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Brackets and products from centres in extension categories

Domenico Fiorenza, Niels Kowalzig

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.

Brackets and products from centres in extension 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.
Paper Structure (33 sections, 20 theorems, 225 equations)

This paper contains 33 sections, 20 theorems, 225 equations.

Table of Contents

  1. Operations in extension categories
  2. The hash operation
  3. The tensor product
  4. The cup product
  5. The mystic hexagon
  6. Cocycle representatives
  7. The cup product in terms of cocycle representatives
  8. The bracket $\{\,\space,\,\}$ in terms of cocycle representatives
  9. An explicit approach for bialgebroids
  10. ${\rm Ext}$ groups and extension categories over bialgebroid modules
  11. The external cup product
  12. A cocycle representative for the cup product
  13. The cup product as internal operadic composition
  14. The external operadic composition
  15. The internal operadic composition
  16. ...and 18 more sections

Key Result

Theorem 1

Let $(U,A)$ be a left bialgebroid, $Z$ a braided commutative monoid in the left weak centre of the monoidal category of left $U$-modules, and $X$ a braided cocommutative comonoid in its right weak centre such that $(X,Z)$ constitutes a commuting pair. Then the cochain complex $\operatorname{Hom}_U(\

Theorems & Definitions (62)

  • Conjecture
  • Theorem : Theorem \ref{['extobroid']}
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • ...and 52 more