CALCULUS III TAYLOR SERIES Thomas G. Goodwillie Brown University Abstract. We study functors from spaces to spaces or spectra that preserve * *weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which * *may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n - 1* *)-excisive part, can be classified: they correspond to symmetric functors of n variables that are 1-excisive in* * each variable. We discuss some important examples, including the identity functor and Waldhausen's algebra* *ic K-theory. Introduction This paper should have been finished many years ago. It is a continuation of * *[1] and [2] (which were themselves a little late in coming). The author has no excuse for this procras* *tination and wishes to apologize to anyone who has been inconvenienced, especially his students and fo* *rmer students. As in [1] and [2] we are concerned with functors from C, which may be the cat* *egory of spaces or more generally spaces over a fixed space, to D, which may be either the category of * *spaces or the category of spectra. It is clear that one could extend the ideas to many other settings. Th* *e functors will always be homotopy functors in the sense that they preserve weak homotopy equivalences. W* *e are also concerned with (natural) maps between such functors. We call such a map of functors F -! * *G a weak equivalence (or for emphasis an objectwise weak equivalence, or for brevity an equivalence)* * if for every object X the map F(X) -! G(X) is a weak equivalence. The goal is to shed light on homotopy f* *unctors in general (and on particular ones) by systematically comparing them with homotopy functor* *s of some very special kinds, namely those which satisfy "nthorder excision". In taking this point of view we are led naturally to speak of the category F * *= F(C, D) of homotopy functors. Some of our conclusions refer to the homotopy category hF of homotopy* * functors, meaning the universal example of a category equipped with a functor from F that takes e* *very weak equivalence to an isomorphism. For example, some of our results are most neatly expressed * *as statements to the effect that such and such a functor from one category of functors to another in* *duces an equivalence of homotopy categories. Homotopy categories in this sense can be made by the "gene* *rators and relations" construction: a morphism in the homotopy category is an equivalence class of wo* *rds, where each letter in a word is either a forward arrow (a morphism in F) or a backward arrow (a fo* *rmal inverse to a weak equivalence in F) and two words are declared to represent the same morphism if * *they are related by a sequence of basic moves (composing two forward or two backward arrows, deleting* * an identity arrow, cancelling a forward arrow and its formal inverse). ___________ 1991 Mathematics Subject Classification. 55P99,55U99. Key words and phrases. homotopy functor, excision,Taylor tower. Typeset by AM* * S-TEX 1 2 THOMAS G. GOODWILLIE The author has recently been reminded that there are set-theoretic objections* * to bandying this kind of language about too freely, but he does not want to stop doing so. Some ways of * *resolving this difficulty will be discussed at the end of this introduction. Of course, if one wants to prove theorems about a homotopy category then one * *should be prepared to do most of the work in the original category. For example, commutative diagr* *ams in the category of functors will be a ubiquitous tool here, but commutative diagrams in the homoto* *py category are relatively useless. We now briefly recall the definition of nthorder excision, from section 3 of * *[2]. Whereas ordinary, or first order, excision has to do with the behavior of a functor on certain sq* *uare diagrams, nthorder excision has to do with the behavior of a functor on certain cubical diagrams o* *f dimension n + 1. We briefly recall the definitions now and refer to [1] and [2] for details. Basic * *facts about cubical diagrams are in section 1 of [2]. The homotopy functor F is excisive if it takes homotopy pushout squares (also* * called homotopy cocartesian squares) to homotopy pullback squares (also called homotopy cartesi* *an squares). It is reduced if it takes the final object to an object weakly equivalent to the final object* *. It is linear if it is both excisive and reduced. A cubical diagram is called strongly homotopy cocartesian if all of its two-d* *imensional faces are homo- topy pushouts. It is called homotopy cartesian if it is a homotopy pullback in * *the sense that the "first" object is equivalent (by the obvious map) to the homotopy limit of all the othe* *rs. In this paper we have generally omitted the word öh motopy" in the expressions "(strongly) homotopy (* *co)cartesian". F is said to be n-excisive, or to satisfy nthorder excision, if it takes strongly homotop* *y cocartesian (n + 1)-cubical diagrams to homotopy cartesian diagrams. An n-excisive functor is always (n + 1* *)-excisive. In [1] we studied the approximation of homotopy functors by excisive functors* * and codified this in the notion of (first) derivative of a functor. We calculated the derivative, first * *in some basic examples and then in the example which had given rise to the whole project: stable pseudoiso* *topy theory (and with it Waldhausen's algebraic K-theory functor A). In [2] we introduced the class of analytic functors. These are homotopy funct* *ors whose deviation from being n-excisive is bounded in a certain way for all n. Some functors are more * *analytic than others; an analytic functor is æ-analytic for some integer æ, where a smaller æ means stro* *nger bounds. We showed that æ-analytic functors behave rather rigidly on the category of æ-connected s* *paces (and more generally on the category of spaces equipped with (æ + 1)-connected maps to a fixed space* *), in the sense, roughly, that if a map between two such functors induces an equivalence of first derivat* *ives everywhere then its homotopy fiber is constant up to homotopy within the category of æ-connected sp* *aces. In [3] these results were used to relate A to some functors built out of the * *free loopspace. Here we move from excisive approximation to n-excisive approximation, obtaini* *ng functors PnF which can be thought of as Taylor polynomials of F. We show that if F is æ-analytic a* *nd X is æ-connected then as n tends to infinity the nthapproximation converges to F(X) in the strong sen* *se that the connectivity of a certain map F(X) -!(PnF)(X) tends to infinity. In other words, the number * *æ gives a sort of radius of convergence. What we were doing in [2] was like showing that a function f(x) is determined* *, in some region, by f(0) and f0(x). Continuing with the same analogy, what we are doing here is lik* *e showing that f(x) is determined by f(0), f0(0), f00(0), . ... Here is a sketch of the main results of this paper, presented in a rather dif* *ferent order from the one in which they will be proved. First consider homotopy functors T -F!Sp from based spaces to spectra. For an* *y such functor, and for any n 0, we can make an n-excisive functor T -PnF--!Sp with a map F -!PnF* * that is universal (in a homotopy category) among maps from F to n-excisive functors. The functors {Pn* *F}n 0 fit together to form a tower, and F maps into the limit of the tower. CALCULUS III TAYLOR SERIES * * 3 If the role of nthTaylor polynomial is being played by PnF, then the role of * *nthterm in the series is played by the (homotopy) fiber of the map PnF -! Pn-1F, wich will be denoted by* * DnF. We refer to PnF as the nthstage of the tower and to DnF as the nthlayer. These homogeneous * *polynomial functors are the focus of much of the paper. The constant term (P0F)(X) is the same, up to natural weak equivalence, as th* *e constant functor F(*). Up to the same kind of equivalence, the linear (or homogeneous of degree one)* * functor (D1F)(X) necessarily has the form C1^ X for some fixed spectrum C1, at least when restri* *cted to finite complexes X. The coefficient spectrum C1, which of course is (D1F)(S0), is called the de* *rivative of F at the one-point space. Likewise the homogeneous quadratic functor (D2F)(X) is necessarily given by (C2^ (X ^ X))h 2 where 2 is the symmetric group, C2 is some spectrum with 2-action, X ^X has t* *he obvious action, and the subscript h 2 denotes homotopy orbit spectrum. The coefficient spectrum C2,* * with its 2-action, is called the second derivative of F at the one-point space. The explanation of why homogeneous quadratic functors always have this form i* *nvolves symmetric bilinear functors. If H is any 2-excisive functor (still from based spaces to s* *pectra), then its second order cross-effect, defined as the total fiber (= homotopy fiber of homotopy fibers) * *of the square diagram H(X1_ X2) ----! H(X1) ?? ? y ?y H(X2) ----! H(*), will be a functor L(X1, X2) of two variables, linear in each variable, and symm* *etric with respect to interchanging the variables. It turns out that the homogeneous quadratic part o* *f H can be recovered as L(X, X)h 2. A bilinear functor necessarily has the form L(X1, X2) = C ^ (X1^ X2), and a symmetry on L means a 2-action on C. This pattern persists. For any n 1 an nthdegree homogeneous functor H must * *have the form H(X) = L(X, . .,.X)h n where L, the nthorder cross-effect of H, is a symmetric multilinear functor of * *n variables. Thus (DnF)(X) will have the form (Cn ^ (X ^ . .^.X))h n at least for finite X. The object Cn, a spectrum with an action of n, will be * *called the nthderivative of F at the one-point space. We also refer to it as the nthcoefficient of the m* *ultilinear functor, or of the homogeneous functor. It is worth emphasizing what brand of stable equivariant homotopy theory is a* *ppearing here. Let G be a finite group. To do serious homotopy theory in the category of G-spaces* * one usually takes the weak equivalences to be those equivariant maps which induce weak equivalenc* *es on spaces of H- fixed points for all subgroups H. (For CW objects this is equivalent to saying* * that the map has an inverse up to G-homotopy.) There is also a very weak notion of equivalence: equ* *ivariant maps which are nonequivariantly weak equivalences. Using these as the weak equivalences leads * *to a theory that is easier 4 THOMAS G. GOODWILLIE and less interesting, but which has its uses. For example, a very weak equivale* *nce of G-spaces always induces a weak equivalence of homotopy orbit spaces and also of homotopy fixed * *point spaces. Every G-space is equivalent in the very weak sense to a free G-space (product with EG* *), and for free G-spaces the two kinds of equivalence coincide. In the course of investigating homogeneous functors we encounter spectra with* * finite groups acting on them, and this leads us to an obvious stable analogue of this easier brand of e* *quivariant homotopy theory: Make a category whose objects are spectra equipped with a G-action and whose ma* *ps are maps of spectra respecting the action, and call such an equivariant map a (very) weak equivalen* *ce if it is nonequivariantly a weak equivalence of spectra. It is these objects (with G = n) up to this kin* *d of equivalence which correspond to homogeneous functors of degree n up to weak natural equivalence. In order to extend the main ideas of serious equivariant homotopy theory to t* *he stable setting, May and his collaborators created a beautiful and powerful theory of G-spectra [18]* *. We mention this only to say that we do not need it here. From that sophisticated point of view our s* *pectra with G-action are what are sometimes called naive G-(pre)spectra, namely G-spectra involving only* * trivial actions of G on suspension coordinates. The G-spectrum expert will know what else to say about * *the place of the naive theory in the sophisticated one, but neither the expert nor the novice should h* *ave to think about the sophisticated theory to read this paper (unless the expert just cannot help thi* *nking about it). Returning to the towers, a simple and familiar example is the "Snaith splitti* *ngö fQthe functor F(X) = 1 X. Its nthhomogeneous part is (DnF)(X) ~ 1 (X^n), and its tower splits: P* *nF ~ 1 k nDkF. The limit of the tower is the product of all the layers. If X is connected then* * the tower converges to F(X) in the sense that the map from F(X) to the homotopy limit of the tower is * *a weak equivalence, and in fact in the stronger sense that the map from F(X) to (PnF)(X) has a conn* *ectivity tending to infinity with n. A related example is F(X) = 1 X. Here we have (DnF)(X) ~ n 1 (X^n). (We will give a quick proof of this using the previous example and the general * *fact Pn(F O ) ~ (PnF)O . See Example 1.20 below.) The nthderivative is the wedge of n factorial copies * *of the spectrum S-n permuted transitively by n. The tower does not split. It converges (in the str* *ong sense again) when the space X is 1-connected. A more general example is F(X) = 1 Map*(K, X) where K is a based finite comp* *lex and Map* means the space of based maps. Here the nthderivative is the S-dual of a based * *complex with n-action, namely the quotient of the smash product K^n by the fat diagonal. F is dim(K)-a* *nalytic, and the tower converges to F(X) if X is dim(K)-connected. Arone [4] thoroughly explored this * *class of examples, giving a function-space description of the stages PnF and not just the layers DnF. The nthcoefficient of Waldhausen's A(X) is again the S-dual of a based finite* * complex with n-action, namely ( n)+ ^CnSn-1, where Cn n is the transitive cyclic subgroup of order * *n and the sphere Sn-1 is the one-point compactification of the reduced regular representation of Cn. So far we have been concerned with functors from based spaces to spectra. We* * now discuss three different variants of this setup: functors from unbased spaces, functors from s* *paces over a fixed space, and functors to spaces. In these new settings the Taylor tower construction goe* *s through with no change, but some additional work is needed to understand what a homogeneous functor loo* *ks like. The switch from functors T -! Sp of based spaces to functors U -! Sp of unbas* *ed spaces is fairly innocuous. If T -OE!U is the forgetful functor then (PnF) O OE = Pn(F O OE) and* * (DnF) O OE = Dn(F O OE). A snag appears when one tries to relate homogeneous functors to symmetric mul* *tilinear functors, since the definition of the cross-effect requires basepoints. The good news is * *that in the end this does not matter: homogeneous functors U -!Sp of any degree extend uniquely (in an approp* *riate up-to-natural- weak-equivalence sense) to T , and the same is true for symmetric multilinear f* *unctors in any number CALCULUS III TAYLOR SERIES * * 5 of variables. The proof involves the left adjoint _ of OE, in other words the f* *unctor that adds a disjoint basepoint to an unbased space. For example, although the usual definition of th* *e suspension spectrum of a space X requires X to be based, there is a well-known extension to unbased sp* *aces. It associates to X the homotopy fiber of 1 _(X) -! 