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Consider a monoidal category which is at the same time abelian with enoughprojectives and such that projectives are flat on the right. We show that thereis a $B_{\infty}$-algebra which is $A_{\infty}$-quasi-isomorphic to the derivedendomorphism algebra of the tensor unit. This $B_{\infty}$-algebra is obtainedas the co-Hochschild complex of a projective resolution of the tensor unit,endowed with a lifted $A_{\infty}$-coalgebra structure. We show that in theclassical situation of the category of bimodules over an algebra, this newlydefined $B_{\infty}$-algebra is isomorphic to the Hochschild complex of thealgebra in the homotopy category of $B_{\infty}$-algebras.

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