - 1)-cube, so by induction, the map from Y(;) to limY(U) is su* *rjective. That is, there exists a y 2 Y(;) with Im U(y) = zU. Mapping y to z 2 X (;) giv* *es an element z with ImU (z) = zU for U {1, . .,.n -}1. As above, if {n} U, then * *zU = 0, so ImU (z) = zU in this case as well. Therefore, we have produced an element z * *mapping to each coherent collection of elements zU, as desired. 7. Main Theorem: Outline To establish the main theorem, we use induction on n, beginning with the case* * n = 1. We further break down the induction into äC se I", where ?n+1 F (X) ' 0, and äC se* * II", where ALGEBRAIC GOODWILLIE CALCULUS 10 ? n+1F (X) 6' 0. Our proof will be a ladder induction, with Case I depending on* * Case II for smaller values of n, and Case II depending on Case I for the same value of n. In Section 8, we treat the case when ? n+1F (X) ' 0. In this case, we show di* *rectly that the fiber of the fibration sequence we obtain from induction, |? *n(? nF()X)| ! F (X) ! Pnalg-1F (X), is a homogeneous degree n functor. This implies that F (X) ' PnalgF (X) in this* * case. In Section 9, we treat the case when ?n+1F (X) 6' 0. In this case, we conside* *r the auxiliary diagram: AF(X) _________//|? *n+1(? n+1F()X)|ffl___//F (X) | | | | | | fflffl| fflffl| fflffl| PnalgAF(X)____//_Pnalg|? *n+1(? n+1F()X)|_//_PnalgF (X) where AF is defined as the homotopy fiber of the map ffl in the top row, and th* *e bottom row is shown to be a quasifibration as well (Propositon 9.8 and 9.6). We show that ?n+* *1 AF(X) ' 0 (Lemma 9.10), and hence the case ? n+1F ' 0 shows that there is an equivalen* *ce of the fibers, so the square on the right is Cartesian. Then it is not hard to es* *tablish that Pnalg|? *n+1(? n+1F()X)|' 0 (Lemma 9.9), so that (3.2)is actually a quasifibrat* *ion. 8. Main Theorem, Case I: ? F ' 0 In this section, the goal is to establish that when the (n + 1)-st cross effe* *ct of F vanishes, F is equivalent to its n-additive approximation, PnalgF . Proposition 8.1. If F is a reduced functor that has either connected values (Hy* *pothesis 3.8) or group values (Hypothesis 3.9) on coproducts of X, and ? n+1F (X) ' 0, then F* * (X) ' PnalgF (X). Since PnalgF (X) is defined by evaluating the functor Pn(LFX ) at the space S* *0, and LFX commutes with realizations, Proposition 8.1 is an immediate corollary of the fo* *llowing propo- sition. Proposition 8.2. Suppose that F commutes with realizations and has connected va* *lues (3.8) or group values (3.9). If ? n+1F ' 0, then F ' PnF . The proof of Proposition 8.2 will be given in Section 8.3. 8.1. Additivization and the bar construction. The case n = 1 is the work of Seg* *al. Lemma 8.3. Suppose F is a reduced functor that has either connected values (Hyp* *othe- sis 3.8) or group values (Hypothesis 3.9) and commutes with realizations. If ? * *2F ' 0, then F ' O F O . Proof.Let X be a space. Under these hypotheses,Wthe map F (X _X) '-!F (X)xF (X)* * is an equivalence, so we can regard [n] 7! F ( nX) as a -space. Segal's work [10, P* *roposition 1.4] then shows that F (X) ' BF (X), where B denotes the bar construction, and note* * that BF (X) ' |F (X ^ S1.)| ' F (S1 ^ X), because F commutes with realizations. ALGEBRAIC GOODWILLIE CALCULUS 11 Corollary 8.4. If F has either connected values (Hypothesis 3.8) or group value* *s (Hypoth- esis 3.9), and ? n+1F ' 0, then, as a symmetric functor of n variables, ? nF is* * the infinite loop space of a symmetric functor to connective spectra ? nF: ?n F ' 1 (? nF). Proof.When F satisfies the hypotheses above, then ? nF also satisfies these hyp* *otheses. Since ? n+1F ' 0, we know that applying ? 