1 _(*). If X is based then this is naturally (with respect to based maps) equivalent to* * 1 X, and on the other hand if U -L!Sp is any linear functor such that L O OE is (naturally equivalent* * to) 1 then L must be as defined above. This was explained in [1] and is generalized to higher degrees h* *ere. Note that for inhomogeneous functors this goes very wrong. For example, the 1* *-excisive functor 1 _(*) _ hofiber( 1 _(X) -! 1 _(*)). from U to Sp is genuinely different from 1 _(X), but the difference disappears* * after restriction to T . (The empty set is the only space X at which the functors disagree, but even for* * nonempty spaces the equivalence cannot be chosen to be natural with respect to unbased maps.) An ex* *ample of a 2-excisive functor T -! Sp which does not extend to U at all is the suspension spectrum of F(X) = ((X ^ X)= X ) 2, the orbit space for the 2-action on the quotient of X ^ X by the diagonal. If * *X is the disjoint union of a point and a circle then the rational homology of F(X) depends on where the ba* *sepoint is placed in X. The second switch, from functors of spaces to functors of spaces over a fixed* * space, is something like the switch from MacLaurin series to general Taylor series. Instead of building,* * for each space X, a tower that has F(*) at the bottom and attempts to converge to F(X), one can build, fo* *r each map of spaces X -!Y , a tower that has F(Y ) at the bottom and attempts to converge to F(X). * *If we fix Y and think of everything in sight as a functor of spaces over Y then the nthstage of the t* *ower is n-excisive. As in the case Y = *, there are two options, each with its own technical advantages: * *the category UY of plain spaces over Y and the category TY of spaces over Y equipped with a section. We * *sometimes refer to the latter as fiberwise based spaces over Y . The correspondence between homogeneous and symmetric multilinear functors wor* *ks the same for spaces over Y as it does for spaces, but the business of describing a multiline* *ar functor by coefficient spectra becomes more complicated when Y has more than one point. As a trivial e* *xample, to describe a linear functor of spaces over the two-point space {y1, y2} one needs two spec* *tra. In general a linear functor L of spaces over Y determines a spectrum for each point y 2 Y , namely * *the coefficient L(Y _yS0) of the linear functor Z 7! L(Y _y Z) from T to Sp, where Y _y Z means the wedge* * sum of (Y, y) with the based space Z, viewed as a space over Y . This spectrum depends öc ntinuous* *lyö n the point y in some sense. If Y is path-connected then up to weak homotopy equivalence the spe* *ctrum is independent of the point; but there is a twisting that must not be overlooked. In [1] we defined the differential DYF of a functor U -F!Sp at a space Y to b* *e a linear functor from spaces over Y to spectra, the linear approximation to the functor (X -!Y ) 7! hofiber(F(X) -!F(Y ) We also defined the derivative to be a spectrum @yF(Y ) depending on a space Y * *and a point y 2 Y , namely (DYF)(Y _yS0); it might be called the y coefficient of DYF or the partia* *l derivative of F at Y in the y direction. The relationship between the differential DYF (a linear functo* *r of spaces over Y ) and the derivative @yF(Y ) (a spectrum) is something like the relationship between the * *differential of a function (a linear function on the tangent space) and a partial or directional derivativ* *e (a number, which specifies 6 THOMAS G. GOODWILLIE the behavior of that linear function on a certain one-dimensional tangent subsp* *ace). The spectrum @yF(Y ) records the derivative of F at Y in the "direction" given by y. Here we* * make the multilinear generalization, defining the nthdifferential to be the symmetric multilinear fu* *nctor corresponding to the nthlayer of the Taylor tower and defining the nthderivative @(n)y1,...,ynF(Y ) to be its value at (Y _y1S0, . .,.Y _ynS0). The third switch, from spectrum-valued to space-valued functors, brings a rea* *l surprise. The con- struction of the Taylor series goes through with no change. If the functor F ha* *ppens to be 1 G for some spectrum-valued functor G, then we have PnF = 1 PnG and DnF = 1 DnG. The* * surprise is that although most functors are not of the form 1 G this is not reflected at t* *he homogeneous level: every homogeneous functor T -! T of degree n 1 is infinitely deloopable, in f* *act in a unique and functorial way. Thus DnF always has the form 1 ((Cn ^ X^n)h n) and even in the* * space-valued case we can introduce a spectrum called the nthderivative of the functor, or the coe* *fficient spectrum of the nthhomogeneous layer. A central example of a functor T -! T that is interesting from this point of * *view is the identity. Let us call it I. Its first derivative is the sphere spectrum. It is easy to see, i* *n any of a number of ways, that the nthderivative is equivalent to the wedge sum of (n - 1)! copies of the (1 -* * n)-sphere spectrum, with n acting in such a way that the subgroup n-1 freely permutes the summands. To* * identify DnI one has to know the action of the full group n. Johnson [5] gave an explicit finit* *e complex with n-action whose S-dual is the answer. Arone and Mahowald [6] gave a different answer of t* *hat kind, showed that it was equivalent to Johnson's, and used it to make some very interesting calcu* *lations. A basic example of an inhomogeneous functor that does not deloop is P2I, the * *2-excisive approximation of the identity functor. The functor P1I is of course Q = 1 1 . (P2I)(X) is a* *nother standard object, the homotopy fiber of the James-Hopf map QX -!Q((X ^ X)h 2). (which is not induced by a spectrum map from 1 X to 1 (X ^ X)h 2). One notable feature of this last example turns out to be quite general: the f* *ibration sequence DnF -!PnF -!Pn-1F can always be extended one step to the right, as long as the constant functor P* *0F is contractible. This falls out of the proof of (and is essentially equivalent to) the deloopability * *of homogeneous functors. The construction of the Taylor tower also goes through for functors taking va* *lues in unbased instead of based spaces, but in that case it is nonsense to speak of the layers DnF as * *functors, since that would involve using "the" fiber of a map of unbased spaces. This small fussy point is* * really not so small. The identity functor I above was from based spaces to based spaces. Let J be the i* *dentity functor from unbased spaces to unbased spaces. Then (P1J)(X) is the homotopy fiber of Q(X+) -!QS0, but not with respect to the usual basepoint in QS0. In particular (P1J)(;) is e* *mpty, so that there is no natural basepoint in (P1J)(X). This seriously interferes with defining D2J. P1J* * is excisive and reduced, but perhaps too badly twisted to be rightly called linear. For such reasons we * *hesitate to even speak of homogeneous functors to unbased spaces. The paper is organized as follows: CALCULUS III TAYLOR SERIES * * 7 x1 defines the tower {PnF} in the general case, proves that PnF is n-excisive* *, establishes the universal mapping property of PnF, and notes the convergence of the tower in the case of * *an analytic functor F. x2 shows that homogeneous space-valued functors can be functorially delooped * *and concludes that they correspond precisely to homogeneous spectrum-valued functors. x3 establishes the correspondence between homogeneous functors and symmetric * *multilinear functors in the case of functors from fiberwise based spaces over Y to spectra. By x2 t* *his result extends to space-valued functors. x4 shows that for homogeneous functors, and also for symmetric multilinear fu* *nctors, it does not matter whether the domain category is (fiberwise) based or unbased spaces, so that som* *e of the results of x3 extend to functors of plain spaces over Y . x5 establishes notation for the symmetric multilinear functors that (accordin* *g to x2-x4) encode the homogeneous layers of a Taylor tower, and develops the idea of coefficient spec* *tra for multilinear functors. x6 establishes a useful tool for working out examples. The key point is that * *the nthcross effect of the homogeneous functor DnF can be made by üm ltilinearizing" the nthcross effect o* *f F itself. x7 works out the nthderivatives of functors like 1 Map*(K, -). x8 recalls and discusses known results on the Taylor tower of the identity. x9 indicates how to get the nthderivative of Waldhausen's A, taking the first* * derivative as starting point. A few words about set theory are in order. We all know that we must not speak* * of the set of all sets or the set of all topological spaces; set theory, as formulated to avoid Russel* *l-type paradoxes, does not allow it. And since the category C of spaces is not small, there may be objecti* *ons to speaking of the category of all functors, or homotopy functors, from C to D in that case. Even * *if C and D are both small, the functor category will not be small in general, so that it is illegal to mak* *e a new category by using generators and relations to invert some morphisms. We choose to dodge this as follows. As usual when basing mathematics on set t* *heory, we begin by fixing a universe U of sets. Now by topological spaces we mean those whose poin* *t sets are sets in this strict sense. The category of such spaces is not a small category, any more tha* *n the set of all (U-)sets is a (U-)set. Nevertheless, if we are willing to work in two universes [8,I.6], we* * are not in such bad shape. Introduce a larger universe U0 in which the set of all U-sets is a set. The cat* *egory of all (U-)spaces is then U0-small. In U0there can be no objection to speaking of the category of al* *l functors from spaces to spaces, or of the full subcategory of homotopy functors. To invert the weak equ* *ivalences by generations and relations, one can always pass to a third universe in which the category of* * homotopy functors is small. This solution will not suit all tastes. It may be a bit wasteful and it may b* *e a bit crude. For some more refined purposes it will be inadequate. For example, if one wants to intro* *duce some sort of space of maps between two homotopy functors such that the set of components will be t* *he set of morphisms in the homotopy category, then it will certainly be a drawback to find that this ü* *f nction space" is not a space in the original sense. In general, if one wants to work very seriously wi* *th the homotopy category of functors then one will probably want to introduce a closed model structure on t* *he functor category, with all the benefits that that brings. (In fact, although we have not attempted to * *do so, there are certainly many reasons for reworking this whole theory in the context of closed model cat* *egories. The objects of study should be functors from one (small?) model category to another, subject t* *o some mild axioms, and the category of homotopy functors should turn out to be a model category, too.) On the other hand, in this paper the role of the homotopy categories is a mod* *est one: they are used here mainly as shorthand, to make some sentences briefer and more readily compr* *ehensible than they otherwise would be. The üm ltiple universes" device allows us to use this short* *hand without telling any lies, but without making the catogory of spaces artificially small. 8 THOMAS G. GOODWILLIE x1. The Taylor tower Let C F-!D be a homotopy functor, where C = CY is either UY or TY and D is ei* *ther U, T , or Sp. Let n 0. The n-excisive approximation PnF will be constructed by the infinite iteratio* *n of another construction Tn which is designed to bring the functor F a little closer to being n-excisive* *. The special case when n = 1 was treated in [1, Def. 1.10], where T1 and P1 were called T and P. Thus * *if Y = * and F(*) = * then T1F ~ F and PnF ~ hocolimi 0iF i. The definition of Tn uses the fiberwise join over Y , as introduced in the pr* *oof of [2, 5.1]. Let X be a space over Y and let U be a space. (In most applications U will be a discrete f* *inite set.) The fiberwise join is the space X *Y U = hocolim(X X x U ! Y x U) considered as a space over Y . The name signifies that the functor öj in with U* *" is being applied to all the fibers of X -!Y . If U has one element, then X *Y U -! Y is the fiberwise cone of X over Y (the* * mapping cylinder of X -! Y considered as a space over Y ). If U has two elements then X *Y U -! * *Y is the fiberwise suspension and will sometimes be denoted by YX. It should be noted that this construction has the best of both worlds, in tha* *t on the one hand each fiber of X *Y U -!Y is homeomorphic to the join with U of the corresponding fib* *er of X -!Y , and on the other hand each homotopy fiber is weakly homotopy equivalent to the join with U* * of the corresponding homotopy fiber of X -!Y . Recall that even if C is TY rather than UY then X *Y U is still in C; it inhe* *rits a section from X. In other words, if X is a fiberwise based space over Y then X *Y U is also cano* *nically fiberwise based, without for example choosing a point in U. The object X*Y U depends (bi-)functorially on X and U. Note also that there i* *s a natural isomorphism of spaces over Y : (X *Y U) *Y V ~=X *Y (U * V ) where U * V is the ordinary join of two spaces. Let P(n_+_1_) be the poset of subsets of n_+_1_= {1, . .,.n + 1}. Any object * *X 2 C yields an (n + 1)- dimensional cubical diagram in C. P(n_+_1_) ! C U 7! X *Y U Consider the composed functor U 7! F(X *Y U), a cubical diagram in D. The holim* *of its restriction to P0(n_+_1_), the poset of nonempty subsets of n_+_1_, will be called (TnF)(X)* *. Clearly this yields a homotopy functor C TnF---!D. There is a natural map F -tnF-!TnF, since any cubi* *cal diagram determines a map from the "initialö bject to the homotopy limit of the others (see [2, De* *f 1.2]): F(X) = F(X *Y ;) tnF--!holim F(X *Y U) = (TnF)(X) U2P0(n+1_) Let (PnF)(X) be the sequential homotopy colimit of the diagram 2F* *)(X) F(X) (tnF)(X)------!(TnF)(X) (tnTnF)(X)--------!(T2nF)(X) (tnTn-* *-------!... . Clearly PnF is a homotopy functor and we have a natural map F -pnF-!PnF. The cubical diagram U 7! X *Y U is strongly cocartesian for all X. Therefore* * if the functor F is n-excisive then the maps tnF and pnF will be weak equivalences for all X . In * *this sense n-excisive functors are unchanged by Tn and Pn. CALCULUS III TAYLOR SERIES * * 9 We will see below that PnF is always n-excisive, and that it is the best n-ex* *cisive approximation to F in a (homotopy-) categorical sense. 1.1 Remark. If the map X -!Y is m-connected, then for nonempty U the map X *Y U* * -!Y is (m + 1)- connected, and therefore (TinF)(X) depends only on the behavior of F on objects* * whose structural maps to Y are (m + i)-connected. In this sense the Taylor approximations of a functo* *r CY -F!D depend only on the restriction of the functor to objects ä rbitrarily close to Y " (just as* * the Taylor expansion of f(x) in powers of x - y depends only on the behavior of f(x) in an arbitrarily small* * neighborhood of y). A related comment is that Pn commutes with fiberwise suspension: Pn(F O Y) = (PnF) O Y. This follows from the natural isomorphism Y(X *Y U) = ( YX) *Y U. Before giving the general proof that PnF is n-excisive, we give a different p* *roof under the added assumption that F is stably n-excisive ([2, 4.1]), and in this case we show tha* *t PnF approximates F not only in a categorical sense but also from the point of view of connectivity. The first task is to show that if F is stably n-excisive then TnF is more nea* *rly n-excisive than F is, and that TnF agrees with F to nthorder in the following sense: 1.2 Definition. A map u : F ! G between two functors from CY to D is said to sa* *tisfy On(c, ~) if, for every k ~, for every object X of CY such that the map X ! Y to the final obje* *ct is k-connected, the map uX : F(X) ! G(X) is (-c + (n + 1)k)-connected. We say that F and G agree to* * order n (via the map u) if this holds for some constants c and ~. 