A diagram of algebras is a functor valued in a category of associative algebras. I construct an operad acting on the Hochschild bicomplex of a diagram of algebras. Using this operad, I give a direct proof that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra. I also show that the total complex is an $L_\infty$-algebra. The same results are true for the asimplicial subcomplex and its cohomology. This structure governs deformations of diagrams of algebras through the Maurer-Cartan equation.
|Publication status||Unpublished - 3 Feb 2020|
Bibliographical note60 pages
- 18D50 (primary) 16E40 (Secondary)