Linearization, Dold-Puppe stabilization, and Mac Lane’s 𝑄-construction

Abstract

Abstract. In this paper we study linear functors, i.e., functors of chain complexes of modules which preserve direct sums up to quasi-isomorphism, in order to lay the foundation for a further study of the Goodwillie calculus in this setting. We compare the methods of Dold and Puppe, Mac Lane, and Goodwillie for producing linear approximations to functors, and establish conditions under which these methods are equivalent. In addition, we classify linear functors in terms of modules over an explicit differential graded algebra. Several classical results involving Dold-Puppe stabilization and Mac Lane's Q-construction are extended or given new proofs. IntroductionDerived functors are a fundamental tool in homological algebra. However, to have a good notion of derived functors, one imposes certain conditions (additivity, right exactness) on the functor with which one starts. These conditions raise some natural questions-for example, can one give a well-defined notion of a derived functor in the absence of these conditions? One solution to this problem is described by Dold and Puppe in [D-P]. For a non-additive functor F from an additive category (with enough projectives) to an abelian category, Dold and Puppe replace F by a functor that preserves direct sums up to quasi-isomorphism and agrees with F up to a certain degree in homology. With these properties, the Dold-Puppe stabilization of F can be treated as a linear approximation to F . Some logical next questions to raise are "Can we construct degree n approximations to F , and what kind of functors would such approximations be?"The answer to the existence question is provided in a series of papers by Tom Goodwillie ([G1], [G2], [G3]) in which he develops the theory of calculus of homotopy functors. In particular, for a functor F of topological spaces satisfying certain connectivity conditions, Goodwillie constructs a tower of functors and natural transformations as shown in Figure 1, in which each P n F is a degree n functor and approximates F in a range that tends to infinity with n. Although this is a result about functors of topological spaces, rather than functors of additive categories, we will see that any functor of R-modules for a ring R can be extended to a functor of spaces for which such a tower exists. In this way, we can establish that the degree n approximations to such functors exist by virtue of existence in the topological case.Unfortunately, this means of proving the existence of the degree n approximations is not very satisfactory to the algebraist. Our goal is to develop Goodwillie'sReceived by the editors July 16, 1996. 1991 Mathematics Subject Classification. Primary 18G99, 18E25, 55P65, 55U99. This work was supported by National Science Foundation grant # 1-5-30943. . . .calculus of functors in a purely algebraic fashion. In this paper, we will concentrate on the case where n = 1 and will provide two descriptions of P 1 F as well as a classification of them. In the first of two sequels to this paper ([J-M1]), we will prove the existence and...

Models for the Maclaurin tower of a simplicial functor via a derived Yoneda embedding

“…For example, if F evaluated on the final object is weakly equivalent to the final object, F( * ) ∼ * , then P 1 F(X) = Ω ∞ F(Σ ∞ X). This is the analog of the Dold-Puppe stabilization in the homological setting (see [18]). As n varies we form the 'Maclaurin tower' of a functor.…”

Directional derivatives and higher order chain rules for abelian functor calculus

“…When F is strictly reduced, there is a functor F : ChB → ChChA defined by simply applying the functor directly to each term in the chain complex. When F is also linear, [JM1,Lemma 5.4] show that the two prolongations Tot( F) and Tot(Ch(F)) are quasi-isomorphic. Lemma B.6 shows that when F is a linearization D 1 H of some functor H : B A, in which case Lemma 5.6(i) shows that F is strictly reduced and linear in the sense of Definition 5.5 (preserving finite direct sums up to natural chain homotopy equivalence), these procedures are in fact chain homotopy equivalent.…”

Formal groups and stable homotopy of commutative rings

“…4] or [27, 6.2] for the definition. A convenient reference for the relationship between Dold-Puppe stabilization and MacLane's cubical construction is [27]. The cubical construction QG of a functor G ∈ F is a chain complex of functors, concentrated in non-negative dimensions, with the following properties:…”

“…(c) In dimension zero, (QG) 0 = G and in positive dimensions QG is a finite sum of higher order cross-effects (see [15,Sec.9] or [27,Sec. 7]) of G.…”