1.3 Remark. This is analogous to saying of two functions f and g (say real func* *tions of one or several variables) that they agree to nthorder at y if there are constants C and K such* * that |f(x) - g(x)| C|x - y|n+1 for all x such that |x - y| K. (The letter O stands for ö sculati* *on".) The condition En(c, ~), stable nthorder excision, was defined in [2,Def. 4.1]. 1.4 Proposition. If F satisfies En(c, ~), then (1)TnF satisfies En(c - 1, ~ - 1) and (2)tnF : F ! TnF satisfies On(c, ~). Proof. The second conclusion is immediate from the definitions. For the first,* * note that the functor X 7! X *Y U from C to C preserves strongly cocartesian cubes and (unless U is e* *mpty) increases the connectivity of maps. It follows immediately that for each nonempty U the funct* *or X 7! F(X *Y U) from C to D satisfies En(c - (n + 1), ~ - 1). By [2, 1.20], TnF satisfies En(c * *- 1, ~ - 1). 1.5 Proposition. If F is stably n-excisive, then (1)PnF is n-excisive and (2)F agrees with PnF to order n (via pnF). Proof. Suppose that F satisfies En(c, ~). By 1.4(1) and induction, TinF satisfi* *es En(c - i, ~ - i). This easily implies that PnF is n-excisive. We also find, by 1.4(2), that tnTinF sat* *isfies On(c - i, ~ - i), and in particular On(c, ~), for all i. It follows easily that On(c, ~) is also sati* *sfied by the composed maps (tnTinF) O . .O.tnF and in the limit by pnF. 10 THOMAS G. GOODWILLIE 1.6 Proposition. Let F -u!G be a map between homotopy functors. If F and G agre* *e to nthorder via u, then the induced map PnF -Pnu-!PnG is an equivalence. The converse holds if * *F and G are stably n-excisive. Proof. Suppose u satisfies On(c, ~). For each nonempty finite set U the resulti* *ng map of functors F(- *Y U) ! G(- *Y U) satisfies On(c - (n + 1), ~ - 1); again we have used the fact that the functor * *X 7! X *Y U from C to C preserves strongly cocartesian cubical diagrams and increases the connectivity * *of maps. Therefore, using * * iu [2,1.20] as in the proof of 1.4, TnF -Tnu-!TnG satisfies On(c - 1, ~ - 1). By i* *nduction, TinF -Tn-!TinG satisfies On(c - i, ~ - i). It follows, letting i tend to infinity, that PnF -!* *PnG is an equivalence. For the converse, use 1.5(2) and the commutative diagram F --u--! G ? ? pnF?y ?ypnG PnF -Pnu---!PnG Note that when F is stably n-excisive PnF can be characterized, up to natural* * equivalence, as the only n-excisive functor that agrees to nthorder with F. The remaining results of x1 (except for the last sentence of 1.13) have nothi* *ng to do with connectivity. In particular, functors are not assumed to satisfy any kind of stable excision * *hypothesis. 1.7 Proposition. Up to equivalence, (1)Tn commutes with holim (2)Pn commutes with finite holim (3)Tn and Pn commute with hofiber. (4)Tn and Pn commute with filtered (in particuloar sequential) hocolim. (5)for spectrum-valued functors, both Tn and Pn commute with any hocolim. Proof and explanation. The main point is that holimcommutes with holim, hocolim* *commutes with hocolim, and, up to equivalence, finite holimcommutes with filtered hocolim. (1) This one is true up to isomorphism: If {Fff} is any sort of diagram of ho* *motopy functors C Fff--!D and F is given by F(X) = holimffFff(X) then we have (TnF)(X)= holimUholimffFff(X *Y U) = holimffholimUFff(X *Y U) = holimff(TnFff)(X) (2) In the same situation we have a natural map (PnF)(X) = hocolimi(TinF)(X) = hocolimiholimff(TinFff)(X) ! holimffhocolimi(TinFff)(X) = holimff(PnFff(X) CALCULUS III TAYLOR SERIES * * 11 It is an equivalence in the case when {Fff} is a finite diagram. (3) This follows from (1) and (2). We have Pnhofiber(F -!G)= Pnholim(F -!G - *) ~ holim(PnF -!PnG - Pn*) ~ holim(PnF -!PnG - *) = hofiber(PnF -!PnG) and likewise for Tn. (4) Tn is a finite holimand Pn is a hocolimof finite holim's. (5) Since hocolim's commute, hocolimpreserves cocartesian cubes. Thus hocolim* *of spectra preserves cartesian cubes. This implies (using [2, 1.19]) that Tn commutes with hocolimup* * to equivalence. There- fore the same holds for Pn = hocolimiTi. Let F(C, D) be the category whose objects are the homotopy functors from C to* * D, and whose mor- phisms are the natural maps. Let hF(C, D) be its homotopy category; it has the * *same objects and is obtained by formally inverting the equivalences. Because the functors Tn and Pn* * from F(C, D) to it- self take equivalences to equivalences, they give rise to functors from the hom* *otopy category to itself. Morphisms in hF(C, D) are called weak maps. Any functor weakly isomorphic to an* * n-excisive functor is n-excisive. 1.8 Theorem. For any homotopy functor C -F!D, the functor PnF is n-excisive. I* *n the homotopy category hF(C, D), pnF is the universal map from F to an n-excisive functor. The key to 1.8 is the following: 1.9 Lemma. Let X be any strongly cocartesian (n_+_1_)-cube in C, and let F be a* *ny homotopy functor. Then the map of cubes F(X ) (tnF)(X)------!(TnF)(X ) factors through some cartesian cube. Proof of 1.8, assuming 1.9. Let X be any strongly cocartesian (n_+_1_)-cube in * *C, and consider the diagram of cubes F(X ) ! (TnF)(X ) ! (T2nF)(X ) ! . . . which leads by hocolimto the cube (PnF)(X ). By 1.9 each of the maps of cubes d* *isplayed above factors through some cartesian cube. Therefore the cube (PnF)(X ) is equivalent to a s* *equential hocolimof cartesian cubes, and so it is itself cartesian. This shows that PnF is n-excisi* *ve. For the existence half of the universal mapping property, let F -u!P be any w* *eak map to an n-excisive functor. The diagram F --u--! P ? ? pnF?y pnP?y PnF -Pnu---!PnP shows that u factors through pnF, since pnP as a weak map is invertible. For the uniqueness we must show that if P is n-excisive then a weak map PnF -* *v!P is determined by the composition v O pnF. It suffices if in the diagram of weak maps F --pnF--!PnF --v--! P ? ? ? pnF?y pnPnF?y~ pnP?y~ PnF Pn(pnF)-----!~PnPnFPnv----!PnP 12 THOMAS G. GOODWILLIE the marked (~) arrows are invertible, for then v is determined by Pnv, which is* * determined by Pnv O Pn(pnF) = Pn(v O pnF), which is of course determined by v O pnF. The marked vertical arrows are equivalences because PnF and P are n-excisive.* * In order for Pn(pnF) to be an equivalence, it will be enough (by 1.7(4)) if Pn(tnF) is an equivalenc* *e. Define the functor JUF by (JUF)(X) = F(X *Y U). In the diagram PnF -Pn(tnF)----!PnTnF = Pn( holim JUF) U2P0(n+1_) ~-! holim P J F = holim J P F U2P0(n+1_)nU U2P0(n+1_)Un the second arrow is an equivalence by 1.7(2). The composition is also an equiva* *lence; this simply means that for every X the functor PnF takes the cube {X *Y U} to a cartesian cube, a* *nd it is true because PnF is n-excisive. It follows that Pn(tnF) is an equivalence. The fact that Pn(pnF) = pnPnF in the homotopy category follows from 1.8; if w* *e had had it before 1.8, then we could have skipped the last part of the proof of 1.8. 1.10 Remark. Theorem 1.8 is one of several statements in this paper for which w* *e have two different proofs: an older one which requires some kind of stable excision hypothesis, an* *d a newer one which has nothing to do with connectivity. Another such statement is Theorem 2.1, which a* *lso depends (near the end of the proof of 2.2) on 1.9. Another is 3.1, and another is 6.1. The older * *proofs have the advantage of a certain common-sense quality, and if what we really care about here is con* *vergent Taylor towers then the older proofs are good enough; but the fact that the theorems are still* * true without connectivity hypotheses is striking and the newer proofs are perhaps worth looking at, too. Proof of 1.9. (Any reader who, like the author, finds this proof a little opaqu* *e, may wish to look at it again after reading the proof of 3.2, which is related but simpler.) Let us wri* *te n instead of n + 1. For subsets T, U1, . .,.Un of n_, define ^X(T, U1, . .,.Un) to be a a hocolim(X (T) - (X (T) x Us) -! (X (T [ {s}).x Us)) 1 s n 1 s n This can also be described as the union, along X (T), of the spaces X (T) *X(T[* *{s})Us. Clearly ^Xis a functor from P(n_) x P(n_)n to C. We have X (T) = ^X(T, ;, . .* *,.;), and there is a natural map ^X(T, U, . .,.U) -!X (T) *Y U that equals the identity map when U = ;. Let E be the set of all (U1, . .,.Un) 2 P0(n_)nsuch that, for at least one s * *2 n_, s 2 Us. Since E contains the image of the diagonal map P0(n_) -!P0(n_)n, we can factor the map tn-1F(X (* *T)) as follows: F(X (T))! holim F(X^(T, U1, . .,.Un)) (U1,...,Un)2E -! holimF(X^(T, U, . .,.U)) U2P0(n_) -! holimF((X (T) *Y U) U2P0(n_) We claim that the cube T 7! holim F(X^(T, U1, . .,.Un)) (U1,...,Un)2E is cartesian (for any homotopy functor F). Let E* be the set of all (U1, . .,.Un) 2 P0(n_)nsuch that, for some s, Us= {s* *}. This poset is left cofinal in E (exercise; see [2, page 298] for the definition), and it follows that the * *restriction map from the holim CALCULUS III TAYLOR SERIES * * 13 over E to the holimover E* is an equivalence. Therefore it is enough if, whenev* *er (U1, . .,.Un) 2 E*, the cube T 7! F(X^(T, U1, . .,.Un)) is cartesian. In fact, it is cartesian for a very basic reason: if Us0= {s0} th* *en the map X^(T, U1, . .,.Un) -!X^(T [ {s0}, U1, . .,.Un) is an equivalence. To see this, examine the diagram ` ` X (T) ---- s(X (T) x Us) ----! s(X (T [ {s}) x Us)) ?? ? ? y ?y ?y ` ` X (T [ {s0})---- s(X (T [ {s0}) x Us)----!s(X (T [ {s, s0}) x Us)) The fact that the induced map from hocolimof top row to hocolimof bottom row is* * an equivalence follows from the trivial fact that for s 6= s0 the square X (T) x Us ----! X (T [ {s}) x Us ?? ? y ?y X (T [ {s0}) x Us----!X (T [ {s, s0}) x Us is cocartesian, plus the even more trivial fact that the square X (T) ---- X (T) x Us0 ?? ? y ?y X (T [ {s0})----X (T [ {s0}) x Us0 is cocartesian. 1.11 Corollary. If 0 m n then the map Pm F -Pm(pnF)-----!Pm PnF is an equivalence. Proof. This is formal; using the universal mapping properties of Pm and Pn and * *the fact that m-excisive functors are n-excisive, one sees that Pm PnF has the universal mapping propert* *y that characterizes Pm F. We now collect the äT ylor polynomials" PnF into a äT ylor series" by showing* * that there is a natural map PnF -qnF-!Pn-1F satisfying qnF O pnF = pn-1F. An effortless way to produce qnF as a weak map would be to reason as in the p* *roof of 1.11: pnF is the universal map from F to an n-excisive functor and (n - 1)-excisive functors are* * n-excisive, so there is a unique map qnF in hF(C, D) such that qnF O pnF = pn-1F. On the other hand, it i* *s desirable to define qnF in such a way that qnF O pnF = pn-1F on the nose and not just weakly. Moreo* *ver, the explicit construction for qnF will be useful in its own right in proving Lemma 2.2. 14 THOMAS G. GOODWILLIE We will make a (commutative) diagram 2F F --tnF--!TnF -tnTnF---!T2nF -tnTn---!. . . ? ? ? (1.12) ?y qn,1?y qn,2?y tn-1T2n-1 F -tn-1F---!Tn-1Ftn-1Tn-1F-------!T2n-1F------!. . . and then define qnF as the induced map of horizontal homotopy colimits. We must* * define qn,iand then verify that the squares commute. Notice that TinF is naturally isomorphic to a holim indexed by a product of i* * copies of the partially ordered set P0(n_+_1_): (TinF)(X) = holim F(X *Y (U1* ... * Ui)) (U1,...,Ui)2P0(n+1_)i From this point of view there is an obvious map TinF -qn,i-!Tin-1F, induced b* *y the inclusion of P0(n_)i in P0(n_+_1_)i. The first square in (1.12) now obviously commutes. The (i + 1)s* *tsquare will commute if both squares commute in iF TinF -tnTn---!Ti+1nF _______Ti+1nF ? ? ? qn,iF?y Tnqn,iF?y qn,i+1F?y Tin-1F------! TnTin-1F-------! Ti+1n-1F tnTin-1F qn,1Tin-1F The left square commutes because tnF is natural in F. The other commutes becaus* *e it is induced by a commutative diagram of posets P0(n_+_1_)i+1_______P0(n_+_1_)i+1 x? x ? ?? P0(n_+_1_) x P0(n_)i----P0(n_)i+1 Summing up, we have: 1.13 Theorem. A homotopy functor F from spaces over Y to either spaces, based s* *paces, or spectra, CALCULUS III TAYLOR SERIES * * 15 determines a tower of such functors .. . ?? y PnF ?? yqnF Pn-1F ?? yqn-1F .. . ?? yq2F P1F ?? yq1F P0F and a map F -{pnF}---!limnPnF. Each Pn is a homotopy functor (from homotopy fun* *ctors to homotopy functors), and pn and qn are natural. The functor PnF is always n-excisive, an* *d (in the homotopy category of homotopy functors) pnF is universal among maps from F to n-excisive* * functors. If F is æ-analytic and the structural map X ! Y is (at least) (æ + 1)-connected, then t* *he connectivity of the map F(X) -pnF-!(PnF)(X) tends to +1 with n, so that F(X) is equivalent to the h* *omotopy limit (P1 F)(X ! Y ) of the tower. Proof. The last statement is the only new one. Recall from [2, 4.2] that æ-anal* *yticity means that there is a number q such that, for all n 0, F satisfies En(næ - q, æ + 1). If X 2 C is* * such that the map X ! Y is k-connected with k > æ then by the proof of 1.5 the connectivity of the map * *F(X) pnF--!(PnF)(X) is at least (q + k + n(k - æ)), which tends to +1 with n. If PnF is analogous to an nthTaylor polynomial, then the homotopy fiber DnF = hofiber(PnF -qnF-!Pn-1F) is analogous to the nthterm in a Taylor series. Notice that the definition of D* *nF is meaningful if the category D is either based spaces or spectra, but not if it is unbased spaces. * *From Pn(F O Y) = (PnF)O Y (Remark 1.1) we have (1.14) Dn(F O Y) = (DnF) O Y. 1.15 Definition. A homotopy functor F : C ! D is called n-reduced if Pn-1F ~ *.* * It is called n-homogeneous, or homogeneous of degree n, if it is both n-excisive and n-reduc* *ed. Thus 1-reduced means reduced, and 1-homogeneous means linear. 1.16 Remark. If n > 1 then it is not easy in general to test whether a functor * *F is n-reduced. Perhaps the main difficulty is that Pn-1F ~ * does not imply Tn-1F ~ *. On the other ha* *nd, by 1.6 a sufficient condition for F to be n-reduced is that F agrees to order n - 1 with the consta* *nt functor *, in other 16 THOMAS G. GOODWILLIE words that the connectivity of F(X) tends to infinity at least n times faster t* *han the connectivity of the map X -!Y . (If F is analytic, or even just stably n-excisive, then this condit* *ion is also necessary for F to be n-reduced.) Thus, for example, for any spectrum C that is bounded below (* *k-connected for some k) the n-excisive functors X 7! C ^ X^n from based spaces to spectra and X 7! * *1 (C ^ X^n) from based spaces to based spaces are homogeneous. In fact this holds for all spectr* *a C, either by expressing C as a homotopy colimit of bounded-below spectra or by 3.1 below. 1.17 Proposition. DnF is always n-homogeneous. Proof. The functor DnF = holim(PnF -! Pn-1F - *) is n-excisive because it is a* * holimof n-excisive functors. To see that it is also n-reduced, use 1.7(4) to identify Pn-1DnF with* * the homotopy fiber of the map Pn-1(PnF) Pn-1(qnF)-------!Pn-1(Pn-1F); by (1.11) this map is an equivalence. 