Distributive laws and Koszulness
Annales de l'Institut Fourier, Volume 46 (1996) no. 2, p. 307-323
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.
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.
     author = {Markl, Martin},
     title = {Distributive laws and Koszulness},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {2},
     year = {1996},
     pages = {307-323},
     doi = {10.5802/aif.1516},
     zbl = {0853.18005},
     mrnumber = {97i:18008},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1996__46_2_307_0}
Markl, Martin. Distributive laws and Koszulness. Annales de l'Institut Fourier, Volume 46 (1996) no. 2, pp. 307-323. doi : 10.5802/aif.1516. http://www.numdam.org/item/AIF_1996__46_2_307_0/

[1] J. Beck, Distributive laws, Lecture Notes in Mathematics, 80 (1969), 119-140. | MR 39 #2842 | Zbl 0186.02902

[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 95j:14002 | Zbl 0820.14037

[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 90c:16016 | Zbl 0676.16022

[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 96a:18004 | Zbl 0855.18006

[7] M. Markl, Models for operads, Communications in Algebra, 24(4) (1996), 1471-1500. | MR 96m:18012 | Zbl 0848.18003

[8] J.P. May, The Geometry of Iterated Loop Spaces, volume 271 of Lecture Notes in Mathematics, Springer-Verlag, 1972. | MR 54 #8623b | Zbl 0244.55009