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Functional equations and their related operads


Author: Vahagn Minasian
Journal: Trans. Amer. Math. Soc. 357 (2005), 4413-4443
MSC (2000): Primary 55U15; Secondary 18D50, 55P99
DOI: https://doi.org/10.1090/S0002-9947-05-03974-7
Published electronically: June 9, 2005
MathSciNet review: 2156716
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Abstract: Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed, and their Taylor towers are computed. We also show that these functors factor through objects enriched over the homology of little $n$-cubes operads and discuss the relationship between functors defined via functional equations and operads. In addition, we compute the differentials of the forgetful functor from the category of $n$-Poisson algebras in terms of the homology of configuration spaces.


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  • 1. G.Bredon. Topology and Geometry. New York: Springer-Verlag, 1993 MR 1224675 (94d:55001)
  • 2. F.Cohen. The Homology of $\mathcal{C}_{n+1}$-spaces, $n \geq 0$ in ``The homology of iterated loop spaces''. Lecture Notes in Math., 533, 207-351. 1976 MR 0436146 (55:9096)
  • 3. A.D.Elmendorf, I.Kriz, M.A.Mandell and J.P.May. Rings, Modules, and Algebras in Stable Homotopy Theory. Mathematical Surveys and Monographs. Vol. 47. AMS. 1996 MR 1417719 (97h:55006)
  • 4. E.Fadell and L.Neuwirth. Configuration Spaces. Math. Scan., 10, 119-126. 1962 MR 0141126 (25:4537)
  • 5. E.Getzler and J.D.S.Jones. Operads, Homotopy Algebra, and Iterated Integrals for Double Loop Spaces. Preprint. 1993
  • 6. T.Goodwillie. Calculus I: The First Derivative of Pseudoisotopy Theory. K-theory, 4, 1-27. 1990 MR 1076523 (92m:57027)
  • 7. T.Goodwillie. Calculus II: Analytic Functors. K-theory, 5, 295-332. 1992 MR 1162445 (93i:55015)
  • 8. T.Goodwillie. Calculus III: The Taylor Series of a Homotopy functor. Geometry and Topology, 7, 645-711. 2003 MR 2026544 (2005e:55015)
  • 9. B.Johnson and R.McCarthy. Deriving Calculus with Cotriples. Trans. Amer. Math. Soc., 356, 757-804. 2004 MR 2022719
  • 10. I.Kriz and J.P.May. Operads, Algebras, Modules and Motives. Astérisque, 233, 1995 MR 1361938 (96j:18006)
  • 11. R.McCarthy and V.Minasian. On Triples, Operads and Generalized Homogeneous Functors. Preprint, 2004
  • 12. J.Milnor and J.C.Moor. On the Structure of Hopf Algebras. Ann. of Math. (2), 81, 211-264. 1965 MR 0174052 (30:4259)
  • 13. V.Minasian. André-Quillen spectral sequence for $THH$. Topology Appl., 129, 273-280. 2003 MR 1962984 (2004c:55018)
  • 14. D.G.Quillen. On the (co-)homology of commutative rings. AMS Proc Symp. Pure Math., 17, 65-87. 1970 MR 0257068 (41:1722)
  • 15. A.Robinson. Gamma homology, Lie representations and $E_{\infty}$ multiplications. Invent. Math., 152, 331-348. 2003 MR 1974890 (2004c:55020)
  • 16. C.A.Weibel. An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, 38. 1994 MR 1269324 (95f:18001)

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Additional Information

Vahagn Minasian
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912-1917
Email: minasian@math.brown.edu

DOI: https://doi.org/10.1090/S0002-9947-05-03974-7
Keywords: Little $n$-cubes operads, Goodwillie calculus, functional equations
Received by editor(s): June 16, 2003
Published electronically: June 9, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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