1.18 Proposition. Up to equivalence, (1)Dn commutes with finite holim. (2)Dn commutes with hofiber. (3)Dn commutes with filtered hocolim. (4)for spectrum-valued functors, Dn commutes with arbitrary hocolim. Proof. This follows easily from 1.7. 1.19 Example. Let the functors Fa and Fb be a-homogeneous and b-homogeneous res* *pectively, with a < b, and let Fa f-!Fbbe any natural map. A simple example is the diagonal inc* *lusion QX ! Q(X ^X). Then the functor F(X) = hofiber(Fa f-!Fb) is b-excisive. Its tower has only two nontrivial layers DaF ~ Fa and DbF ~ Fb;* * we have PnF ~ * n < a PnF ~ Fa a n < b PnF ~ F n b All of this follows from PnF ~ hofiber(PnFa -!PnFb). Note that when a > b there* * can be no interesting natural map Fa ! Fb, since by (1.8) any such map factors (in hF(C, D)) through * *Pa-1Fa ~ *. 1.20 Example. There is a weak equivalence ("Snaith splitting") Y (1.21) 1 X ~ 1 X^n n 1 for based connected spaces X. The functorQ 1 X^n of X is n-homogeneous. The mth* *Taylor polynomial of the right-hand side of 1.21 is 1 n m 1 (X^n), by 1.6, since this finite pr* *oduct agrees with the infinite product to order m. The same therefore holds for the left-hand side, b* *y 1.1. In particular the nth homogeneous layer of 1 X is 1 X^n. We will find later that this is enough t* *o determine the nth homogeneous layer of F(X) = 1 X. In fact, using 1.14 we have (DnF)( X) ~ (Dn(F O ))(X) ~ 1 X^n ~ n 1 ( X)^n, CALCULUS III TAYLOR SERIES * * 17 and by 3.8 this will imply (DnF)(X) ~ n 1 X^n. Incidentally, the naturality of Taylor towers gives a quick way to get from t* *he James model of X to that splitting of 1 X. The space JX, free monoid on the based space X, is* * naturally filtered by word length as an increasing union * = J0X J1X . .i.n such a way that the s* *ubquotient Jn=Jn-1 is X^n. It follows that for each n there is a natural fibration sequence 1 Jn-1X ! 1 JnX ! 1 X^n One sees by induction that 1 Jn is n-excisive, since it fibers over a (homogen* *eous) n-excisive functor and the fiber is ((n - 1)-excisive, hence) n-excisive. The sequence above must * *split. Indeed, any natural fibration sequence F(X) ! G(X) ! H(X) of spectra in which H is n-homogeneous and F is (n - 1)-excisive must split: a * *retraction from G to F (in the homotopy category of homotopy functors) is given by the diagram F ----! G ? ? pn-1F?y~ ?ypn-1G Pn-1F ----!~Pn-1G The left arrow is an equivalence because F is (n - 1)-excisive; the lower arrow* * because Pn-1H is con- tractible. 1.21 Remark. The Taylor tower construction extends easily to functors CY1x . .x.CYn F-!D of several variables. Let us say that F is (d1, . .,.dn)-excisive [resp. (d1, .* * .,.dn)-reduced] if for 1 j n it is dj-excisive [resp. dj-reduced] as a functor of the jthvariable. There is * *an n-variable Taylor polynomial construction which will be denoted Pd1,...,dnF, and which gives the universal (* *d1, . .,.dn)-excisive functor under F (in the homotopy category of homotopy functors). It may be defined eith* *er as P(1)d1.P.(.n)dnF where P(j)dis Pd with respect to the jth variable, or directly as the homotopy * *colimit of (Td1,...,dn)kF where (Td1,...,dnF)(X1, . .,.Xn) = holim Q F(X1*Y1U1, . .,.Xn *YnUn* *); (U1,...,Un)2 jP0(dj+1_) these are naturally equivalent. A functor of n variables will be called multili* *near if it is linear in each variable, in other words both (1, . .,.1)-excisive and (1, . .,.1)-reduced. If * *F is (1, . .,.1)-reduced then the functor P1,...,1F is multilinear and may be called the multilinearization o* *f F. Loosely, this is the homotopy colimit of k1+...knF( k1Y1X1, . .,. knYnXn) as (k1, . .,.kn) -!(1, . .,.1). 18 THOMAS G. GOODWILLIE x2. Delooping homogeneous functors We will show that all homogeneous space-valued functors arise from homogeneou* *s spectrum-valued functors. The main step is to show that they have natural deloopings. Let Hn(C, D) be the category of homogeneous functors of degree n from C to D,* * a full subcategory of F(C, D). The homotopy category hHn(C, D)) is obtained from Hn(C, D) by form* *ally inverting the (objectwise) equivalences. 1 The functor Sp --! T preserves both weak equivalences and cartesian cubes, an* *d therefore composing with it yields a functor 1 Hn(C, Sp) --! Hn(C, T ) which itself takes weak equivalences to weak equivalences. 1 2.1 Theorem. The functor Hn(C, Sp) --! Hn(C, T ) has an inverse up to weak equi* *valence. Proof. The key is to get a functor F 7! BF from Hn(C, T ) to itself such that * *BF is naturally equivalent to F. This will be given by the next lemma. Assume for now that we have this. Then any object F of Hn(C, T ) yields a seq* *uence {BpF} of such objects related by equivalences BpF ~ Bp+1F. Call the resulting spectrum-value* *d functor B1 F. The fact that B1 F is an object of Hn(C, Sp) follows easily from the fact that each* * BpF is an object of Hn(C, T ). The fact that the functor B1 takes equivalences to equivalences fol* *lows from the fact that each functor Bp does so. Clearly 1 B1 is naturally equivalent to the identity* *. To check that this is also true for B1 1 , let F be an object of Hn(C, Sp) and write F(X) = {Fq(X)}.* * The bispectrum {BpFq(X)} shows that the two spectra {BpF0(X)} = {Bp 1 F(X)} = B1 1 F(X) and {B0Fq(X)} = {Fq(X)} = F(X) are naturally equivalent. The next result provides the desired delooping of F, namely RnF. As usual CY * *is either UY or TY. 2.2 Lemma. Let n > 0. If F : CY ! T is any reduced (F(Y ) ~ *) homotopy functor* *, then up to natural equivalence there is a fibration sequence PnF --qnF--!Pn-1F----! RnF in which the functor RnF is n-homogeneous. Proof. More precisely, we will obtain a natural diagram of homotopy functors PnF -qnF---!Pn-1F x x ~ ?? ??~ (2.3) P^nF ----! ~Pn-1F ?? ? y ?y KnF ----! RnF CALCULUS III TAYLOR SERIES * * 19 in which the marked arrows are equivalences, the lower square is cartesian, RnF* * is n-homogeneous, and KnF is contractible. (If F were not reduced, then in fact KnF would be equival* *ent to the constant functor F(Y ).) The proof is based on a close examination of the maps qn,iF which were used i* *n defining qnF. The first step is to define, for each i 0, a diagram TinF ----! Sin-1F ? ? (2.4(i)) ?y ?y Kn,iF----! Rn,iF Roughly, this will become the lower square of (2.3) when i goes to infinity. Define posets Bn = P0(n_+_1_) - {{n + 1}} An,i= P0(n_+_1_)i- P0(n_)i. Define the functor Sn-1F by (Sn-1F)(X) = holimU2BF(X *Y U) n Note that the inclusions P0(n_+_1_) Bn P0(n_) induce maps (2.5) TnF ----! Sn-1F --~--!Tn-1F whose composition is qn,1F. The second map is an equivalence because P0(n_) is * *left cofinal in Bn. Now let (2.4(i)) be obtained from the posets and inclusions: P0(n_+_1_)i---- Bn i x x (2.6(i)) ?? ?? An,i ---- An,i\ Bni by forming the holimof (U1, . .,.Ui) 7! F(X *Y (U1* . .*.Ui)) over each poset. Diagram (2.4(i)) is cartesian by [2, 1.9], since An,iand Bniare concave and t* *heir union is P0(n_+_1_)i. 2.7 Claim. Kn,iF ~ *. Proof. Compare (Kn,iF)(X), a holimover An,i, with the corresponding holimover t* *he smaller poset A*n,i= P0(n_+_1_)i- Bni On the one hand, the comparison map is an equivalence, because A*n,iis left cof* *inal in An,i. On the other hand, the holimof F(X *Y (U1* . .*.Ui)) over A*n,iis contractible: for each (U1* *, . .,.Ui) 2 A*n,i, we have |Uj| = 1 for some j, so that U1* . .*.Uiis contractible and F(X *Y (U1* . .*.Ui)) ~ F(Y ) ~ * 20 THOMAS G. GOODWILLIE 2.8 Claim. Rn,iF is n-reduced. Proof. There is a natural equivalence Pn-1Rn,iF ~ Rn,iPn-1F by 1.7, so it will be enough if Rn,iF is contractible whenever F is (n - 1)-exc* *isive (and reduced). There is an isomorphism of posets An,i\ Bni~=P0(n_)ix P0(i_) given by (U1, . .,.Ui)7! (V1, . .,.Vi, W) Vi= Ui- {n + 1} W = {i : n + 12}Ui Therefore (Rn,iF)(X) can be written as holim F(X *Y (U1* . .*.Ui)) (V1,...,Vi,W)2P0(n_)ixP0(i_) Analyze this as an iterated holim: First fix (V2, . .,.Vi, W) and take holimwit* *h respect to V1. Since F is (n - 1)-excisive, this yields, up to equivalence, F(X *Y (e1* U2* . .*.Ui)) * *where e1 is {n + 1} or ; according as 1 is or is not in W. Next take holimwith respect to V2, then V3, a* *nd so on through Vi, obtaining (Rn,iF)(X) ~ holim F(X *Y (e1* . .*.ei)) W2P0(i_) This is contractible because for each W 2 P0(i_)the space e1* . .*.eiis contrac* *tible. (It is a simplex of dimension |W| - 1 0.) The next step ought to be to take mapping telescopes, letting i tend to 1 in * *(2.4(i)). This is not possible, since we do not have natural maps Kn,iF ! Kn,i+1F or Rn,iF ! Rn,i+1F.* * We do, however, have maps defined up to homotopy, and with a little care these will suffice. Here are two variations on (2.6(i+1)): P0(n_+_1_)i+1---- P0(n_+_1_)x Bin x x (2.9(i+1)) ?? ?? P0(n_+_1_)x An,i----P0(n_+_1_)x (An,i\ Bin) P0(n_+_1_)i+1---- P0(n_+_1_)x Bin x x (2.10(i+1)) ?? ?? An,i+1 ---- An,i+1\ (P0(n_+_1_)x Bin) We have maps of square diagrams of posets (2.9(i + 1)) ! (2.10(i + 1)) (2.6(i + 1)). From each of the three diagrams we get a square diagram of functors by taking t* *he holimof F(X *Y (U1* . .*.Ui)). From (2.6(i+1)) we get (2.4(i+1)). From (2.9(i+1)) we get what might* * be called Tn(2.4(i))). From (2.10(i+1)) we get something new; call it (2.11(i+1)). These are related * *by maps (of square diagrams) . .!.2.4(i) ! Tn(2.4(i)) 2.11(i + 1) ! 2.4(i + 1) ! . . . Now consider the (pointwise) homotopy colimit of this, another square diagram. * *This will be the lower half of (2.3). In view of the following, it is essentially a limit of the carte* *sian squares 2.4(i) and therefore it is itself cartesian: CALCULUS III TAYLOR SERIES * * 21 2.12 Claim. The backwards arrow Tn(2.4(i)) 2.11(i + 1) is an equivalence (in * *all four corners of the square). Proof. In the upper corners it is an isomorphism. In the lower corners it is in* *duced by the inclusions P0(n_+_1_)x An,i ----! An,i+1 P0(n_+_1_)x (An,i\ Bin)----!An,i+1\ (P0(n_+_1_)x Bin) In each of the two cases the larger poset is the union of the smaller one, whic* *h is concave (in the sense of [2, page 298]), with the concave set Q = An,1x Bin and in each case the intersection is Q0 = An,1x (An,i\ Bin) Thus by [2, 0.2] it will be enough if the inclusion Q0 ! Q induces an equivalen* *ce of holims. But each of these holims is contractible. (The argument is as in the proof of 2.7; replace * *Q by A*n,1x Binand Q0 by A*n,1x (An,i\ Bin).) It follows from 2.7 and 2.12 that KnF is contractible; it is the hocolimof a * *diagram . .-.---!Kn,iF --tn--!TnKn,iF--~-- ? ----! Kn,i+1F--tn--!. . . of contractible objects. It follows from 2.8 and 2.12 that RnF is n-reduced, being the hocolimof the f* *unctors . .-.---!Rn,iF--tn--!TnRn,iF--~-- ?----! Rn,i+1F--tn--!. . . Moreover, RnF is n-excisive, by Lemma 1.9; if X is any strongly cocartesian (n * *+ 1)-cube in C then (RnF)(X ) is cartesian because for each i the map (Rn,iF)(X ) tn-!(TnRn,iF)(X ) factors through a cartesian cube. Finally we construct the upper half of (2.3). Note that ^PnF is the hocolimof . .-.---! TinF--tn--!Ti+1nF-=---Ti+1nF--=--!Ti+1nF--tn--!. . . Eliminating the identity maps we obtain an equivalence from ^PnF to the hocolim* *of . .-.---!TinF--tn--!Ti+1nF-tn---!. . . which is PnF. This is the upper left arrow in (2.3). Likewise ~Pn-1F is the hoc* *olimof . .-.---!Sin-1F--tn--!TnSin-1F--=-- TnSin-1F----! Si+1n-1Ftn----!. . . and so has an equivalence to the hocolimof the upper row in . .-.---!Sin-1F--tn--!TnSin-1F----! Si+1n-1F----!. . . ? ? ? ~?y ~ ?y ~?y . .-.---!Tin-1F--tn--!TnTin-1F_______Ti+1n-1F----!. . . This in turn has an equivalence (2.7) to the hocolimof the lower row, and hence* * to Pn-1F. The composed map is the upper right arrow in (2.3). The square commutes. This concludes the * *proof of (2.2). 22 THOMAS G. GOODWILLIE x3.Symmetric multilinear functors Let C -! Cn be the diagonal functor. It was shown in [2, 3.4] that the compos* *ed functor F O is (d1 + . .+.dn)-excisive if the functor Cn -F!D is dj-excisive in the jthvariabl* *e, or in the language of 1.21 (d1, . .,.dn)-excisive. (The latter is easier to write but harder to prono* *unce.) In particular F O is n-excisive if F is (1, . .,.1)-excisive. 3.1 Lemma. If Cn F-!D is a (1, . .,.1)-reduced homotopy functor, then F O is * *n-reduced. Thus L O is n-homogeneous if Cn L-!D is multilinear. The proof resembles that of 1.8. The key is: 3.2 Lemma. If Cn -F!D is a (1, . .,.1)-reduced homotopy functor, then for any X* * 2 C the map (F O )(X) tn-1(FO-)------!Tn-1(F O )(X) factors through a (weakly) contractible ob* *ject. Proof of 3.1, assuming 3.2. Let X 2 C. The object Pn-1(F O )(X) is defined as * *the sequential hocolim of a diagram whose (i + 1)stmap is tn-1Tin-1(FO )i+1 Tin-1(F O )(X)----------!Tn-1(F O )(X) It is enough if each of these maps factors through a weakly contractible object* *. Lemma 3.2 takes care of this, not only for the first map but also for the others, since the functor Tn-* *1(F O ) is equal to G O for a functor G that satisfies the same hypotheses as F. Proof of 3.2. Let E and E* be as in the proof of 1.9. Since E contains the imag* *e of the diagonal map P0(n_) -!P0(n_)n, the map tn-1(F O )(X) can be factored F(X, . .,.X)= F(X *Y ;, . .,.X *Y ;) ! holim F(X *Y U1, . .,.X *Y Un) (U1,...,Un)2E ! holim F(X *Y U, . .,.X *Y U) U2P0(n_) But holim F(X *Y U1, . .,.X *Y~Un)holim F(X *Y U1, . .,.X *Y Un) (U1,...,Un)2E (U1,...,Un)2E* and this is a holimof weakly contractible objects: if (U1, . .,.Un) 2 E* then s* *ome Us is a one-element set, giving X *Y Us~ Y and F(X *Y U1, . .,.X *Y Un) ~ *. The functor L : Cn ! D is symmetric if it has additional structure consisting* * of isomorphisms L(ß) : L(X1, . .,.Xn) ! L(Xß(1), . .,.Xß(n)) for all ß 2 n, with L(oe O ß) = L(ß) O L* *(oe). (In other words L is extended from Cn to a wreath product category.) If L is symmetric and multiline* *ar then the homogeneous functor LO has a n-action. In the case of spectrum-valued functors (that is, * *when D = Sp), the object ( nL)(X) = L(X, . .,.X)h n is then again an n-homogeneous functor of X, by 1.7(5), since homotopy orbit sp* *ace (or spectrum) is a special case of hocolim. We are headed toward proving that all n-homogeneous fu* *nctors arise in just this way. CALCULUS III TAYLOR SERIES * * 23 The inverse of n will be provided by a construction called the nthcross-effe* *ct, which takes a ho- motopy functor F and produces a symmetric homotopy functor crnF of n variables.