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

TL;DR

This paper constructs a $B_ fty$-algebra structure on the derived endomorphism algebra of the tensor unit in certain monoidal categories, linking it to Hochschild complexes and extending algebraic structures.

Contribution

It introduces a $B_ fty$-algebra structure on the derived endomorphism algebra in a broad class of monoidal categories, generalizing classical Hochschild complex results.

Findings

The $B_ abla$-algebra is $A_ abla$-quasi-isomorphic to the derived endomorphism algebra.

In bimodule categories, the $B_ abla$-algebra matches the Hochschild complex.

The construction applies to categories with enough projectives and flat projectives on the right.

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.