Deriving calculus with cotriples
Authors:
B. Johnson and R. McCarthy
Journal:
Trans. Amer. Math. Soc. 356 (2004), 757803
MSC (2000):
Primary 18G30; Secondary 55P65, 55U15
Published electronically:
August 21, 2003
MathSciNet review:
2022719
Fulltext PDF Free Access
Abstract 
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Abstract: We construct a Taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors. The tower was inspired by Goodwillie's Taylor tower for functors of spaces, and is related to Dold and Puppe's stable derived functors and Mac Lane's construction. We study the layers, , and the limit of the tower. For the latter we determine a condition on the cross effects that guarantees convergence. We define differentials for functors, and establish chain and product rules for them. We conclude by studying exponential functors in this setting and describing their Taylor towers.
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 [A]
 G. Arone, The MitchellRichter filtration of loops on Stiefel manifolds stably splits, Proc. Amer. Math. Soc. 129 (2001), 12071211. MR 2002d:55004
 [AM]
 G. Arone and M. Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999), no. 3, 743788. MR 2000e:55012
 [Be]
 S. Betley, Stable derived functors, the Steenrod algebra and homological algebra in the category of functors, Fund. Math. 168 (2001), no. 3, 279293. MR 2002f:18025
 [Bo]
 A. K. Bousfield, Homogeneous functors and their derived functors, unpublished manuscript.
 [BM]
 M. Basterra and R. McCarthy, homology, topological AndréQuillen homology and stabilization, Topology Appl. 121 (2002), no. 3, 551566. MR 2003d:13013
 [CCGH]
 G. Carlsson R. Cohen, T. Goodwillie, and W.C. Hsiang, The free loop space and the algebraic theory of spaces, Theory 1 (1987), no. 1, 5382. MR 88i:55002
 [DP]
 A. Dold and D. Puppe, Homologie nichtadditiver Funktoren. Anwendungen, Ann. Inst. Fourier (Grenoble) 11 (1961), 201312. MR 27:186
 [EM1]
 S. Eilenberg and S. Mac Lane, Homology theories for multiplicative systems, Trans. Amer. Math. Soc. 71 (1951), 294330. MR 13:314c
 [EM2]
 S. Eilenberg and S. Mac Lane, On the groups , II, Ann. of Math. 60 (1954), 49139. MR 16:391a
 [BKMM]
 A. D. Elmendorf, I. Kriz, M. A. Mandell, and S. P. May, Rings, modules, and algebras in stable homotopy theory, Math. Surveys and Monographs, Vol. 47, Amer. Math. Soc., Providence, RI, 1997.
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 T. Goodwillie, Calculus III: The Taylor series of a homotopy functor, in preparation.
 [JP]
 M. Jibladze and T. Pirashvili, Cohomology of algebraic theories, J. Algebra 137 (1991), 253296. MR 92f:18005
 [JM1]
 B. Johnson and R. McCarthy, Linearization, DoldPuppe stabilization, and Mac Lane's construction, Trans. Amer. Math. Soc. 350 (1998), 15551593. MR 98i:18007
 [JM2]
 B. Johnson and R. McCarthy, Taylor towers for functors of additive categories, J. Pure Appl. Algebra 137 (1999), 253284. MR 2000b:18018
 [JM3]
 B. Johnson and R. McCarthy, A classification of degree n functors, to appear in Cahiers Topologie Géom. Différentielle Catég.
 [KMc]
 R. Kantorovitz and R. McCarthy, The Taylor towers for rational algebraic theory and Hochschild homology, Homology Homotopy Appl. 4 (2002), no. 1, 191212.
 [KM]
 I. Kriz and J. P. May, Operads, algebras, modules and motives, Astérisque 233, Société Mathématique de France, 1995. MR 96j:18006
 [MO]
 A. MauerOats, Algebraic Goodwillie calculus and a cotriple model for the remainder, preprint.
 [M1]
 R. McCarthy, Relative algebraic theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 197222. MR 99e:19006
 [M2]
 R. McCarthy, Dual calculus for functors to spectra, Homotopy Methods in Algebraic Topology (Boulder, CO, 1999), 183215, Contemp. Math., 271, Amer. Math. Soc., Providence, RI, 2001. MR 2002c:18009
 [M]
 V. Minasian, AndréQuillen spectral sequence for THH, Topology Appl. 129 (2003), no. 3, 273280.
 [P]
 T. Pirashvili, Higher additivizations, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk. Gruzin. SS2 91 (1988), 4454. (Russian) MR 90k:18012
 [Q1]
 D. Quillen, Homotopical algebra, Lecture Notes in Mathematics, vol. 43, SpringerVerlag, Berlin, 1967. MR 36:6480
 [Q2]
 D. Quillen, On the (co)homology of commutative rings, Proc. Sympos. Pure Math. XVII, Amer. Math. Soc., Providence, RI, 1970, 6587. MR 41:1722
 [R]
 B. Richter, An AtiyahHirzebruch spectral sequence for topological AndréQuillen homology, J. Pure Appl. Algebra 171 (2002), no. 1, 5966. MR 2003c:13012
 [S]
 D. Simson, Stable derived functors of the second symmetric power functor, second exterior power functor and Whitehead gamma functor, Colloq. Math. 32 (1974), 4955. MR 50:10020
 [ST]
 D. Simson and A. Tyc, Connected sequences of stable derived functors and their applications, Dissertationes Math. 111 (1974), 171. MR 51:14030
 [We]
 C. Weibel, An introduction to homological algebra, Cambridge University Press, Cambridge, 1994. MR 95f:18001
 [W]
 M. Weiss, Orthogonal Calculus, Trans. Amer. Math. Soc. 347 (1995), no. 10, 37433796. MR 96m:55018
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Additional Information
B. Johnson
Affiliation:
Department of Mathematics, Union College, Schenectady, New York 12308
Email:
johnsonb@union.edu
R. McCarthy
Affiliation:
Department of Mathematics, University of Illinois at UrbanaChampaign, 1409 W. Green St., Urbana, Illinois 61801
Email:
randy@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S000299470303318X
PII:
S 00029947(03)03318X
Received by editor(s):
January 20, 1999
Received by editor(s) in revised form:
February 18, 2003
Published electronically:
August 21, 2003
Additional Notes:
The second author was supported by National Science Foundation grant # 1530943 and a Sloan Fellowship
Article copyright:
© Copyright 2003 American Mathematical Society