* * To see how this inverse construction should go, we recall an algebraic analogue. n is analogo* *us to the construction in algebra which uses a symmetric multilinear function l(x1, . .,.xn) to make a* * homogeneous function f(x) = l(x, . .,.x)=n!. For example, if f is a degree two polynomial then from * *the bilinear form l(x1, x2) = f(x1+ x2) - f(x1) - f(x2) + f(0) we recover the purely quadratic part of f as l(x, x)=2. The second cross-effect* * of a functor F of based spaces will take the based spaces X1 and X2 to the total homotopy fiber of the * *diagram F(X1_ X2) ----! F(X1) ?? ? y ?y F(X2) ----! F(*) Strictly speaking, in order to make crnF preserve weak equivalences we must fir* *st replace each Xi by an equivalent object having nondegenerate basepoint, perhaps by using "whiskers* *". In the general case (functors of fiberwise based spaces over Y ) the wedge X1_ X2 is replaced by a * *categorical sum in TY. Again, before forming the sum we should use fiberwise whiskers, replacing each * *object X by the mapping cylinder of the structural coretraction Y -! X. Thus the nthcross-effect is defined as follows: Let TY -F!D be a homotopy fun* *ctor, D = T or Sp. For objects (X1, . .,.Xn) of TY, let S(X1, . .,.Xn) be the evident n-cube takin* *g n_- T to the (whiskered) sum, in TY, of the objects Xs for s 2 T. Define the cross-effect by first apply* *ing F to the cube and then taking the total homotopy fiber: (crnF)(X1, . .,.Xn) = tfiberF(S(X1, . .,.Xn)). It is easy to see that crnF is a homotopy functor (in each variable), and sym* *metric and (1, . .,.1)- reduced. The first cross-effect is what we have sometimes called the "reduced functor"* *: (cr1F)(X) = fiber(F(X) -!| F(Y )). The 0thcross-effect, should we ever need it, is a functor of no variabl* *es: the object F(Y ). 3.3 Proposition. If F is n-excisive then for 0 m n the functor crm+1F is (n* * - m)-excisive in each variable. In particular, the nthcross-effect of an n-excisive functor is symmet* *ric multilinear and the nth cross-effect of an (n - 1)-excisive functor is trivial (equivalent to a point). Proof. Induction on m. The case m = 0 is clear. To pass from m - 1 to m, write (crm+1F)(X1, . .X.m, A) = (crm F+A)(X1, . .,.Xm ), where F+A(X) = hofiber(F(X + A) -! F(X)) and + denotes whiskered sum in TY. Us* *e the rather obvious fact that F+A is (n - 1)-excisive if F is n-excisive. In general, then, crnDnF = crnhofiber(PnF -!Pn-1F) ~ hofiber(crnPnF -!crnPn-1F) ~ crnPnF. Thus if F is n-excisive then crnF is naturally equivalent to crnDnF; the nthcro* *ss-effect of an n-excisive functor "seesö nly the n-homogeneous part. The following simple result suggest* *s that it sees it quite clearly, at least in the spectrum-valued case. 24 THOMAS G. GOODWILLIE 3.4 Proposition. If F and G are n-homogeneous functors TY -! Sp, then any (natu* *ral) map F -! G that induces an equivalence crnF -!crnG must be an equivalence itself. Proof. Let H be the homotopy fiber of F -! G. Thus H is n-homogeneous and crnH * *is the homotopy fiber of crnF -!crnG. Since a map of spectra must be an equivalence if its homo* *topy fiber is contractible, we have only to show that, for an n-homogeneous functor TY -H!Sp, crnH ~ * impl* *ies H ~ *. In fact we will do a little better: assuming only that H is n-excisive and crnH ~ *, we* * will show that H is (n - 1)-excisive. If X is any strongly cocartesian n-cube in TY, we must show that H(X ) is car* *tesian. Because we are dealing with spectra, it will be enough if we show that tfiberH(X ) ~ *. By assumption, this holds in the case where X is the cube S(X1, . .,.Xn) for * *any objects X1, . .,.Xn of TY. It also holds for the related cube, call it S*(X1, . .,.Xn), which has the sa* *me objects but with reversed arrows, sending T rather than n_- T to the sum of the objects Xs for s 2 T. In * *fact we have tfiberH(S*(X1, . .,.Xn)) ~ ntfiberH(S(X1, . .,.Xn)) = n(crnH)(X1, . * *.,.Xn). This takes care of all cubes X in which X (;) ~ Y , because such a cube is natu* *rally equivalent to S*(X ({1}), . .,.X ({n})) (see [2, 2.2]). Given an arbitrary strongly cocartesian cube X , put X 0(T) = hocolim(Y - X (;) ! X (T)). The evident map of n-cubes X ! X 0is a strongly cocartesian (n + 1)-cube, so th* *e resulting cube H(X ) ! H(X 0) is cartesian. The cube H(X 0) is cartesian by the case already t* *reated. Therefore by [1, 1.6] H(X ) is cartesian. Let Ln(C, D) be the category of symmetric multilinear functors L : Cn ! D. Th* *e maps are the natural maps that respect the symmetry. As usual the homotopy category hLn(C, D) means * *the category obtained by inverting the (objectwise) equivalences. If D is either Sp or T then there is the cross-effect functor Hn(TY, D) crn--!Ln(TY, D) If C is either UY or TY then there is the functor Ln(C, Sp) -n-!Hn(C, Sp) going the other way. Both crn and n preserve weak equivalences and so induce f* *unctors on homotopy categories. 3.5 Theorem. The functors Hn(TY, Sp) crn--!Ln(TY, Sp) Ln(TY, Sp) -n-!Hn(TY, Sp) are mutual inverses up to natural (weak) equivalence. Proof of 3.5. To prove that the composition Ln(TY, Sp) -n-!Hn(TY, Sp) crn--!Ln(TY, Sp) CALCULUS III TAYLOR SERIES * * 25 is equivalent to the identity, we look to the algebraic analogue. If l is a sym* *metric multilinear function of n variables and f is the homogeneous polynomial f(x) = l(x, . .,.x)=n!, then l can be recovered from f. It is given by l(z1, . .,.zn) = f(z1+ . .+.zn) - f(z1. .+.zn-1) - . .+.(-1)nf(0) (an alternating sum of 2n terms). One sees this, of course, by expanding the ex* *pression l(x1+ . .+.xn, . .,.x1+ . .+.xn) as a sum of nn terms, cancelling all except the permutations of l(x1, . .,.xn) * *and dividing by n factorial. Here is a corresponding categorical argument: We have (crn nL)(X1, . .,.Xn)= tfiber(( nL) O S(X1, . .,.Xn)) a a = tfiber(n_- T 7! L( Xs, . .,. Xs)h n) s2T s2T ~ (tfiber(X ))h n, ` ` ` where X (n_- T) = L( s2TXs, . .,. s2TXs) and denotes whiskered sum in TY. Th* *e evident equiva- lence Y X (n_- T) ~-! L(Xß(1), . .,.Xß(n)) n_ß-!T * * Q is natural with respect to T. It can be interpreted as an equivalence of cubes * *X -! n ß Yß where * * _-!n_ Yß(n_- T)= L(Xß(1), . .,.Xß(n)), if ß(n_) T Yß(n_- T)= *, otherwise. For any ß that is not a permutation and therefore not surjective, the cube Yß i* *s cartesian. (Viewed in one way it is an isomorphism of (n - 1)-cubes: if s =2ß(n_) then all maps Yß(T)* * -! Yß(T [ {s}) are isomorphisms.) For any permutation ß we have tfiber(Yß) = L(Xß(1), . .,.Xß(n)) Therefore Y tfiber(X ) ~-! L(Xß(1), . .,.Xß(n)) ß2 n This map respects the symmetry if the group is made to permute the factors of t* *he right-hand side in the evident way, and so it leads to an equivalence of homotopy orbit spectra Y (crn nL)(X1, . .,.Xn) ~ ( L(Xß(1), . .,.Xß(n)))h n ~ L(X1, . .,.* *Xn) ß2 n Note for future reference that an explicit inverse equivalence L(X1, . .,.Xn) `-!(crn nL)(X1, . .,.Xn) 26 THOMAS G. GOODWILLIE is the map of total homotopy fibers induced by an obvious map of cubes Y1 -!X -!Xh n, where n_1-!n_is the identity map and the map Y1(;) -!X (;) is the map L(X1, . .,.Xn) i-!L(Z, . .,.Z) ` induced by the inclusions Xj-! Z = 1 j nXj. The following diagram commutes: L(X1, . .,.Xn) ---i-! L(Z, . .,.Z) ? ? `?y ?y (crn nL)(X1, . .,.Xn)ffl----!L(Z, . .,.Z)h n. Here, and throughout this proof, ffl denotes the projection from the total homo* *topy fiber of a cubical diagram to the "initialö bject in the diagram. The remaining task is to exhibit an equivalence ncrnF -fl!F for any n-homogeneous TY -F!Sp. In fact we will define fl for any n-excisive F * *and then show that it induces an equivalence crn ncrnF -crn(fl)---!crnF. By 3.4 this suffices. To define fl we use a map ^fl, defined for Z 2 TY as the composition a F(f) (crnF)(Z, . .,.Z) ffl-!F( Z) ---!F(Z) 1 j n where f is the öf ld" map which takes each copy of Z identically to Z. The map * *^flis equivariant with respect to the obvious n-actions. (The action on F(Z) is trivial.) Define fl a* *s the composition ( ncrnF)(Z) = ((crnF)(Z, . .,.Z))h^nflh-n--!F(Z)h n = F(Z) ^ (B n)+ -!F* *(Z) where the last arrow is induced by the nontrivial based map (B n)+ ! S0. To see that crn(fl) is an equivalence we examine the composition crnF -`!~crn ncrnF -crn(fl)---!crnF. It will be enough if it coincides with the identity, at least on the level of h* *omotopy groups. In fact it will be enough if the composition (crnF)(X1, . .,.Xn) crn(fl)O`-----!(crnF)(X1, . .,.Xn) ffl-!F* *(Z) ` is equal to ffl, since ffl is a split injection. (Here again Z = 1 j nXj.) CALCULUS III TAYLOR SERIES * * 27 That it is is a direct consequence of the following facts: The diagram (crnF)(X1, . .,.Xn)--i--! (crnF)(Z, . .,.Z) ? ? `?y ?y (crn ncrnF)(X1, . .,.Xn)ffl----!(crnF)(Z, . .,.Z)h n ? ? crn(fl)?y fl?y (crnF)(X1, . .,.Xn)--ffl--! F(Z); commutes; the composition of the right-hand vertical maps above is ^fl, which i* *s also the composition of the lower horizontal maps below; the diagram crnF(X1, . .,.Xn)ffl----!F(Z) ? ? i?y F(D)?y ` F(f) crnF(Z, . .,.Z)-ffl---!F( 1 j nZ)----!F(Z), ` commutes, where Z D-! 1 j nZ sends the copy of Xiin Z to the copy of Xiin the i* *thcopy of Z; and the composition f O D is the identity map. 3.6 Corollary. The functor Hn(TY, T ) crn--!Ln(TY, T ) has an inverse up to wea* *k equivalence. Proof of 3.6. Using the commutative diagram 1 * Hn(TY, T )---- Hn(TY, Sp) ? ? crn?y crn?y 1 * Ln(TY, T )---- Ln(TY, Sp) this will follow from 3.5, 2.1, and the next result. 1 * 3.7 Proposition. The functor Ln(C, Sp) ---!Ln(C, T ) has an inverse up to weak * *equivalence. Proof. As in proving 2.1, we need a delooping functor from Ln(C, T ) to itself.* * This is much easier than 2.2. We refer to Remark 1.21 for notation. If L is symmetric multilinear then ther* *e are natural equivalences L(X1, . .,.Xn) ~-!(T1,...,1L)(X1, . .,.Xn) ~- nL( YX1, . .,. YXn* *). We will need these equivalences to respect the n-symmetry. In order for this t* *o be true, permutations of the loop coordinates must be built into the symmetry of the last expression;* * a better expression is VnL( YX1, . .,. YXn), where Vn is the standard n-dimensional representation* * of n. Since this contains a trivial one-dimensional representation, we have what we need: if Vn * *= R ~Vnthen L ~ BL where BL is defined by (BL)(X1, . .,.Xn) = ~VnL( YX1, . .,. YXn). We can now justify an assertion made in 1.20. Let D be either T or Sp. 28 THOMAS G. GOODWILLIE 3.8 Corollary. If TY -F!D is n-homogeneous then F is determined by F O Y. Proof. According to 3.5 and 3.6 F is determined by the cross-effect crnF, and t* *his satisfies (crnF)( YX1, . .,. YXn) = (crn(F O Y))(X1, . .,.Xn). On the other hand, for any symmetric multilinear functor L we have L(X1, . .,.Xn) ~ VnL( YX1, . .,. YXn), so that (crnF)(X1, . .,.Xn) ~ Vn(crnF)( YX1, . .,. YXn) = Vncrn(F O Y)(X1, .* * .,.Xn) x4.The role of the base point The cross-effect construction applies to functors TY -!D but not to functors * *UY -!D. In spite of this we now show that the classification of homogeneous functors extends without cha* *nge to the UY case. Let TY -OE!UY be the forgetful functor. Because OE preserves equivalences an* *d cocartesian square diagrams, composition with OE yields functors * Hn(UY, D)OE-!Hn(TY, D) * Ln(UY, D)OE-!Ln(TY, D). 4.1 Theorem. If D is T or Sp then both of the two functors OE* above have inver* *ses up to weak equiva- lence. 4.2 Corollary. The functor Ln(UY, Sp) -n-!Hn(UY, Sp) has an inverse up to weak * *equivalence. Proof of 4.2. Use 3.5 and the diagram Ln(UY, Sp)--n--!Hn(UY, Sp) ? ? OE*?y OE*?y Ln(TY, Sp)--n--!Hn(TY, Sp). Proof of 4.1. We can assume D = Sp, since by 2.1 and 3.7 the spectrum-valued ca* *se of 4.1 implies the space-valued case. Let UY -_!TY be the left adjoint of OE, so that if X is a space over Y then _* *(X) is the disjoint union of X and Y viewed as a fiberwise based space over Y . Like OE, this functor*pre* *serves equivalences and cocartesian square diagrams. Therefore if TY -F!Sp is n-excisive then UY -_-F-!* *Sp will be n-excisive. Unlike OE, _ does not preserve the final object Y , so _*F need not be homogene* *ous if F is. We will show that * Hn(TY, Sp) DnO_----!Hn(UY, Sp) is a weak inverse for * Hn(UY, Sp) OE-!Hn(TY, Sp). CALCULUS III TAYLOR SERIES * * 29 The unit map X -!OE_X of the adjoint pair induces a map F -!_*OE*F. For any m* *orphism X -!X0 in UY the square diagram X ----! OE_X ?? ? y ?y X0----! OE_X0 is cocartesian. Therefore if X is any strongly cocartesian n-cube in UY the uni* *t yields a strongly cocarte- sian (n + 1)-cube X -!OE_X . It follows that if UY -F!Sp is n-excisive then the fiber of F -! _*OE*F is (n -* * 1)-excisive. Thus if F is n-homogeneous there are natural equivalences F -~!PnF -~ DnF -~!Dn_*OE*F (The last map is an equivalence because its fiber is contractible. This implica* *tion relies on the fact that these are spectrum-valued functors.) The counit _OEX -!X, like the unit, yields a cocartesian square for every map* * and a strongly cocarte- sian (n + 1)-cube for every strongly cocartesian n-cube. Therefore for any n-ho* *mogeneous UY -F!Sp it yields equivalences F -~!PnF -~ DnF -~ DnOE*_*F = OE*Dn_*F. This completes the proof in the homogeneous case. We sketch the proof in the multilinear case, which is much the same. The inve* *rse of * Ln(UY, Sp) OE-!Ln(TY, Sp). is * Ln(TY, Sp) rO_---!Ln(UY, Sp), where r is the operation of "reducingä symmetric homotopy functor of n variab* *les in all variables simultaneously. For example, when n = 2 then (rL)(X1, X2) is the total homotopy* * fiber of L(X1, X2)----! L(X1, Y ) ?? ? y ?y L(Y, X2)----! L(Y, Y ). The key point is that the maps L -!OE*_*L and _*OE*L -!L induce equivalences rL* * -!rOE*_*L = OE*r_*L and r_*OE*L -!rL. For example, in the composition L(X1, X2) -!L(X1, __X2) -!L(_OEX1, _OEX2) the fiber of the first map becomes contractible upon reducing with respect to X* *2 while the fiber of the second map becomes contractible upon reducing with respect to X1, and so they b* *oth become contractible upon applying r. The proof of 4.1 suggests a variant of the notion of multilinear functor, rel* *ated to it as unreduced homology is related to reduced homology. 30 THOMAS G. GOODWILLIE 4.3 Definition. A functor UnY-L!Sp is unreduced-multilinear if it is 1-excisive* * in each variable and if it satisfies L(X1, . .,.Xn) ~ * whenever Xs is the initial object ; for some s.* * The category of symmetric unreduced-multinear functors is ^Ln(UY, Sp). Since _(;) = Y , the functor _* maps Ln(TY, Sp) into ^Ln(UY, Sp). * 4.4 Proposition. The functor Ln(TY, Sp) _-!L^n(UY, Sp) has an inverse up to wea* *k equivalence. Proof. The proof of 4.1 shows that OE*O r is an inverse. 4.5 Warning. Although Definition 4.3 could be extended verbatim to functors int* *o based spaces, the resulting category ^Ln(UY, T ) would not have the expected property: the corres* *ponding variant of Propo- sition 4.4 would be false. An instructive example is obtained by adding a disjo* *int basepoint to the excisive functor U -P1J-!U mentioned near the end of the introduction. 4.6 Remark. 