Abstract
We establish a connection between abstract clones and operads, which implies that both clones and operads are particular instances of a more general notion. The latter is called W-operad (due to a close resemblance with operads) and can be regarded as a functor on a certain subcategory W, of the category of finite ordinals, with some rather natural properties. When W is a category whose morphisms are the various bijections, the variety of W-operads is rationally equivalent to the variety of operads in the traditional sense. Our main result claims that if W coincides with the category of all finite ordinals then the variety of W-operads is rationally equivalent to the variety of abstract clones.
operad abstract clone variety rational equivalence
Preview
Unable to display preview. Download preview PDF.
References
- 1.Skornyakov L. A. (ed.), General Algebra. Vol. 2 [in Russian], Nauka, Moscow (1991).Google Scholar
- 2.May J. P., The Geometry of Iterated Loop Spaces, Springer-Verlag, Berlin (1972). (Lecture Notes in Math.; 271).Google Scholar
- 3.Boardman J. M. and Vogt R. M., Homotopy Invariant Algebraic Structures on Topological Spaces [Russian translation], Mir, Moscow (1977).Google Scholar
- 4.Tronin S. N., “Abstract clones and operads,” in: Abstracts: Logic and Applications: The International Conference on the Occasion of the Sixtieth Birthday of Academician Yu. L. Ershov, Novosibirsk, 4–6 May 2000, Novosibirsk, 2000, p. 100.Google Scholar
- 5.Artamonov V. A., “Clones of polylinear operations,” Uspekhi Mat. Nauk, 24, No. 1, 47–59 (1969).Google Scholar
- 6.Ginzburg V. and Kapranov M., “Koszul duality for operads,” Duke Math. J., 76, No. 1, 203–272 (1994).Google Scholar
- 7.Ginzburg V. and Kapranov M., “Erratum to ‘Koszul duality for operads’,” Duke Math. J., 80, No. 1, 293 (1995).Google Scholar
- 8.Operads: Proc. of Renaissance Conferences. V. 202, J.-L. Loday, J. D. Stasheff, A. A. Voronov (Eds.), Contemp. Math., 1996.Google Scholar
- 9.Kapranov M., “Operads and algebraic geometry,” Proc. Intern. Congr. Math., Berlin, Aug. 18–27, 1998, V. II: Invited Lectures, Berlin, 1998. pp. 277–286. (Documenta Math. Extra Volume ICM. II).Google Scholar
- 10.Smirnov V. A., Simplicial and Operad Methods in Algebraic Topology, Amer. Math. Soc., Providence, R.I. (2001). (Translations of Math. Monographs; 198.)Google Scholar
- 11.Johnstone P. T., Topos Theory [Russian translation], Mir, Moscow (1986).Google Scholar
- 12.Mal'tsev A. I., “Structure characterization of some classes of algebras,” Dokl. Akad. Nauk SSSR, 120, No. 1, 29–32 (1958).Google Scholar
- 13.Pinus A. G., “Invariants of rational equivalence,” Sibirsk. Mat. Zh., 41, No. 2, 430–436 (2000).Google Scholar
- 14.Tronin S. N., “On varieties defined by polylinear identities,” in: Abstracts: XIX All-Union Algebraic Conference, 9–11 September 1987. Part 2, L'vov, 1987, p. 280.Google Scholar
- 15.Tronin S. N., “On some properties of finitary algebraic theories,” in: Abstracts: V Siberian School on Varieties of Algebraic Systems, 1–5 July 1988, Barnaul, 1988, pp. 68–70.Google Scholar
- 16.Tronin S. N., “On some properties of algebraic theories of varieties of linear algebras. I. Varieties defined by polylinear identities,” submitted to VINITI August 11, 1988, No. 6511–B88.Google Scholar
- 17.Tronin S. N., On Retractions of Free Algebras and Modules [in Russian], Dis. Kand. Fiz.-Mat. Nauk, Kishinív (1989).Google Scholar
- 18.Tronin S. N., “Varieties of superalgebras and linear operads,” in: The Theory of Functions, Its Applications, and Related Problems: Proceedings of the School-Conference Dedicated to the Occasion of the 130 Birthday of D. F. Egorov, Kazan', 13–18 September 1999, Kazansk. Mat. Obshch., Kazan', 1999, pp. 224–227.Google Scholar
- 19.Tronin S. N. and Kopp O. A., “Matrix linear operads,” Izv. Vuzov Mat., 6, 52–63 (2000).Google Scholar
Copyright information
© Plenum Publishing Corporation 2002