Brackets and products from centres in extension categories

TL;DR

This paper generalizes the Gerstenhaber algebra structure on Ext groups from associative algebras to broad monoidal categories, using categorical and operadic methods to extend classical results.

Contribution

It introduces a vast generalization of Schwede's construction of Gerstenhaber algebras on Ext groups to monoidal categories with coefficients in (weak) monoidal centres.

Findings

Provides explicit algebraic structures in categories of modules over bialgebroids.

Proves the Gerstenhaber algebra structure via operadic methods.

Suggests a conjectural extension to arbitrary monoidal categories.

Abstract

Building on Retakh's approach to Ext groups through categories of extensions, Schwede reobtained the well-known Gerstenhaber algebra structure on Ext groups over bimodules of associative algebras both from splicing extensions (leading to the cup product) and from a suitable loop in the categories of extensions (leading to the Lie bracket). We show how Schwede's construction admits a vast generalisation to general monoidal categories with coefficients of the Ext groups taken in (weak) left and right monoidal (or Drinfel'd) centres. In case of the category of left modules over bialgebroids and coefficients given by commuting pairs of braided (co)commutative (co)monoids in these categorical centres, we provide an explicit description of the algebraic structure obtained this way, and a complete proof that this leads to a Gerstenhaber algebra is then obtained from an operadic approach. This, in particular, considerably generalises the classical construction given by Gerstenhaber himself. Conjecturally, the algebraic structure we describe should produce a Gerstenhaber algebra for an arbitrary monoidal category enriched over abelian groups, but even the bilinearity of the cup product and of the Lie-type bracket defined by the abstract construction in terms of extension categories remain elusive in this general setting.