3.8 is valid for functors of unbased spaces, in view of 4.1 and the* * fact that OE commutes with Y up to isomorphism. x5. The nthdifferential and the nthderivative We can summarize the main results of x2-x4 by saying that the following eight* * categories of functors are equivalent at the homotopy category level: 1* Hn(UY, Sp)----! Hn(UY, T ) ? ? OE*?y OE*?y 1* Hn(TY, Sp)----! Hn(TY, T ) 1* Ln(UY, Sp)----! Ln(UY, T ) ? ? OE*?y OE*?y 1* Ln(TY, Sp)----! Ln(TY, T ) In addition to the arrows displayed, there is also crn from the upper square to* * the lower in each of the two (TY, -) cases and n from the lower square to the upper in each of the (-, * *Sp) cases. We have explicitly inverted enough of these arrows to show that all of them are inverti* *ble: We inverted the four called 1*in 2.1 and 3.7; we inverted the left hand OE* of each square in 4.1; * *and in 3.1 we connected the two squares by showing that crn and n are inverses in the (TY, Sp) case. Let CY -F!D be a homotopy functor from spaces over Y to either based spaces o* *r spectra, and suppose that we wish to describe the homogeneous layer DnF in its Taylor tower for some* * n 1. By the results above, knowing DnF is the same as knowing a certain symmetric multilinear funct* *or. This will be called the n-fold differential of F. 5.1 Definition. The object in Ln(CY, D) corresponding to the homogeneous functo* *r DnF 2 Hn(CY, D) is called the n-fold differential of F and is denoted by D(n)F. Specifically, D(n)F determines DnF by the rule: (DnF)(X) ~ (D(n)F)(X, . .,.X)h n in the case when D = Sp, or (DnF)(X) ~ 1 ((B1 D(n)F)(X, . .,.X)h n) CALCULUS III TAYLOR SERIES * * 31 in the case when D = T . Here B1 is the inverse (up to natural weak equivalence* *) to 1 Ln(CY, Sp) -*-!Ln(CY, T ) provided by 3.6. Conversely, to obtain D(n)F from DnF one simply takes the nthcross-effect in * *the case when CY = TY, while in the case when CY = UY one knows that OE*D(n)F determines D(n)F by 4.2 * *and is given by OE*D(n)F = D(n)OE*F = crnDnOE*F. In practice we often work with a functor UY -FY-!D that is the restriction of* * a functor U -F!D, meaning the composition of F with the forgetful functor UY -!U. 5.2 Definition. In this case D(n)FY is called the n-fold differential of F at Y* * and denoted by D(n)YF. The next theme to be developed is the description of multilinear functors by * *their öc efficients". We discuss this briefly in the important special case when Y is the one-point spac* *e, and then a little more elaborately in the general case. If C is a spectrum then the functor L(X1, . .,.Xn) = C ^ (X1^ . .^.Xn) from (n-tuples of) based spaces to spectra is multilinear, and if C has an acti* *on of the symmetric group n then L is symmetric. Conversely, if L is a multilinear functor from based spaces to spectra then, * *taking C to be L(S0, . .,.S0), we have (essentially, see [1, page 5]) a natural assembly map C ^ (X1^ . .^.Xn) -!L(X1, . .,.Xn). This is an equivalence when Xj = S0 for all j, and it follows (see 5.8 below), * *using the multilinearity of both functors, that it is an equivalence when the Xj are finite complexes. If L* * satisfies a suitable limit axiom (5.10 below), then this even holds for all Xj. If L is symmetric then C g* *ets a n-action and the assembly map respects the symmetry. In short, symmetric multilinear functors from finite based spaces to spectra * *correspond precisely, in the sense of an equivalence of homotopy categories, to spectra with n-action. * *The spectrum (with n- action) is called the coefficient of the symmetric multilinear functor, or of t* *he corresponding homogeneous functor. Thus if F is a homotopy functor from based spaces to based spaces or spectra * *then for every n > 0 the n-homogeneous layer DnF of its Taylor tower is governed by a certain spectrum w* *ith n-action. This will be called the nthderivative of F at the one-point space and denoted by @(n)F(*)* *. In the spectrum-valued case we have @(n)F(*) = (D(n)F)(S0, . .,.S0) and (DnF)(X) = (@(n)F(*) ^ X^n)h n. In the space-valued case we have 1 @(n)F(*) = (D(n)F)(S0, . .,.S0) and (DnF)(X) = 1 ((@(n)F(*) ^ X^n)h n). 32 THOMAS G. GOODWILLIE To fully describe @(n)F(*) in the space-valued case, we examine the proof of 3.* *7 and see that the ithspace in the spectrum is i~Vn(D(n)F)(Si, . .,.Si) 5.3 Remark. By the proof of 3.8 we have @(n)F(*) ~ Vn@(n)(F O )(*) 5.4 Remark. If a subgroup G n acts on a spectrum C then the functor (C ^ X^n* *)hG is homogeneous. When written in standard form it is ((( n)+ ^G C) ^ X^n)h n; the coefficient is the "induced spectrum" ( n)+ ^G C. We now pass from functors of spaces to functors of spaces over Y while doing * *our best to retain the principle that a multilinear functor is determined by its behavior on a small c* *lass of objects. Here is some notation for naming special objects of TY: If Z is a based space* * and y is a point in Y then let Y _yZ be the union of Y and Z with y identified to the basepoint of Z.* * This is to be viewed as a space over Y with all of Z being mapped to y. More generally if several point* *s y1, . .,.yn and several based spaces Z1, . .,.Zn are given then Y _y1Z1_ . ._.ynZn is the space obtaine* *d from Y by attaching Zj at yj for all j. Again this is to be viewed as an object of TY. Let F be a homotopy functor from unbased spaces to either based spaces or spe* *ctra. 5.6 Definition. Let Y be a space and let y1, . .,.yn be points in Y . The nth* * derivative of F at (Y, y1, . .,.yn), denoted @(n)y1,...,ynF(Y ), is the coefficient spectrum of the multilinear functor (Z1, . .,.Zn) 7! (D(n)YF)(Y _y1Z1, . .,.Y _ynZn). Thus in the spectrum-valued case we have @(n)y1,...,ynF(Y ) = (D(n)YF)(Y _y1S0, . .,.Y _ynS0), @(n)F has a symmetry with respect to permutations of (y1, . .,.yn) and is als* *o functorial in Y . To be more precise, we have here a functor whose domain is the category in which a* *n object is a "space with n basepoints" (Y, y1, . .,.yn) and a morphism (Y, y1, . .,.yn) -!(W, w1, .* * .,.wn) is a pair (f, ß) with Y -f!W being a continuous map and ß a permutation such that f(yß(j)) = wj for a* *ll j. The points yj are not assumed distinct. It is true in various senses, beginning with 5.9 below, that the nthderivativ* *e of a functor determines the behavior of the nthdifferential, at least on finite objects. 5.7 Definition. An object Y -i!X -r!Y of TY is finite if (X, i(Y )) is a finite* * CW pair. An object X -r!Y of UY is finite if the space X is finite CW. CALCULUS III TAYLOR SERIES * * 33 5.8 Proposition. Let L1 g-!L2 be a map between symmetric multilinear functors f* *rom TY to Sp. In order that g L1(X1, . .,.Xn) -!L2(X1, . .,.Xn) should be an equivalence whenever the objects Xj are finite, it is enough if g * *is an equivalence in the special case when each Xj is Y _ yjS0 for an arbitrary point yj2 Y . Proof. The symmetry is irrelevant here, and it is clear that the case n = 1 imp* *lies the general case. We therefore give a proof for the case of a map L1 g-!L2 between linear functors f* *rom TY to spectra. Define TY -L!Sp by letting L(X) be the homotopy fiber of L1(X) g-!L2(X). The * *functor L äv nishes" at Y _y S0, for every point y in Y , in the sense that L(Y _y S0) ' *. We also * *know that L is linear; it preserves weak equivalences, it takes cocartesian squares to cartesian squares,* * and it vanishes at Y . We must show that L vanishes at every finite object X. We name some more objects of TY: If Z is a space and Z f-!Y is a map then let* * Y +fZ be the disjoint union of Y and Z, considered as an object of TY in the evident way. If X is an object of TY obtained by attaching an m-cell to another object X0,* * then there is a cocartesian diagram Y +f|@Sm-1 ----! X0 ?? ? y ?y Y +f Dm ----! X for some Dm -f!Y , and thus a cartesian diagram L(Y +f|@Sm-1) ----! L(X0) ?? ? y ?y L(Y +f Dm ) ----! L(X). L(X) will be contractible if the other three spectra are. Thus an induction on * *the number of cells in X - i(Y ) will be possible as soon as we have dealt with the cases X = Y +fDm a* *nd X = Y +fSm . The sphere case follows from the disk case by induction on m (beginning with the ca* *se Y +f S-1 = Y ), and the disk case is taken care of by a weak equivalence Y _yS0 -!Y +f Dn obtained by choosing a point y 2 f(Dn). Note that if Y is path-connected then it is only necessary to verify the hypo* *thesis of 5.8 for one choice of (y1, . .,.yn), since a path I f-!Y from y to y0in Y yields a diagram of equi* *valences in TY: Y _yS0 ~-!Y +f I ~- Y _y0S0. There is a variant of 5.8, with the same proof, for unreduced-multilinear fun* *ctors from UY to spectra, with a point over yj replacing Y _yjS0. The proof of 5.8 also generalizes rathe* *r easily to prove that a map between d-excisive homotopy functors from TY [resp. UY] to Sp must be an * *equivalence for all objects if it is an equivalence for all objects of the form Y +f S [resp. S] wh* *ere S is a discrete set of at most d points with a map to X. For us the main consequence of 5.8 is: 34 THOMAS G. GOODWILLIE Corollary 5.9. If a map F -!G of homotopy functors U -!T induces an equivalence* * of nthderivatives @y1,...,ynF(Y ) -!@y1,...,ynG(Y ) for every point (y1, . .,.yn) 2 Y nthen it induces an equivalence of nthdiffere* *ntials DnFY(X) -!DnGY(X) for every finite object X of UY or TY. Proof. Apply 5.8 to D(n)FY -!D(n)GY, noting that the behavior of DnFY on finite* * objects is determined by the behavior of D(n)FY on finite objects. Note that these statements definitely require spectra rather than spaces as t* *he output of the functors. For example, if C is a spectrum and L is the linear functor from based spaces t* *o based spaces given by L(X) = 1 (C ^ X), then the statement that L(S0) is contractible means only that the homotopy grou* *ps ßj(C) are trivial for j 0. The restriction to finite objects in the results above can be removed if F sa* *tisfies a suitable limit axiom. 5.10 Definition. A homotopy functor C F-!D is finitary if it preserves filtered* * homotopy colimits up to weak equivalence, that is, if the natural map hocolimffF(Xff) -!F(hocolimffXff) is a weak equivalence whenever {Xff} is a diagram in C indexed by a filtering c* *ategory. In the case of linear functors from spaces to spaces, this condition means th* *at the corresponding homology theory satisfies Milnor's wedge axiom. It is clear that the nth Taylo* *r approximation of a finitary functor is itself finitary. 5.11 Remark. This was called the limit axiom in [2], but it seems useful to hav* *e an adjective available. The term öc ntinuous functor" has been used, but we would rather reserve that f* *or something else (a functor that behaves continuously on morphisms). Since every space is equivalent to a filtered hocolimof finite CW complexes, * *a finitary functor of spaces is determined (up to natural weak equivalence) by its behavior on finite comple* *xes. More generally a finitary functor of objects in CY is determined by its behavior on objects that* * are finite in the sense of 5.7. According to 5.9 the nthdifferential (which knows all about the nthlayer of t* *he Taylor tower) is in some sense controlled by the nthderivative. This is only a weak sense, however,* * since 5.9 cannot produce an equivalence between the differentials of F and G unless a map between F and * *G is already given. It would be better to have a way of building the nthdifferential from the nthderiv* *ative. The nthderivative of F at Y gives a spectrum for each ordered n-tuple of poin* *ts in Y . To identify the nthdifferential, one needs just a little more information. We will sketch one o* *f several possible answers to the following vague question: 5.13 Vague question. How can D(n)F be assembled from the spectra @(n)y1,...,ynF* *(Y )? Answers in the case n = 1 tend to generalize easily to the general case, so w* *e will concentrate on that case. CALCULUS III TAYLOR SERIES * * 35 A useful point of view is that a (finitary) linear functor of spaces over Y c* *orresponds to a öc efficient systemö n Y which assigns a spectrum Ey to each point y. The spectrum Ey must * *depend on the point y continuously in some sense, so the linear functor cannot really be specified * *by merely giving Ey for each y. (That would be like trying to specify a vector bundle by giving all of its f* *ibers, a mere collection of vector spaces.) One can give rigorous sense to this idea by making the following definition: * *A system of spectra on Y consists of objects {En} in TY such that the structural maps En -!Y are fibrati* *ons, related by maps YEn -!En+1. We may write Ey,nfor the fiber of En -! Y over y 2 Y . The spaces {Ey,n} for fi* *xed y constitute a spectrum Ey, the fiber of E over y. Such a system E determines a prespectrum whose nthspace is the homotopy cofib* *er of the structural map Y -! En. The associated spectrum, which may be thought of as the homology o* *f Y with coefficients in E, will be denoted by Z Eydy. y2Y Of course this notation is deceptive, since it appears not to matter how the va* *rious spectra Ey are related. The integral signs are not meant to suggest antidifferentiation. We draw attention to the familiar special case when E is a "trivial bundle of* * spectraö ver Y : Let C be a spectrum and take En to be Y x Cn. Then Z Z Eydy = C dy = C ^ Y. y2Y y2Y In the general case there is a spectral sequence of Atiyah-Hirzebruch type, with Z E2p,q= Hp(Y ; ßq(Ey)) ) ßp+q Eydy. y2Y One constructs it by taking the direct limit over n of a spectral sequence with E2p,q= Hp(Y ; ßSq+n(En,y)) ) ßSp+q+nhocofiber(Y -! En). Systems of spectra pull back: a map X -f!Y and a system E on Y determine a sy* *stem f*E on X given by (f*E)n = En xY X, whose fiber at x 2 X is Ef(x). Thus a system E on Y * *gives a functor from UY to Sp: Z X = (X, X f-!Y ) 7! Ef(x)dx. x2X This is a homotopy functor, and it is finitary and 1-excisive; all of these ass* *ertions can proved by spectral sequence comparison arguments. I;t vanishes at the empty set, so it is not line* *ar but rather unreduced- linear (4.5). By reducing it one gets a linear functor, which sends X to the ho* *motopy fiber of Z Z Ef(x)dx -! Eydy. x2X y2Y It is fairly clear that one could classify the finitary linear functors on TY* * along these lines, although we will not pursue that here. The generalization to n > 1 involves systems of s* *pectra over Y nwith a n-symmetry, or alternatively systems of spectra over Y nx n E n. 36 THOMAS G. GOODWILLIE If Y is equivalent to the classifying space of a group G then systems of spec* *tra on Y are the same (at the homotopy category level) as spectra with G-action. This idea can be extende* *d to simplicial groups G, so that it applies to all based connected Y . The idea of constructing linear functors by systems of spectra over a space w* *as implicitly present in sections 2 and 3 of [1]. The systems that arose there were mainly "fiberwise su* *spension spectra" in the following sense: If W is an object of TY whose structural map W -!Y is a fibrat* *ion, then the repeated fiberwise suspensions nYW form a system E whose fiber Ey is the suspension spe* *ctrum of the fiber of W over y. There is also a dual construction, in which E is used as coefficients for twi* *sted cohomology rather than twisted homology. If E is a system of spectra on a CW complex K then the spectr* *um Z k2K Ekdk. is defined by letting the nthspace be the space of sections of the fibration En* * -! Y . If K is locally compact then there is also the compactly supported version Z k2K Ekdk, c made out of spaces of compactly supported sections. If X 7! E(X) is a homotopy functor from spaces to systems of spectra on the f* *inite complex K then "differentiation under the integral" is valid; there is an equivalence Z k2K Zk2K @y Ek(Y ) dk ~ @yEk(Y ) dk. The proof is by induction on the number of cells in K. This extends to the comp* *actly supported case if for example the one-point compactification of K is a finite complex. We will also need the following simple principle: If P -ß!B is a principal G* *-bundle (G being a topological group) and E is a system of spectra on the locally compact space B,* * then when G is made to act on the spectrum Z p2P Eß(p)dp in the obvious way the homotopy fixed point spectrum is equivalent to Zb2B Ebdb. Confession. We should really distinguish between (fiberwise) unreduced suspensi* *on and (fiberwise) re- duced suspension. The former, (fiberwise) join with a two-point set, is what w* *e ordinarily denote by Y here, and it has the pleasant feature that it takes fibrations to fibrations* *. The latter, on the other hand, is much better for making spectra. When they are inequivalent, we have to* * choose the former, but this must be paid for by growing whiskers. We have not systematically imposed a* * solution of this small technical difficulty on the reader, because we do not have a neat solution. x6. Multilinearized cross-effects Before working on some examples, we need one more tool. CALCULUS III TAYLOR SERIES * * 37 6.1 Theorem. Let D be T or Sp. The nthdifferential D(n)F = crnDnF of a homotopy* * functor TY -F!D is (naturally, weakly) equivalent to the multilinearization of the nthcross-eff* *ect crnF of F itself. This means that (D(n)F)(X1, . .,.Xn) is essentially the homotopy colimit of i1+...