2 in any input to crnF is contracti* *ble, so Lemma 8.3 gives ? nF as the first space of a connective -spectrum. Corollary 8.5. Suppose F is a reduced functor that commutes with realizations a* *nd has either connected values (Hypothesis 3.8) or group values (Hypothesis 3.9), and * *? 2F ' 0, then F ' P1F . Proof.We need to show that F ! nOF O n is an equivalence for all n. This almos* *t follows from Lemma 8.3, but one needs to check that O F O commutes with realization* *s. This follows from Theorem 4.5 because F (S1 ^ -) is always connected. 8.2. Iterated cross effects produce homogeneous functors. This section establis* *hes that ? *n(? nF()X) has the structure of a homogeneous functor (see Equation (4.* *1)). Lemma 8.6. If F has either connected values (Hypothesis 3.8) or group values (H* *ypothe- sis 3.9) on coproducts of X, and ? n+1F (X) ' 0, then |? *n(? nF()X)| ' ? nF(X) ^ n E +n, where ? nF denotes the lift to spectra of ? nF , as in Corollary 8.4. We write +nfor the space n with a disjoint basepoint added. Let H(X1, . .,.Xn) be a symmetric functor from spaces to spectra, and suppose* * H is addi- tive in each variable separately. In practice, such an H will arise as ? nF fro* *m Corollary 8.4. Using the additivity in each variable and then the stable equivalence of copr* *oducts and products, we can identify ?n H with +n^ H. Here we have X1 = . .=.Xn, but labe* *l them differently to be able to see the action of the symmetric group more clearly. Y ? nH(X1, . .,.Xn)' H(Xff(1), . .,.Xff(n)) ff2 n (8.1) ? nH(X1, . .,.Xn)' +n^ H(X1, . .,.Xn). This identification goes back at least to [5, Theorem 9.1]. The identification in Equation 8.1 can be made equivariant with respect to th* *e action of n on both H and ? n(-) as follows. The action induced by permuting the inputs * *of ? nH (i.e., from the fact that ? n(-) is a symmetric functor) is sent to multiplicat* *ion on the n factor. The action on ?n H induced by the n action on H is sent to the same ac* *tion on H on the other side. With this model, the map ffl : ?n H ! H is given by oe ^ x 7* *! x. We are now in a position to understand ? *n?nF. The issue of how multiplicat* *ion by oe arises from the ffl above is somewhat subtle, so we spell it out in detail. * * Applying (8.1) repeatedly at each level, we have ?kn?nF (X) ' +n^ . .^. +n^? nF (X). ______-z_____" k factors ALGEBRAIC GOODWILLIE CALCULUS 12 Recall that the face maps from dimension n to n - 1 are given by di = ? inffl ?* *n-in. In dimension k, the face map dk = ffl ?knjust drops the first element: dk(gk ^ . .^.g1 ^ y) = gk-1^ . .^.g1 ^ y. To compute the others, note that for any f, the map ? n(f) is equivariant with * *respect to the action of n on ?n (by permuting inputs), so in particular ?n(ffl) : ?n(? n* *F ) ! ?n F is equivariant with respect to the action on of n on the leftmost ? n, so ? nffl(g ^=y)?nffl(g * (1 ^ y)) = g * ?n ffl(1 ^ y) = g * y, where the last follows since the degeneracy ffi : ?n F ! ?2nF given by ffi(y) =* * 1^y is a section to the face map ? nffl. This argument shows that all of the face maps dj with * *0 j < k are given by multiplying gj+1 by the next coordinate to the right (either gj if* * j > 0 or y if j = 0). This is a standard model for E +n^ n ? nF(X), so we have proven Lemma 8.6. 8.3. Proof Of Proposition 8.2. Proof of Proposition 8.2.When n = 0, the hypothesis that F is reduced makes the* * result trivial. When n = 1, Corollary 8.5 shows that F (X) ' P1F (X), so that establis* *hes the true base case in our induction. Finally, when n > 1 we apply Proposition 9.1 with one smaller n to produce a * *fibration sequence: (8.2) |? *n(? nF|)! F ! Pn-1F, where the map F ! Pn-1F is surjective on ß0. (Recall that since F commutes with* * realiza- tions, Pnalg-1F ' Pn-1F .) We now show that the fiber here is an n-excisive functor. From Corollary 