Brackets and products from centres in extension categories
Domenico Fiorenza, Niels Kowalzig

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,…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Topics in Algebra · Algebraic structures and combinatorial models