+in(crnF)( i1YX1, . .,. inYXn) over (i1, . .,.in). The latter may also be described as hocolimi iVn(crnF)( iYX1, . .,. iYXn). In terms of coefficients it means that the spectrum @(n)y1,...,ynF(Y ) may be o* *btained from the prespectrum whose ithspace is i~Vn(crnF)(Y _y1Si, . .,.Y _ynSi). This can be a key tool for identifying DnF in examples. While the definitions* * of PnF and DnF are difficult to use for explicit calculation when n is greater than one, we do kno* *w how to recover DnF from crnDnF, and 6.1 says that this in turn can be obtained rather directly from F i* *tself. 6.2 Remark. Like 1.7 and 2.1 , 6.1 has an easier proof in the case when the fun* *ctor F is analytic. In fact, in that case using 1.5(2) one sees easily that if the maps Xj-! Y are all k-con* *nected then the map (crnF)(X1, . .,.Xn) -!(crnPnF)(X1, . .,.Xn) induced by pnF is ((n + 1)k - c0)-connected for some constant c0, from which it* * follows that the canonical map from iVn(crnF)( iX1, . .,. iXn) to iVn(crnPnF)( iX1, . .,. iXn) ~ (crnPnF)(X1, . .,.Xn) has a connectivity tending to 1 with i. For that matter, in applying 6.1 to a p* *articular F one often uses this same kind of reasoning again: One identifies the multilinearization of crn* *F with a given functor L by exhibiting a (natural and symmetry-preserving) map (crnF)(X1, . .,.Xn) -!L(X1, . .,.Xn) and checking that it is ((n + 1)k - c0)-connected for some constant c0when the * *Xj-! Y are k-connected. Therefore some readers may prefer to skip to x7 after the proof of 6.3. A consequence of 6.1 is that what we are calling a second derivative (that is* *, the coefficient of the bilinear functor corresponding to a 2-homogeneous layer) can actually be seen a* *s the derivative of a derivative. More generally, we have the following useful interpretation of 6.1: 6.3 Corollary. @(p+q)y1,...,yp+qF(Y ) ~ @(p)y1,...,yp@(q)yp+1,...,yp+qF* *(Y ). Explanation and sketch of proof. If F is a homotopy functor from spaces to eith* *er based spaces or spectra, then, as we have already observed, the spectrum @(q)yp+1,...,yp+qF(Y )* * depends functorially on Y = (Y, yp+1. .,.yp+q) with appropriate definitions. In stating 6.3 we are exte* *nding our usage of the äp rtial derivative" notation to cover functors of "spaces with several base po* *ints". Thus, for example, 38 THOMAS G. GOODWILLIE if F is a functor of based spaces then @yF(Y, y0) depends functorially on (Y, y* *, y0) and is defined as the coefficient spectrum of the linear part of the functor Z 7! F(Y _yZ). Here y0 2 Y is serving as basepoint in Y _y Z for purposes of applying the func* *tor F while y is serving as basepoint in Y for wedging with Z. In view of 6.1, to prove 6.3 one has only to see that for a functor of p + q * *based spaces, such as (Z1, . .,.Zp+q) 7! F(Y _y1Z1, . .,.Y _ynZp+q), the following two processes are equivalent: (1) reducing in all variables follo* *wed by multilinearizing in all variables, (2) reducing and multilinearizing in the last q variables, follo* *wed by reducing and multilin- earizing in the first p variables. This is easy. Another consequence of 6.1 is: 6.4 Corollary. F is m-reduced if and only if F is reduced and for every 0 < n <* * m the multilinearization of the nthcross-effect of F is contractible. Proof. This is clear from 6.1 and the fact that crnDnF determines DnF. The proof of 6.1 is connected with the idea of üm ltivariable Taylor series" * *(Remark 1.21). A key observation is that the product category`CY1x . .x.CYn is itself the category o* *f spaces over a space: it is equivalent to CY1` ...`Ynwhere the " " denotes disjoint union. In this equival* *ence of categories, n-tuples of weak equivalences correspond to weak equivalences and n-tuples of cocartesia* *n cubes correspond to cocartesian cubes. When it is necessary to distinguish between a functor CY1x . .x.CYn F-!D of n variables and the associated functor CY1` ...`Yn-!D of one variable, we will denote the latter by ~F. 6.1 is proved using the following statement, which will be proved at the end * *of x6: 6.5 Lemma. If CY1x . .x.CYn G-!D is (1, . .,.1)-reduced then ~P1,...,1G ~ Pn~G. Proof of 6.1. Applying 6.5 in the special case Y = Y1 = . .=.Yn, with G = crnF,* * we find that what we need is a natural equivalence Pn~crnF ~ ~crnPnF. This will follow from a natural equivalence Tn~crnF ~ ~crnTnF, which in turn wi* *ll follow from a natural equivalence JU~crnF ~ ~crnJUF, where (as in the proof of 1.8) JU is composition* * with the fiberwise join with a finite set U. ~ O crn is a three-step process: Compose with the (wh* *iskered) sum TYn-+!TY, then reduce in all variables (this was called r in the proof of 4.2), then comp* *ose with the equivalence of categories TY `...`Y-! TYn. Each of these steps commutes with JU up to natural equivalence. To get to 6.5 we must revisit and strengthen some results that were discussed* * in x3. The following is a strengthening of [2, 3.4]. CALCULUS III TAYLOR SERIES * * 39 6.6 Lemma. If a homotopy functor CY1x. .x.CYn F-!D is (d1, . .,.dn)-excisive th* *en ~F is (d1+. .+.dn)- excisive. Proof. In fact the proof of [2, 3.4] becomes a proof of 6.6 if one uses (X1, . * *.,.Xn) in place of (X, . .,.X) throughout. In particular ~F is n-excisive if F is (1, . .,.1)-excisive. We need to know * *also that ~F is n-homogeneous if F is multilinear. The following is a strengthening of 3.1. 6.7 Lemma. If CY1x . .x.CYn F-!D is (1, . .,.1)-reduced, then ~F is n-reduced. Proof. This follows from the next statement as 3.1 followed from 3.2. 6.8 Lemma. If CY1x . .x.CYn F-!D is (1, . .,.1)-reduced, then the map ~F -tn-1~F----!Tn-1~F factors through a weakly contractible functor. Proof. Again, the proof of 3.2 applies with no change except (X1, . .,.Xn) for * *(X, . .,.X). We need this partial converse to 6.6: 6.9 Lemma. If CY1x . .x.CYn L-!D is (1, . .,.1)-reduced and ~L is n-excisive, t* *hen L is multilinear. Proof. We have to show that L is 1-excisive in each variable. It is sufficient * *to consider the last variable. Fix an object Xj of CYjfor each 1 j n - 1and let A ----! B ?? ? y ?y C ----! D be any cocartesian square in CYn. We have to show that the square L(X1, . .,.Xn-1,-A)---!L(X1, . .,.Xn-1, B) ?? ? y ?y L(X1, . .,.Xn-1,-C)---!L(X1, . .,.Xn-1, D) is cartesian. Define an (n + 1)-cube in CY1x . .x.CYn as follows: For each subset S of {1, * *. .,.n - 1}, let Xj(S) be Yjif j 2 S and Xjif j =2S. Then {(X1(S), . .,.Xn-1(S))} constitutes an (n-1)-cu* *be in CY1x. .x.CYn-1 and our (n + 1)-cube will consist of the squares (X1(S), . .,.Xn-1(S),-A)---!(X1(S), . .,.Xn-1(S), B) ?? ? y ?y (X1(S), . .,.Xn-1(S),-C)---!(X1(S), . .,.Xn-1(S), D). The (n + 1)-cube is strongly cocartesian, so L yields a cartesian cube. On the* * other hand, for each nonempty S the square L(X1(S), . .,.Xn-1(S),-A)---!L(X1(S), . .,.Xn-1(S), B) ?? ? y ?y L(X1(S), . .,.Xn-1(S),-C)---!L(X1(S), . .,.Xn-1(S), D) 40 THOMAS G. GOODWILLIE is cartesian; in fact it is made up of contractible objects because L is reduce* *d in each variable. It follows by 1.6 of [2] that the square corresponding to S = ; is also cartesian. Proof of 6.5. Now it is convenient to drop the distinction between F and ~F. P1* *,...,1F is the universal example of a (1, . .,.1)-excisive functor under F, and is also n-excisive (by 6* *.6). PnF is the universal example of an n-excisive functor under F, and is also (1, . .,.1)-excisive (by * *6.9). It follows that they are the same. The reader, looking at 6.6 and 6.7, might have wondered about: 6.10 Lemma. If a homotopy functor CY1x. .x.CYn F-!D is (d1, . .,.dn)-reduced th* *en ~F is (d1+. .+.dn)- reduced. In fact this is true, and it can be deduced from 6.4. x7. Example: Suspension spectra of mapping spaces For an unbased space X let 1+X be the suspension spectrum of the based space* * X+ obtained by adding an extra point to X. We will call this the unreduced suspension spectrum* * (and hope that this does not lead anyone to confuse the unreduced suspension of X with S1 ^ X+). S* *imilarly, if X is a space fibered over Y we can speak of its unreduced fiberwise suspension spectru* *m, meaning the fiberwise suspension spectrum of the fiberwise based space _X, where as in x4 this means * *the disjoint union of X and Y considered as a an object of TY. For a finite CW complex K, the functor F(X) = 1+XK is analytic by [2, 4.4], * *and its first derivative was found in [1, 2.4]. We now find its nthderivative. We begin by recalling wha* *t the formula for the first derivative is and where that formula came from. In the notation of x5 the formula for the first derivative is Zk2K (7.1) @y 1+Y K~ 1+(Y, y)(K,k)dk. Recall that this means that the linearization of the functor Z 7! hofiber( 1+(Y _yZ)K -! 1+Y K) (7.2) ~ 1 hocofiber(Y K-! (Y _yZ)K ) is the functor Zk2K Z 7! Z ^ 1+(Y, y)(K,k)dk. Because K is finite, this last can also be written Z k2K (7.3) 1 (Z ^ (Y, y)(K,k)+) dk. Implicitly in 7.1 we are using a certain system of spectra on K, namely the unr* *educed fiberwise suspension spectrum of a certain space over K, let us call it W, whose fiber over k 2 K is* * Wk = (Y, y)(K,k); W is the subspace of Y Kx K consisting of pairs (f, k) such that f(k) = y. Likewise * *in 7.3 we are using the fiberwise suspension spectrum of a space over K, call it Z ^K _W, whose fiber i* *s Z ^ Wk+, namely colim(Z x W - W -!K). CALCULUS III TAYLOR SERIES * * 41 7.4 Remark. A formula for the differential and not just the derivative was give* *n in [1]. In the present notation it says that the unreduced-linear functor corresponding to DYF takes t* *he object X f-!Y to Z Z k2K 1+(Y, f(x))(K,k)dk dx. x2X In other words, the linear functor is given by a coefficient system on Y which * *may be obtained by "integration over the fiber" from a system on K x Y , the fiberwise unreduced s* *uspension spectrum of the fibration K x Y K-! K x Y whose fiber over (k, y) may be identified with (Y, y)* *(K,k). The method of proof for 7.1 was this: First give a natural map from 7.2 to 7.* *3, then show that its connectivity is roughly twice that of the space Z. To produce the map, it was enough to give a natural map of based spaces Z k2K hocofiber(Y K-! (Y _yZ)K ) -! 1 1 (Z ^ (Y, y)(K,k)+) dk. This was done by means of a tautological map from (Y _yZ)K to the space of sect* *ions of Z ^K _W -!K. f~ To specify this map we say where it sends the map K -!Y _yZ. Let f be the compo* *sed map f~ K -!Y _yZ -!Y Then ~fis sent to the section whose value at k is ~f(k) ^ f 2 Z ^ (Y, y)(K,k)+i* *f ~f(k) 2 Z and otherwise is the (fiberwise) basepoint. It is a good precaution to add a "whisker" to Z, replacing Y _yZ by Y [yI [zZ* *, before making the construction just described, to insure that the map Z ^Y _W -!K is a fibration. The proof that the resulting map from 7.2 to 7.3 is highly connected will not* * be repeated here. There is a variant of 7.1 for spaces of based maps: If K has a basepoint k0, * *then we obtain Zk2K-{k0} @y 1+(Y, y0)(K,k0)~ 1+(Y, y, y0)(K,k,k0)dk. c (This time the whisker is even more important, because it insures that even if * *y = y0 the section being constructed will have compact support.) Now we are in a position to compute a second derivative, using 6.2. We have: Z k22K @y1@y2 1+Y K~ @y1 1+(Y, y2)(K,k2)dk2 Zk22K ~ @y1 1+(Y, y2)(K,k2)dk2 Zk22K Zk12K-{k2} ~ 1+(Y, y1, y2)(K,k1,k2)dk1dk2 c Z(k1,k2)2K(2) (7.5) ~ 1+(Y, y1, y2)(K,k1,k2)d (k1, k2). c The differentiation under the integral sign depends on the hypothesis that K is* * finite. K(2)is the complement of the diagonal in K x K. The last step, replacing a double integral* * by a single integral, is a tautology. 42 THOMAS G. GOODWILLIE The expression in the last line implicitly refers to a system of spectra on K* *(2)whose fiber at (k1, k2) is 1+(Y, y1, y2)(K,k1,k2), namely the unreduced fiberwise suspension spectrum * *of a certain fibration W[2]-!K(2). The space W[2]is the subspace of Y K x K(2)consisting of all (f, k1, k2) such t* *hat f(k1) = y1 and f(k2) = y2, so that the fiber W[2](k1,k2)over (k1, k2) is (Y, y1, y2)(K,k1,k2). This is not enough to determine the quadratic functor D2FY, because we do not* * yet know the 2- symmetry in the second derivative. We can make a good guess about that, because* * the last expression in 7.5 has an obvious symmetry. To verify the guess, we can proceed as follows: We have to study the second-order cross-effect of FY as applied to objects Y * *_yZ, in other words the total homotopy fiber of 1+(Y _y1Z1_y2Z2)K----! 1+(Y _y1Z1)K ? ? (7.6) ?y ?y 1+(Y _y2Z2)K ----! 1+(Y )K . We know that the bilinearization of this is equivalent to Z(k1,k2)2K(2) Z1^ Z2^ 1+(Y, y1, y2)(K,k1,k2)d(k1, k2), c or equivalently Z (k1,k2)2K2- (7.7) 1 (Z1^ Z2^ (Y, y1, y2)(K,k1,k2)+) d(k1, k2). c This last expression refers to the fiberwise suspension spectrum of a certain s* *pace over K(2)whose fibers are Z1^ Z2^ W[2](k1,k2)+. Call it (Z1^ Z2) ^K(2)_W[2]. We are seeking to show that that equivalence can be chosen to preserve the 2* *-symmetry, so we should look for a symmetry-preserving map from the total homotopy fiber of 7.6 to 7.7. The total homotopy fiber of 7.6 can be rewritten as the suspension spectrum o* *f the total cofiber of (Y )K ----! (Y _y1Z1)K ? ? (7.8) ?y ?y (Y _y2Z2)K ----! (Y _y1Z1_y2Z2)K . There is a tautological map from this total cofiber to the zeroth space of 7.7.* * It arises from a tautological map from (Y _y1Z1_y2Z2)K to the space of compactly supported sections of (Z1^ Z* *2) ^Y W[2]-!K(2). f~ To specify this we say where it sends the map K -!Y _y1Z1_y2Z2. Let f be the co* *mposed map f~ K -!Y _y1Z1_y2Z2 -!Y Then ~fis sent to the section whose value at (k1, k2) is ~f(k1) ^ ~f(k2) ^ (f, * *k1, k2) if ~f(k1) 2 Z1 and f~(k2) 2 Z2 and otherwise the (fiberwise) basepoint. We claim, leaving the remaining details to the reader, that this results in a* * map Z (k1,k2)2K(2) @(2)y1,y2 1 (Y K+) -! 1+(Y, y1, y2)(K,k1,k2)d(k1, k2) c that corresponds to 7.5 under 6.2 and is therefore an equivalence. The same method gives the nthderivative. The conclusion is: CALCULUS III TAYLOR SERIES * * 43 7.10 Theorem. For a finite complex K we have a symmetry-preserving equivalence Zk2K(n) @(n)y1,...yn 1+Y K~ 1+(Y, y1, . .,.yn)(K,k1,...,kn)dk. c Here K(n)is the space of all ordered n-tuples k = (k1, . .,.kn) of distinct p* *oints in K. It is interesting to work out what this says in the case when K is a finite s* *et of cardinality m, and to compare it with the formula m(m - 1) . .