8.4,* * we know that ?n F lifts to a functor to connective spectra ? nF, so that ?n F ' 1 ? nF* *. Using this, we have: |? *n(? nF|)' |? *n( 1 ? nF)|. The functor 1 is a right adjoint, so it preserves homotopy fibers, and hence c* *ommutes with ? n: |? *n( 1 ? nF)| ' | 1 ? *n(?nF)|. Now ? nF is a functor to connective spectra, and hence all applications of ?n t* *o it result in functors to connective spectra, so we can use [1] to move 1 outside of the rea* *lization: | 1 ? *n(?nF)| ' 1 |? *n(?nF)|. Then Lemma 8.6 shows that the term inside the realization computes the homotopy* * orbits of the n action on ? nF: |? *n(? nF|)' ? nF ^ n E +n. ALGEBRAIC GOODWILLIE CALCULUS 13 Combining all of these, we identify the fiber in (8.2)as the infinite loop spac* *e of the preceding homotopy orbit spectrum: + |? *n(? nF|)' 1 ? nF ^ n E n . We now establish that this functor is n-excisive. Since it is known to be th* *e fiber of F ! Pn-1F , this will imply that it is actually homogeneous n-excisive. The functor 1 preserves Cartesian squares, so we need only establish that ? * *nF^ nE +n is n-excisive. The functor ? nF is the diagonal of the cross-effect functor crn* *F. The second cross effect in any variable of this functor is contractible, so Corollary 8.5 * *shows that in each variable crnF is 1-excisive. Then [6, Proposition 3.4] shows that its diag* *onal, ? nF is n-excisive. That is, given any strongly co-Cartesian (n + 1)-cube X , the map ? nF X (;) ! holim ? nFX P0(n+1) is an equivalence. Since n acts natually on ? nF, it is a n-equivariant map, * *so it is still an equivalence after taking homotopy orbits. This establishes that the functor ? n* *F ^ n E +n is n-excisive, as desired. Equation (8.2)is a quasifibration, so we know the natural map |? *n(? nF|)! fib(F ! Pn-1F ) is an equivalence. Applying Pn and using the fact that we have just shown that * *|? *n(? nF|) is n-excisive, we have |? *n(? nF|)' Pn|? *n(? nF|) ' Pn fib(F ! Pn-1F ) ' fib(PnF ! Pn-1F ) ' DnF. When F commutes with realizations, the map F ! Pn-1F is surjective on ß0 (Theor* *em 5.1). This lets us argue that the total space F of the fibration in (8.2)is n-excisiv* *e, since the base and the fiber are. The argument is straightforward but does require a variant o* *f the five- lemma on ß0 to make a conclusion about ß0F . Once we know F is n-excisive, we * *know F ' PnF , as desired. A result that is closely related to Theorem 5.1 is the following, which says * *that Pnalg preserves (good) fibrations that are surjective on ß0. Proposition 8.7. Given a space X and functors A, B, and C, supposeWA(Y ) ! B(Y * *) ! C(Y ) is a fibration sequence for all finite coproducts Y = X of X. If, on fi* *nite coproducts of X, either: (1) C takes connected values (Hypothesis 3.8); or (2) B and C take group values (Hypothesis 3.9), and the map B ! C is a surje* *ctive homomorphism of groups, then LAX (Z) ! LBX (Z) ! LCX (Z) is a fibration sequence for all spaces Z. Furthermore, the sequence is surjecti* *ve on ß0. ALGEBRAIC GOODWILLIE CALCULUS 14 Proof.This is an easy application of Theorem 4.5. Corollary 8.8. Under the conditions of Proposition 8.7, PnalgA ! PnalgB ! PnalgC is a fibration and the map to the base is surjective on ß0. Proof.By definition, PnalgF (X) = Pn(LFX )(S0), so combining Proposition 8.7 wi* *th Theo- rem 5.1 gives the desired result. 