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 
References 
Similar Articles 
Additional Information
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.
 [A]
Greg
Arone, The MitchellRichter filtration of
loops on Stiefel manifolds stably splits, Proc.
Amer. Math. Soc. 129 (2001), no. 4, 1207–1211 (electronic). MR 1814154
(2002d:55004), http://dx.doi.org/10.1090/S0002993900057944
 [AM]
Greg
Arone and Mark
Mahowald, The Goodwillie tower of the identity functor and the
unstable periodic homotopy of spheres, Invent. Math.
135 (1999), no. 3, 743–788. MR 1669268
(2000e:55012), http://dx.doi.org/10.1007/s002220050300
 [Be]
Stanisław
Betley, Stable derived functors, the Steenrod algebra and
homological algebra in the category of functors, Fund. Math.
168 (2001), no. 3, 279–293. MR 1853410
(2002f:18025), http://dx.doi.org/10.4064/fm16834
 [Bo]
A. K. Bousfield, Homogeneous functors and their derived functors, unpublished manuscript.
 [BM]
Maria
Basterra and Randy
McCarthy, Γhomology, topological AndréQuillen
homology and stabilization, Topology Appl. 121
(2002), no. 3, 551–566. MR 1909009
(2003d:13013), http://dx.doi.org/10.1016/S01668641(01)000980
 [CCGH]
G.
E. Carlsson, R.
L. Cohen, T.
Goodwillie, and W.
C. Hsiang, The free loop space and the algebraic 𝐾theory
of spaces, 𝐾Theory 1 (1987), no. 1,
53–82. MR
899917 (88i:55002), http://dx.doi.org/10.1007/BF00533987
 [DP]
Albrecht
Dold and Dieter
Puppe, Homologie nichtadditiver Funktoren. Anwendungen, Ann.
Inst. Fourier Grenoble 11 (1961), 201–312 (German,
with French summary). MR 0150183
(27 #186)
 [EM1]
Samuel
Eilenberg and Saunders
MacLane, Homology theories for multiplicative
systems, Trans. Amer. Math. Soc. 71 (1951), 294–330. MR 0043774
(13,314c), http://dx.doi.org/10.1090/S00029947195100437748
 [EM2]
Samuel
Eilenberg and Saunders
Mac Lane, On the groups 𝐻(Π,𝑛). II. Methods of
computation, Ann. of Math. (2) 60 (1954),
49–139. MR
0065162 (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.
 [G1]
Thomas
G. Goodwillie, Calculus. I. The first derivative of pseudoisotopy
theory, 𝐾Theory 4 (1990), no. 1,
1–27. MR
1076523 (92m:57027), http://dx.doi.org/10.1007/BF00534191
 [G2]
Thomas
G. Goodwillie, Calculus. II. Analytic functors,
𝐾Theory 5 (1991/92), no. 4, 295–332.
MR
1162445 (93i:55015), http://dx.doi.org/10.1007/BF00535644
 [G3]
T. Goodwillie, Calculus III: The Taylor series of a homotopy functor, in preparation.
 [JP]
Mamuka
Jibladze and Teimuraz
Pirashvili, Cohomology of algebraic theories, J. Algebra
137 (1991), no. 2, 253–296. MR 1094244
(92f:18005), http://dx.doi.org/10.1016/00218693(91)90093N
 [JM1]
Brenda
Johnson and Randy
McCarthy, Linearization, DoldPuppe
stabilization, and Mac Lane’s 𝑄construction, Trans. Amer. Math. Soc. 350 (1998), no. 4, 1555–1593. MR 1451606
(98i:18007), http://dx.doi.org/10.1090/S0002994798020674
 [JM2]
Brenda
Johnson and Randy
McCarthy, Taylor towers for functors of additive categories,
J. Pure Appl. Algebra 137 (1999), no. 3,
253–284. MR 1685140
(2000b:18018), http://dx.doi.org/10.1016/S00224049(97)00203X
 [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]
Igor
Kříž and J.
P. May, Operads, algebras, modules and motives,
Astérisque 233 (1995), iv+145pp (English, with
English and French summaries). MR 1361938
(96j:18006)
 [MO]
A. MauerOats, Algebraic Goodwillie calculus and a cotriple model for the remainder, preprint.
 [M1]
Randy
McCarthy, Relative algebraic 𝐾theory and topological
cyclic homology, Acta Math. 179 (1997), no. 2,
197–222. MR 1607555
(99e:19006), http://dx.doi.org/10.1007/BF02392743
 [M2]
Randy
McCarthy, Dual calculus for functors to spectra, Homotopy
methods in algebraic topology (Boulder, CO, 1999) Contemp. Math.,
vol. 271, Amer. Math. Soc., Providence, RI, 2001,
pp. 183–215. MR 1831354
(2002c:18009), http://dx.doi.org/10.1090/conm/271/04357
 [M]
V. Minasian, AndréQuillen spectral sequence for THH, Topology Appl. 129 (2003), no. 3, 273280.
 [P]
T.
I. Pirashvili, Higher additivizations, Trudy Tbiliss. Mat.
Inst. Razmadze Akad. Nauk Gruzin. SSR 91 (1988),
44–54 (Russian, with English summary). MR 1029006
(90k:18012)
 [Q1]
Daniel
G. Quillen, Homotopical algebra, Lecture Notes in Mathematics,
No. 43, SpringerVerlag, BerlinNew York, 1967. MR 0223432
(36 #6480)
 [Q2]
Daniel
Quillen, On the (co) homology of commutative rings,
Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII,
New York, 1968) Amer. Math. Soc., Providence, R.I., 1970,
pp. 65–87. MR 0257068
(41 #1722)
 [R]
Birgit
Richter, An AtiyahHirzebruch spectral sequence for topological
AndréQuillen homology, J. Pure Appl. Algebra
171 (2002), no. 1, 59–66. MR 1903396
(2003c:13012), http://dx.doi.org/10.1016/S00224049(01)001177
 [S]
Daniel
Simson, Stable derived functors of the second symmetric power
functor, second exterior power functor and Whitehead gamma functor,
Colloq. Math. 32 (1974), 49–55. MR 0357552
(50 #10020)
 [ST]
Daniel
Simson and Andrzej
Tyc, Connected sequences of stable derived functors and their
applications, Dissertationes Math. (Rozprawy Mat.)
111 (1974), 67. MR 0377861
(51 #14030)
 [We]
Charles
A. Weibel, An introduction to homological algebra, Cambridge
Studies in Advanced Mathematics, vol. 38, Cambridge University Press,
Cambridge, 1994. MR 1269324
(95f:18001)
 [W]
Michael
Weiss, Orthogonal calculus, Trans. Amer. Math. Soc. 347 (1995), no. 10, 3743–3796. MR 1321590
(96m:55018), http://dx.doi.org/10.1090/S00029947199513215903
 [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.
 [G1]
 T. Goodwillie, Calculus I: The first derivative of pseudoisotopy theory, Theory 4 (1990), 127. MR 92m:57027
 [G2]
 T. Goodwillie, Calculus II: Analytic functors, Theory 5 (1992), 295332. MR 93i:55015
 [G3]
 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
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
18G30,
55P65,
55U15
Retrieve articles in all journals
with MSC (2000):
18G30,
55P65,
55U15
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
