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.
Categories: math.KT