9.Main Theorem, Case II: ? F 6' 0 In this section, the goal is to establish the other side of the äl dder induc* *tion" for Theo- rem 3.10. Proposition 9.1. If F is a reduced functor that has either connected values (Hy* *pothesis 3.8) or group values (Hypothesis 3.9) on coproducts of X, and ?n+1 F (X) 6' 0, then * *the following is a fibration sequence up to homotopy: (9.1) |? *n+1(? n+1F()X)| ffl-!F (X) ! PnalgF (X). Furthermore, the map ß0F (X) ! ß0PnalgF (X) is the universal map to the cokerne* *l of the group homomorphism ß0? n+1F (X) ! ß0F (X). We begin with a definition for the homotopy fiber of the map ffl. Definition 9.2 (AF). Define the functor AF(X) to be the homotopy fiber in the f* *ibration: (9.2) AF(X) ! |? *n+1(? n+1F()X)| ! F (X). To show the proposition, we proceed essentially as outlined in Section 7. We* * actually decompose the problem further, considering functors to discrete groups or conne* *cted spaces. As one would expect, the case of a functor to discrete groups is the pivotal on* *e. 9.1. Functors To Discrete Groups: ? Gab= 0. This section shows that a certain f* *unctor Gabnhas no (n+1)stcross effect, as one would expect in view of the construction* * of Gabn(given below). In this section, we consider a functor G from spaces to discrete groups* * (for example, G(X) = ß0 (X)). Definition 9.3. Given an n > 0 and a functor G to discrete groups, define G0n:=* * Im(ffl : ? n+1G ! G) and Gabn:= coker(ffl). Usually, the n is clear from context, and w* *e will abbreviate these G0and Gab. There is a short exact (fibration) sequence of groups (9.3) G0(X) ! G(X) ! Gab(X), and this sequence is surjective on ß0 (i.e., right exact). Remark 9.4. Note that since ? n+1G is constructed by taking a kernel, the image* * of G0 is normal in G. ALGEBRAIC GOODWILLIE CALCULUS 15 Our motivation for the preceding notation comes from considering the case whe* *n the source and target category under consideration are both the category of groups * *and the functor G is the identity G(H) = H. In this case, the image of ?2 G(H) in G(H) * *is the first derived subgroup of H. The cokernel of the map ? 2G(H) ! G(H) is the abelianiza* *tion, Hab. Lacking a more appropriate name for modding out by higher derived subgrou* *ps, we continue to use the same notation in that case. Lemma 9.5. If F takes values in discrete groups, then with G0and Gabas in Defin* *ition 9.3, ? Gab(X) ' 0. Proof.From the construction of G0, the map ffl : ? G ! G factors through the in* *clusion i : G0! G. Applying ? again results in the following diagram: ? 2G(X) jj_ffi__________________________________* *____ LLL ______________________________________* *_____________ | LLL __________________________________ fflff?fflL&&LLLl||____________________________ ? G0(X) __?i_//?G(X) The map ? ffl has a section, ffi, so it is surjective. Hence ? i is also surjec* *tive. Consider the short exact sequence of functors to discrete groups in Equation (9.3)defining G* *ab. If we show that ? preserves this short exact sequence, then the surjectivity of the m* *ap ? i will imply that ? Gab= 0. Short exact sequences of discrete groups are fiber sequences that are surject* *ive on the base space. The functor ? preserves fiber sequences because the construction in* *volves only taking fibers. The functor ? preserves surjections because all of the maps in* * the cube CR n+1F (X, . .,.X) defining ?n+1 F (X) have sections, and hence taking fibers * *with respect to them does not lower connectivity. 9.2. Pnalgpreserves AF fibration. This section establishes that Pnalgactually p* *roduces a fibration when applied to the fibration defining AF. The results in this sectio* *n also contain a statement about the map from F ! PnalgF , because in the case of F taking value* *s in discrete groups, the proof that this map is surjective on ß0 uses the same technical det* *ails that the proof that we get a fibration. To remind the reader that the functor takes values in discrete groups in the * *next propo- sition, we use the letter G (for group) to denote the functor, instead of the u* *sual F . Proposition 9.6. If G takes values in