Graded Lie structure on cohomology of some exact monoidal categories

For some exact monoidal categories, we describe explicitly a connection between topological and algebraic definitions of the Lie bracket on the extension algebra of the unit object. The topological definition, due to Schwede and to Hermann, involves loops in extension categories. The algebraic definition, due to the first author, involves homotopy liftings of maps. As a consequence of our description, we prove that the topological definition indeed yields a Gerstenhaber algebra structure in this monoidal category setting. This answers a question of Hermann for those exact monoidal categories in which the unit object has a particular type of resolution that is called power flat. For use in proofs, we generalize $A_\infty$-coderivation and homotopy lifting techniques from bimodule categories to these exact monoidal categories.

2010 Mathematics Subject Classification

16E40, 18G15

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The first author was supported by RFBR according to the research project 20-01-00030 and in part by a Young Russian Mathematics Award.

The second author was partially supported by NSF grants DMS-1401016 and DMS-1665286.

Received 5 March 2022

Received revised 31 July 2023

Accepted 6 October 2023

Published 18 September 2024