  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.