discrete groups (so in particular G satis* *fies Hypoth- esis 3.9), then the following is a quasifibration: *+1 alg (9.4) PnalgAG(X) ! Pnalg|? n+1G(X) |! Pn G(X). Proof.Replacing the base G in the definition of AG (Equation (9.2)) with G0from* * Defini- tion 9.3, we have the fibration sequence (9.5) AG(X) ! |? *+1n+1G(X)| ! G0(X), and this sequence is surjective on ß0. The hypotheses of Corollary 8.8 are sati* *sfied by the sequences in (9.3)and (9.5), so applying Pnalgboth are fibration sequences whos* *e maps to ALGEBRAIC GOODWILLIE CALCULUS 16 the base spaces are surjective on ß0: (9.6) PnalgG0(X) ! PnalgG(X) ! PnalgGab(X) (9.7) Pnalg(AG)(X) ! Pnalg(|? *+1n+1G(-)|)(X) ! PnalgG0(X). The aim now is to show that (9.7)remains a fibration when the base PnalgG0(X)* * is re- placed by PnalgG(X). From Lemma 9.5, ? n+1Gab(X) ' 0, so Proposition 8.1 shows* * that PnalgGab(X) ' Gab(X), which is a discrete space. Then, using the long exact se* *quence on homotopy, the fibration in (9.6)gives PnalgG0(X) '-!PnalgG(X) except possibly o* *n ß0, where the map is injective. This is enough to show that changing the base in (9.7)fro* *m PnalgG0(X) to PnalgG(X) still yields a fibration. That is, (9.4)is a fibration (but perhap* *s not surjective on ß0). Proposition 9.7. If G takes values in discrete groups (so in particular G satis* *fies Hypoth- esis 3.9), then ß0PnalgG(X) ~=cokerGps(ß0ffl), and the map ß0G ! ß0PnalgG is the universal map to the cokernel of ß0ffl in the* * category of groups. Proof.As in the preceding Proposition 9.6, we have the following fibration sequ* *ence that is surjective on ß0: Pnalg(AG)(X) ! Pnalg(|? *+1n+1G(-)|)(X) ! PnalgG0(X). Lemma 9.9 shows that the total space in this fibration is contractible, and the* * map to the base is surjective on ß0, so ß0PnalgG0(X) = 0. Also following Proposition 9.6, we have the following diagram in which the ho* *rizonal rows are fibrations that are surjective on ß0: G0(X) ________//G(X)_______//_Gab(X) | | | | | |' fflffl| fflffl| fflffl| PnalgG0(X)____//_PnalgG(X)__//_PnalgGab(X) Since ß0PnalgG0(X) = 0, the long exact sequence for the bottom fibration implie* *s that ß0PnalgG(X) ~= ß0PnalgGab. The right hand vertical map is an equivalence, agai* *n as noted in the preceding proposition, using Lemma 9.5 and Proposition 8.1. Combining t* *hese, we have ß0PnalgG(X)~=ß0PnalgGab(X) ~=ß0Gab(X), which is isomorphic to cokerGps(ß0ffl) because the map ffl : ?n+1 G(X) ! G(X) f* *actors through G0(X). ALGEBRAIC GOODWILLIE CALCULUS 17 Proposition 9.8. If F has connected values (Hypothesis 3.8), then the following* * is a quasi- fibration: *+1 alg PnalgAF(X) ! Pnalg|? n+1F (X)|! Pn F (X), and furthermore the map F (X) ! PnalgF (X) is (trivially) surjective on ß0. Proof.If F has connected values on coproducts of X, then AF(X) ! |? *+1n+1F (X)| ! F (X) is a fibration over a connected base. Therefore, by Corollary 8.8, applying Pn* *algyields a fibration, so Equation (9.4)is a fibration. The map 0 ! F is 0-connected, so Theorem 5.1 shows that 0 ' Pnalg(0) ! PnalgF* * (X) is 0-connected as well. Hence ß0PnalgF (X) = 0. 9.3. If m < n, then Pnalg|? *n(? nF|)' 0. This section establishes the relative* *ly easy fact that for ? n, the part of the Goodwillie tower below degree n is trivial. Lemma 9.9. Let R(X1, . .,.Xn) = |crn (?