The $B_{\infty }$-structure on the derived endomorphism algebra of the unit in a monoidal category

Abstract

Consider a monoidal category which is at the same time abelian with enough projectives and such that projectives are flat on the right. We show that there is a $B_{\infty }$-algebra which is $A_{\infty }$-quasi-isomorphic to the derived endomorphism algebra of the tensor unit. This $B_{\infty }$-algebra is obtained as the co-Hochschild complex of a projective resolution of the tensor unit, endowed with a lifted $A_{\infty }$-coalgebra structure. We show that in the classical situation of the category of bimodules over an algebra, this newly defined $B_{\infty }$-algebra is isomorphic to the Hochschild complex of the algebra in the homotopy category of $B_{\infty }$-algebras.

24 pages, 1 figure