The B-infinity-Structure on the Derived Endomorphism Algebra of the Unit in a Monoidal Category

Abstract

Consider a monoidal category that 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-infinity-algebra that is A(infinity)-quasi-isomorphic to the derived endomorphism algebra of the tensor unit. This B-infinity-algebra is obtained as the co-Hochschild complex of a projective resolution of the tensor unit, endowed with a lifted A(infinity)-coalgebra structure. We show that in the classical situation of the category of bimodules over an algebra, this newly defined B-infinity-algebra is isomorphic to the Hochschild complex of the algebra in the homotopy category of B-infinity-algebras.