*nF )(X1, . .,.Xn)| be a functor of n va* *riables. Define the diagonal of such a functor to be the functor of one variable given b* *y (diagR)(X) = R(X, . .,.X). Then Pmalg(diagR)(X) ' 0 for 0 m < n. Proof.Goodwillie's Lemma 2.1 [7, Lemma 2.1] shows that if H(X1, . .,.Xn) is a f* *unctor of n variables that is contractible whenever some Xi is contractible (this is call* *ed a üm lti- reduced" functor), then Pm (diagH) ' 0 for 0 m < n. Writing out the definitio* *n of Pmalg, it is easy to check that L(diagR)X is multi-reduced, so Goodwillie's result app* *lies. 9.4. The functor AF has no n + 1 cross effect. Having created the functor AF to* * be "F with the cross effect killed", we now need to establish that ? AF ' 0. The main* * issue is the commuting of the ? and the realization. Lemma 9.10. Let F be a functor satisfying Hypothesis 3.8 or Hypothesis 3.9, let* * ? denote ? n for some n, and let AF be the functor given in Definition 9.2. Then AF is c* *ontractible. Proof.Taking cross effects is a homotopy inverse limit construction, and homoto* *py inverse limits commmute, so *+1 ? AF ' hofib? |? F| ! ? F . It is easy to check that if F satisfies Hypothesis 3.8 or Hypothesis 3.9, then * *so does ? F , and hence so does |? *(? F )|. An argument paralleling Lemma 4.9 shows that ? c* *ommutes with the realization in that functor, so ?|? *(? F )| ' |? ? *(? F )|. Finally,* * the existence of ffi : ? F ! ?2 F shows that ? F is the augmentation of the simplicial space ? ?* **(? F ), so the standard "extra degeneracyä rgument [11, Exercise 8.4.6, p. 275] shows that |?* * ? *(? F )| ' ? F , and hence that AF ' hofib(?F ! ? F ) ' 0, as desired. ALGEBRAIC GOODWILLIE CALCULUS 18 9.5. Proof Of Proposition 9.1. Proof.First, suppose that F (X) takes either connected values or discrete group* * values on coproducts of X. Consider the auxiliary diagram created by applying Pnalgto the* * fibration sequence defining AF(X): AF(X) _________//|? *+1n+1F_(X)|ffl_//F (X) | | | | | | fflffl| fflffl| fflffl| PnalgAF(X)____//_Pnalg|? *+1n+1F_(X)|//_PnalgF (X) Proposition 9.8 (in the case of connected values) or Proposition 9.6 (in the ca* *se of discrete group values) shows that the bottom row is a quasifibration. Proposition 9.8 (* *connected values) or Proposition 9.7 (discrete group values) imply that the map F (X) ! P* *nalgF (X) surjective on ß0. Lemma 9.10 shows that ? n+1AF(X) ' 0, so that Proposition 8.* *1 gives AF(X) ' PnalgAF(X), and hence the square on the right is homotopy Cartesian. Le* *mma 9.9 shows that Pnalg|? *+1n+1F (X)|' 0, so this square being Cartesian is equivalen* *t to (9.1) being a quasifibration, as we wanted to establish. We can reduce the general problem when F satisfies Hypothesis 3.9 to the case* *s of con- nected and discrete group values that we have already considered by examining t* *he fibration Fb(X) ! F (X) ! ß0F (X), where bF(X) is the component of the basepoint in F (X). This gives rise to the* * following square: |? *+1n+1bF(X)|___//_bF____//_PnalgbF | || | | | | fflffl| fflffl| fflffl| |? *+1n+1F (X)|___//_F_____//_PnalgF | || | | | | fflffl| fflffl| fflffl| |? *+1n+1ß0F (X)|_//_ß0F___//Pnalgß0F It is straightforward to check that every row and column except the middle row * *is a fibration that is surjective in ß0, and that the composition of the two maps in the middl* *e is null homotopic, and we have shown that the map F ! PnalgF is surjective on ß0. This * *gives us the data required to use the 3 x 3 lemma for fibrations to show that the middle* * row (i.e., (9.4)) is a fibration and surjective on ß0. The statement about ß0 is trivial in the connected case; ß0 of every space in* * the top row is zero (which is trivially a group). This implies that the vertical arrow* *s connecting the second and third rows are ß0-isomorphisms, so the statement about ß0 follow* *s from Proposition 9.7. Appendix A. ? is a cotriple This appendix is dedicated to verifying the technical result that ? is part o* *f a cotriple on the category of homotopy functors from pointed spaces to pointed spaces. To do* * this, we ALGEBRAIC GOODWILLIE CALCULUS 19 actually produce a model for the iterated cross effects, ?k, that