(.m - n - 1)ym-n . for the nthderivative of ym in ordinary calculus. In the case when Y is a one-point space, the right-hand side of 7.10 becomes Z k2K(n) 1 S0dk ~ Map*(K(n)c, 1 S0) = (K(n)c)* c In other words, the nthderivative of F(X) = 1+XK at a point is the S-dual (K(n* *)c)* of a certain based n-space K(n)c, the one-point compactification of K(n)(or the quotient of* * Kn by the fat diagonal Kn - K(n)). The nthhomogeneous functor is (DnF)(X) ~ ((K(n)c)*^ X^n)h n, and this can be identified with Map*(K(n)c, 1 X^n)h n because K(n)cis finite. It can also be identified with Map*(K(n)c, 1 X^n) n because the group action on K(n)cis free. The analogous conclusion for based K says Zk2(K-{k0})(n) @(n)y1,...yn 1+(Y, y0)(K,k0)~ 1+(Y, y0, y1, . .,.yn)(K,k* *0,k1,...,kn)dk. c When Y is a point we find that the nthcoefficient of F(X) = 1+(X, x0)(K,k0)is * *the S-dual of the one-point compactification of (K - {k0})(n). The case when K is a based circle is particularly simple. Since (S1 - {k0})(n* *)is the disjoint union of open n-cells freely and transitively permuted by the group, the formula for the* * nthhomogeneous layer of 1+ X becomes n 1 (X^n), as already pointed out in 1.20. x8. Example: The identity functor The identity functor T I-!T from based spaces to based spaces is a central e* *xample, and its nth derivative @(n)I(*) is a basic object in homotopy theory. This spectrum with n* *-action turns out to be S-dual to a certain finite complex with n-action. We summarize and discuss som* *e known results. Note that determining the nthderivative of the identity is the same as determ* *ining the nthderivative of the functor . In fact, by 5.3 we have @(n) (*) = SV~n^ @(n)I(*) 44 THOMAS G. GOODWILLIE and therefore @(n)I(*) = ~Vn@(n) (*) To begin with the obvious, the first derivative of I is the sphere spectrum. The second derivative can be identified by using the second cross-effect. The* * total homotopy fiber of X1_ X2 ----! X1 ?? ? y ?y X2 ----! * is the homotopy fiber of X1_ X2 -!X1x X2, and the bilinearization of this is Q* *(X1^ X2), with the obvious 2-symmetry (trivial action on the loop coordinate), simply because the* *re is a map (natural and symmetry-preserving) hofiber(X1_ X2 -!X1x X2) -! Q(X1^ X2) that is approximately 3k-connected when X1 and X2 are k-connected. It follows * *that @(2)I(*) is the -1-sphere with trivial action. For the nthderivative, partial information can be obtained from the Hilton-Mi* *lnor theorem [9]. Recall that this describes the space (X _ Y ) as a weak product (direct limit of fin* *ite products) of factors each of which has the form (X^a ^ Y ^b). The factors for a given pair (a, b) * *correspond to certain nested commutator expressions, an integral basis for the bidegree (a, b) summan* *d of a free Lie ring on two generators whose bidegrees are (1, 0) and (0, 1). Iteration yields a descri* *ption of (X1_ . ._.Xn) as a weak product of factors of the form (X^a11^ . .^.X^ann). The cross-effec* *t (crnI)(X1, . .,.Xn) is the product of those factors for which aj 1 for all j. The multilinearized cr* *oss-effect sees only those factors for which aj = 1 for all j. Thus the nthdifferential of the functor * * is a product of copies of Q(X1^ . .^.Xn), and the nthderivative is the product of a corresponding numb* *er of copies of the sphere spectrum. The number of copies is (n - 1)!, and they correspond to a bas* *is for the group Lie(n) generated by all Lie monomials in n variables such that each variable occurs ju* *st once (and of course considered modulo the Jacobi identity and antisymmetry). A standard choice of b* *asis for Lie(n) consists of the monomials [[. .[.[xß(1), xß(2)], xß(3)] . .,.xß(n-1)], xn], one for each permutation ß belonging to the subgroup n-1 n. It follows that* * the nthderivative of , regarded as a spectrum with n-1-action, is the suspension spectrum of the * *finite set ( n-1)+. Of course for determining DnI one needs the full n-action. The method outlin* *ed above even iden- tifies the action of n on the homology of the spectrum @(n)I(*) (a free abelia* *n group of rank (n - 1)! concentrated in degree 1 - n); it is the obvious action of n on Lie(n), twiste* *d by signs of permutations. But this is still insufficient for determining the homogeneous functor. Johnson [5] defined a based finite complex Kn with n-action whose S-dual is * *@(n)I(*). Her Kn was designed to admit an interesting map (crnI)(X1, . .,.Xn) -!Map*(Kn, X1^ . .^.Xn), both natural and symmetry-preserving, and she showed that after multilinearizat* *ion this map leads to an equivalence (D(n)I)(X1, . .,.Xn) -!Map*(Kn, Q(X1^ . .^.Xn)). Arone and Mahowald [6] later came up with another answer to the same question* *. It is defined in terms of the poset of all partitions (equivalence relations) on the set n_= {1,* * . .n.}. The poset has a maximal element (the trivial partition) and a minimal element (the improper par* *tition), so that the nerve CALCULUS III TAYLOR SERIES * * 45 of the poset is (for two reasons) contractible. The Arone-Mahowald version of K* *n can be described as the double suspension of the nerve of the poset of proper nontrivial partitions* *, or alternatively as the quotient of the nerve of all partitions by the union of the nerve of the nontri* *vial partitions and the nerve of the proper partitions. In [6] this model is justified by proving directly that it is (equivariantly)* * homotopy equivalent to Johnson's space, but it was actually discovered from a very different point of * *view, which is worked out in detail in [15]. There is a cosimplicial functor from spaces to spaces which * *has in degree d the functor Qd+1, iterated composition of Q with itself. When applied to 1-connected spaces* * it serves as a resolution of the identity functor. (In general it gives the Bousfield-Kan integral comple* *tion functor.) For each d the functor Qd+1 has a split Taylor tower, which can be read off from the Snait* *h splitting formula. It is rather easy to see that the nthcoefficient of the dthfunctor is (functoriall* *y in d, up to homotopy) the S-dual of the (discrete) space of d-simplices in Kn. (The details worked ou* *t in [15] dispose of that unfortunate ü p to homotopy".) Arone and Mahowald use this description of Kn to investigate the mod p homolo* *gy of the spectrum (@(n)I(*) ^ X^n)h n whose zeroth space is (DnI)(X). Their main results concern * *the case when X is a sphere. For simplicity take it to be an odd sphere. When n is not a power of * *p they find that the layer (DnI)S2m-1 is p-locally trivial. (This is equivalent to the statement tha* *t the homology of n with coefficients in Lie(n) localized at p is trivial.) When n = pk they use Dyer-La* *shof operations to calculate the homology as a module over the Steenrod algebra, finding in particular that * *the layer (DpkI)S2m-1 has trivial vj-periodic homotopy when j < k. For further insight into these matters, see [14]. We close with some remarks about the derivatives of the identity at a general* * space Y . Recall from the discussion following 6.3 that the nthderivative of T -I!T will be a functor* * of spaces with n + 1 basepoints, symmetric with respect to permutations of the last n points. It is * *clear that @y(Y, y0) ~ 1+Pyy0Y. where Pyy0Y is the path space (Y, y0, y)(I,0,1). In general, the nthderivative * *of the identity at an arbitrary space may be described in terms of the nthderivative of the identity at a point* *. For example, given a space Y and points y0, y1, y2, the total homotopy fiber (with respect to y0) of Y _y1Z1_y2Z2 ----! Y _y1Z1 ?? ? y ?y Y _y2Z2 ----! Y is equivalent to the total homotopy fiber of (Py1y0Y+ ^ Z1) _ (Py2y0Y+-^-Z2)--!Py1y0Y+ ^ Z1 ?? ? y ?y Py2y0Y+ ^ Z2 ----! *. (The second square consists essentially of the homotopy fibers over y0 2 Y of t* *he spaces in the first square.) Bilinearizing with respect to Z1 and Z2, we find that (D(2)(Y,y0)I)(Y _y1Z1, Y _y2Z2) ~ (D(2)*I)(Py1y0Y+ ^ Z1, Py2y0Y+ * *^ Z2). The same argument applies for any n and yields a natural and symmetrical equiva* *lence @(n)y1,...,yn(Y, y0) ~ Py1y0Y+ ^ . .^.Pyny0Y+ ^ K*n 46 THOMAS G. GOODWILLIE These observations can be used to give an alternative to the Hilton-Milnor ar* *gument above; if one is willing to settle for n-1- rather than n-symmetry, then by 6.3 one can write @(n)y1,...,yn(Y, y0) ~ @(n-1)y1,...,yn-1 1+Pyny0Y, which by another very slight generalization of 7.10 is equivalent to Zk2(I-{0,1})(n-1) 1+(Y, y0, y1, . .,.yn-1, yn)(I,0,k1,...,kn-1,1)d* *k. c In the case Y = * this becomes the S-dual of the one-point compactification of * *(I - {0,(1})n-1), in other words the S-dual of a wedge of (n - 1)-spheres freely and transitively permuted* * by n-1. In fact, by taking a different point of view, one can see that the nthdiffere* *ntial or derivative of the identity has a n+1-symmetry and not just a n-symmetry, just as if the identit* *y were itself a derivative. We will return to this point in [7]. x9. Example: Waldhausen K-theory Let A be Waldhausen's algebraic K-theory functor from spaces to spectra [10].* * In [1] it was shown that (9.1) @yA(Y ) ~ 1+ Y . We will give similar formulas for the higher derivatives of A. Really what was shown in [1] was (9.2) @yPDiff(Y ) ~ 2 1 Y , where PDiff is stable smooth pseudoisotopy theory. Then 9.1 was a corollary in * *view of Waldhausen's relation (9.3) A(X) ~ WhDiffX x 1+X. where the Whitehead functor WhDiffsatisfies 2 1 WhDiffX ~ PDiffX. A natural map 1 PDiffX -! 2Q(XS =X) played a key role in obtaining 9.2, where XS1=X is the quotient of the free loo* *pspace XS1 by the constant loops. It was then clear that a more direct account of 9.1 ought to involve som* *e analogous map A(X) ø-!L(X), where L(X) = 1+XS1, a map that ought to have a K-theoretic rather than a manif* *old-theoretic de- scription. It is not hard to make such a map (the trace ), using the methods of* * [11]. 9.4 Remark. This was generalized by Bokstedt. The trace map ø is reminiscent of* * the Dennis trace map from the algebraic K-theory of a ring to its Hochschild homology, and this obse* *rvation pointed the way to the generalization. Since A(BG) can be defined as the algebraic K-theory of * *a generalized ring which may be thought of as the group ring of G over the sphere spectrum, and since th* *e Hochschild homology CALCULUS III TAYLOR SERIES * * 47 of a group ring k[G] is the homology of (BG)S1 with coefficients in k, it was n* *atural to imagine that ø should be a special case of a construction K(R) -!THH(R) defined for reasonable ring spectra R, where THH(R) is a kind of öH chschild ho* *mology over the sphere spectrum". Bokstedt invented (and named) the object THH and defined the trace (* *originally for a class of generalized rings called functors with smash product ). See [13]. The trace induces a map @yA(Y ) @yø--!@yL(Y ) By 7.1 we have Z k2S1 1 @yL(Y ) ~ 1+(Y, y)(Sd,k)k. Using rotations of the circle to continuously identify (S1, k) with (S1, 1), wh* *ere 1 is one point in S1, this last spectrum may be identified with Z k2S1 1 1+(Y, y)(Sd,1)k = Map*(S1+, 1+ yY ), and thus split into two factors 1+ yY x 1+ yY . Projecting on the first fact* *or and composing with @yø we get a map @yA(Y ) -! 1+ yY, which is in fact an equivalence. 9.5 Remark. It would have been very tedious to prove this last fact directly fr* *om 9.2. That would have meant combining the proofs of 9.2 and 9.3 and the construction of the trace and* * undoubtedly dealing with several different models for A(X). Instead in [3] we took a short cut, obs* *erving that the functors @yA(Y ) and 1+ yY are abstractly equivalent by 9.1, and then arguing by univer* *sal examples that a natural map between them must be an equivalence for all Y if this is so in some* * special cases where things can be checked. It is possible, and very convenient, to use a homotopy fixed point constructi* *on to single out the factor of @yL(Y ) that is to correspond to @yA(Y ). Consider the obvious action of the* * circle group T on L. The trace can easily be made to factor through the homotopy fixed point spectrum A(Y ) ~ø-!L(Y )hT -!L(Y ), and it was shown in [3] that the map (9.6) @yA(Y ) -!(@yL(Y ))hT resulting from this refined trace ~øis an equivalence. This form of 9.1 will be* * very useful for getting the higher derivatives. We emphasize that the right hand side of 9.6 is not @y((L(Y )hT), and that Lh* *T is not an analytic functor. There is a canonical map @y((L(Y )hT) -!(@yL(Y ))hT, because T acts continuously on the homotopy functor L; but this map does not ha* *ppen to be an equiva- lence. 48 THOMAS G. GOODWILLIE 9.7 Theorem. For all n 1 the (natural and symmetry-preserving) map @(n)y1,...,ynA(Y ) -!(@(n)y1,...,ynL(Y))hT induced by ~øis an equivalence. Proof. In proving this we are allowed to ignore the symmetry; thus for this pur* *pose we can get away with using 6.3 to view nthderivatives as first derivatives of (n - 1)stderivatives. The key point now is that the operations @y and ()hT, which did not commute w* *hen applied to L(Y ), do commute when applied to @(n-1)y2,...,ynL(Y.)Once this is established we can * *argue by induction on n: @y1@(n-1)y2,...,ynA(Y ) ~ @y1((@(n-1)y2,...,ynL(Y))hT) ~ (@y1@(n* *-1)y2,...,ynL(Y))hT. What is needed to establish it is the equation Zk2(S1)(n-1) 1 (9.8) @(n-1)y2,...,ynL(Y~) 1+(Y, y2, . .,.yn)(S ,k2,...,kn* *)dk c (an instance of 7.10), together with the observation that T is acting freely on* * (S1)(n-1). Now apply the principle given at the end of x5, taking G to be T, P to be (S1* *)(n-1)and B to be the orbit space. It is clear from 9.6 that for a suitable system E, functorial in (* *Y, y2, . .,.yn), we have Zp2P @(n-1)y2,...,ynL(Y~) Eß(p)dp equivariantly and Z p2P Z b2B @y(( Eß(p)dp)hT)~ @y Ebdb Zb2B ~ @yEbdb Z p2P ~ ( @yEß(p)dp)hT Z p2P ~ (@y Eß(p)dp)hT. The differentiations under the integral are justified by the finiteness of B an* *d of P. (This step fails when n = 1, basically because the appropriate B would then be the space BT, which is* * not so finite.) In the special case Y = * the conclusion of 9.7 is that the nthderivative of * *A is the homotopy fixed point spectrum for T acting on the S-dual of the one-point compactification of * *(S1)(n), where (S1)(n) has the obvious commuting actions of T and n. Since the T-action is free, the * *answer may be rewritten as the S-dual of the one-point compactification of (S1)(n)=T. As a n-space, (S* *1)(n)=T is isomorphic to nxCn(S1x ~VCn), where the cyclic group Cn acts on S1 as a subgroup of the rota* *tions and acts linearly on the vector space ~VCnby the reduced regular representation. It follows that * *the nthderivative of A at a one-point space is induced from Cn and the nthlayer of the Taylor tower is gi* *ven by (DnA)(X) ~ Map*(S1+ ^ SV~Cn, 1 (X^n))Cn. CALCULUS III TAYLOR SERIES * * 49 9.9 Remark. In the special case when X is a suspension Y this becomes Map*(S1* *+, 1 (X^n))Cn. In fact, in that case the Taylor tower splits: Y A( Y ) ~ A(*) x Map*(S1+, 1 (X^n))Cn n 1 if Y is connected. A proof of this was sketched in [12] and corrected in [3]. * * This special case of the conclusion was used in the short cut mentioned in 9.5. 9.10 Remark. There is also a direct K-theoretic approach to all of this. Dunda* *s and McCarthy [16] generalized 9.1 to (generalized) rings. 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