Distributive laws and Koszulness

Markl, Martin

Annales de l'Institut Fourier, Tome 46 (1996) no. 2, pp. 307-323.

Une loi distributive est une façon de composer deux structures algébriques, disons $𝒰$ et $𝒱$, en une structure algébrique plus complexe $𝒲$. Le but de ce travail est de comprendre les lois distributives en termes d’opérades. Le résultat central dit que si les opérades correspondant à $𝒰$ et $𝒱$ sont de Koszul, alors l’opérade correspondant à $𝒲$ est aussi de Koszul. On donne une application à la cohomologie des espaces de configurations.

Distributive law is a way to compose two algebraic structures, say $𝒰$ and $𝒱$, into a more complex algebraic structure $𝒲$. The aim of this paper is to understand distributive laws in terms of operads. The central result says that if the operads corresponding respectively to $𝒰$ and $𝒱$ are Koszul, then the operad corresponding to $𝒲$ is Koszul as well. An application to the cohomology of configuration spaces is given.

[1] J. Beck, Distributive laws, Lecture Notes in Mathematics, 80 (1969), 119-140. | MR | Zbl

[2] T.F. Fox and M. Markl, Distributive laws, bialgebras, and cohomology, Contemporary Mathematics, to appear.

[3] W. Fulton and R. Macpherson, A compactification of configuration spaces, Annals of Mathematics, 139 (1994), 183-225. | MR | Zbl

[4] M. Gerstenhaber and S.D. Schack, Algebraic cohomology and deformation theory. In Deformation Theory of Algebras and Structures and Applications, pages 11-264. Kluwer, Dordrecht, 1988. | MR | Zbl

[5] E. Getzler and J.D.S. Jones, Operads, homotopy algebra, and iterated integrals for double loop spaces, preprint, 1993.

[6] V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. Journal, 76(1) (1994), 203-272. | MR | Zbl

[7] M. Markl, Models for operads, Communications in Algebra, 24(4) (1996), 1471-1500. | MR | Zbl

[8] J.P. May, The Geometry of Iterated Loop Spaces, volume 271 of Lecture Notes in Mathematics, Springer-Verlag, 1972. | MR | Zbl