obviously for* *ms a simplicial object, and then we proceed to verify the requisite identities in this context. Given a set A U, we write Ac to denote the complement of A in U. We define a "diagonal" to encode the information needed to construct a cube o* *f coproducts and inclusion and projection maps of the type used to define the cross effect. Definition A.1 (Diagonal). For any sets S and U, define the "diagonal" (S, U) * *to be the subsets of P(S x U) that are complements of a singleton in each component. That* * is, given a function f : S ! U, define the set Bf S x U by [ Bf = (s, f(s)c), s2S and then define the diagonal (S, U) to be [ (S, U) = {Bf} . f:S!U Since (S, U) is natually isomorphic to Hom (S, U) by sending Bf to f, it is * *obviously a functor in S and U. Definition A.2 (Free cube). Given sets U and S and a functor g from the discret* *e category (S, U) to a pointed category with coproducts (for example pointed spaces or cu* *bes of pointed spaces), we define Free(S, U, g) to be the (S x U)-cube X with vertices ` X (A) = g(B) {B2 (S,U):A B} Morphisms in X are induced by the maps g(B) ! g(B0) that are the identity if B * *= B0and the zero map otherwise. We now establish that "free cubesä re closed under the pullback operation. Lemma A.3. Let m and n be sets, let the (m x U)-cube X = Free(m, U, g) be a fre* *e cube, and let f : n ! m be a function. The (n x U)-cube P(f x 1)*X is isomorphic to a* * free cube Y = Free(n, U, h) with ` h(B) = g(B0). B02 (f,1)-1(B) Proof.This is a straightforward argument by expanding the definition of h in Y(* *A), com- bining and interchanging the order of quantifiers to turn two coproducts into o* *ne, and then verifying that the resulting indexing set is the same as the indexing set for P* *(f x 1)*X . We are now in a position to identify the relationship ? has to the free cube * *functor. For the rest of this section, we fix a functor F and a space X. Given sets U and S, let cX be the function on (S, U) that has a constant val* *ue X. Let C(S) be the contravariant functor of sets S given by C(S) = hofibF O Free(S, U, cX ). Lemma A.4. C(S) is a contravariant functor of S. ALGEBRAIC GOODWILLIE CALCULUS 20 Proof.Let Y = C(S). Given a function f : S ! T , we can construct the (S x U)-* *cube P(f x 1)*Y. The map in the indexing categories induces [3, xXI.9] a map on the * *homotopy fibers: hofib F Y ! hofib P(f x 1)*Y, P(TxU) P(SxU) so it remains to construct a map hofibF O P(f x 1)*Y ! hofib F O Free(S, U, cX ). P(SxU) P(SxU) Recall from Lemma A.3 that P(f x 1)*Y is a free cube with generating function ` h(B) = X. (f,1)-1(B) To specify the desired map between free cubes, it suffices toWspecify a natural* * transformation between their "generating functions"; the universal map from X to X that is t* *he identity on each X is a natural choice in this case. Lemma A.5. Let n = |U|. The C(k) is equivalent to ? knF (X). Proof.This is a straightforward verification that both are naturally equivalent* * to the homo- topy fiber of a k x U-cube whose vertices are _i=k ! ` ` F X . i=1vi62Vi Lemma A.6. Let n = |U|. Via the equivalence of Lemma A.5, the map C(1) ! C(;) * * W induced by C(i : ; ! 1) is equivalent to the map ? nF (X) ! F (X) induced by F * *( nX ! X). Proof.This is also straightforward. LetWX = Free(1, U, cX ) and observe that th* *e map P(i x 1)* induces the identity on X (;) = nX. Theorem A.7. The functor ? is a cotriple on the category of homotopy functors * *fromW pointed spaces to pointed spaces. The map ffl : ? ! 1 is induced by the öf ld m* *ap" X ! X. Proof.Lemma A.5 identifies ? kand C(k), and Lemma A.6 identifies the map ffl. T* *hen the requisite identities follow from the applying C to the following diagrams: {1,O2}oo__{1}_OOO <{1}