Deriving calculus with cotriples

Authors:
B. Johnson and R. McCarthy

Journal:
Trans. Amer. Math. Soc. **356** (2004), 757-803

MSC (2000):
Primary 18G30; Secondary 55P65, 55U15

Published electronically:
August 21, 2003

MathSciNet review:
2022719

Full-text 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 Mitchell-Richter filtration of loops on Stiefel manifolds stably splits*, Proc. Amer. Math. Soc.**129**(2001), no. 4, 1207–1211 (electronic). MR**1814154**, 10.1090/S0002-9939-00-05794-4**[A-M]**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**, 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 MR**1853410**, 10.4064/fm168-3-4**[Bo]**A. K. Bousfield,*Homogeneous functors and their derived functors*, unpublished manuscript.**[B-M]**Maria Basterra and Randy McCarthy,*Γ-homology, topological André-Quillen homology and stabilization*, Topology Appl.**121**(2002), no. 3, 551–566. MR**1909009**, 10.1016/S0166-8641(01)00098-0**[CCGH]**G. E. Carlsson, R. L. Cohen, T. Goodwillie, and W. C. Hsiang,*The free loop space and the algebraic 𝐾-theory of spaces*, 𝐾-Theory**1**(1987), no. 1, 53–82. MR**899917**, 10.1007/BF00533987**[D-P]**Albrecht Dold and Dieter Puppe,*Homologie nicht-additiver Funktoren. Anwendungen*, Ann. Inst. Fourier Grenoble**11**(1961), 201–312 (German, with French summary). MR**0150183****[E-M1]**Samuel Eilenberg and Saunders MacLane,*Homology theories for multiplicative systems*, Trans. Amer. Math. Soc.**71**(1951), 294–330. MR**0043774**, 10.1090/S0002-9947-1951-0043774-8**[E-M2]**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****[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*, 𝐾-Theory**4**(1990), no. 1, 1–27. MR**1076523**, 10.1007/BF00534191**[G2]**Thomas G. Goodwillie,*Calculus. II. Analytic functors*, 𝐾-Theory**5**(1991/92), no. 4, 295–332. MR**1162445**, 10.1007/BF00535644**[G3]**T. Goodwillie,*Calculus III: The Taylor series of a homotopy functor*, in preparation.**[J-P]**Mamuka Jibladze and Teimuraz Pirashvili,*Cohomology of algebraic theories*, J. Algebra**137**(1991), no. 2, 253–296. MR**1094244**, 10.1016/0021-8693(91)90093-N**[J-M1]**Brenda Johnson and Randy McCarthy,*Linearization, Dold-Puppe stabilization, and Mac Lane’s 𝑄-construction*, Trans. Amer. Math. Soc.**350**(1998), no. 4, 1555–1593. MR**1451606**, 10.1090/S0002-9947-98-02067-4**[J-M2]**Brenda Johnson and Randy McCarthy,*Taylor towers for functors of additive categories*, J. Pure Appl. Algebra**137**(1999), no. 3, 253–284. MR**1685140**, 10.1016/S0022-4049(97)00203-X**[J-M3]**B. Johnson and R. McCarthy,*A classification of degree n functors*, to appear in Cahiers Topologie Géom. Différentielle Catég.**[K-Mc]**R. Kantorovitz and R. McCarthy,*The Taylor towers for rational algebraic**-theory and Hochschild homology*, Homology Homotopy Appl.**4**(2002), no. 1, 191-212.**[K-M]**Igor Kříž and J. P. May,*Operads, algebras, modules and motives*, Astérisque**233**(1995), iv+145pp (English, with English and French summaries). MR**1361938****[MO]**A. Mauer-Oats,*Algebraic Goodwillie calculus and a cotriple model for the remainder*, preprint.**[M1]**Randy McCarthy,*Relative algebraic 𝐾-theory and topological cyclic homology*, Acta Math.**179**(1997), no. 2, 197–222. MR**1607555**, 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**, 10.1090/conm/271/04357**[M]**V. Minasian,*André-Quillen spectral sequence for THH*, Topology Appl.**129**(2003), no. 3, 273-280.**[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****[Q1]**Daniel G. Quillen,*Homotopical algebra*, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR**0223432****[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****[R]**Birgit Richter,*An Atiyah-Hirzebruch spectral sequence for topological André-Quillen homology*, J. Pure Appl. Algebra**171**(2002), no. 1, 59–66. MR**1903396**, 10.1016/S0022-4049(01)00117-7**[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****[S-T]**Daniel Simson and Andrzej Tyc,*Connected sequences of stable derived functors and their applications*, Dissertationes Math. (Rozprawy Mat.)**111**(1974), 67. MR**0377861****[We]**Charles A. Weibel,*An introduction to homological algebra*, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR**1269324****[W]**Michael Weiss,*Orthogonal calculus*, Trans. Amer. Math. Soc.**347**(1995), no. 10, 3743–3796. MR**1321590**, 10.1090/S0002-9947-1995-1321590-3

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 Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61801

Email:
randy@math.uiuc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-03-03318-X

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 # 1-5-30943 and a Sloan Fellowship

Article copyright:
© Copyright 2003
American Mathematical Society