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For some exact monoidal categories, we describe explicitly a connectionbetween topological and algebraic definitions of the Lie bracket on theextension algebra of the unit object. The topological definition, due toSchwede and Hermann, involves loops in extension categories. The algebraicdefinition, due to the first author, involves homotopy liftings of maps. As aconsequence of our description, we prove that the topological definition indeedyields a Gerstenhaber algebra structure in this monoidal category setting. Thisanswers a question of Hermann for those exact monoidal categories in which theunit object has a particular type of resolution that is called power flat. Foruse in proofs, we generalize $A_{\infty}$-coderivation and homotopy liftingtechniques from bimodule categories to these exact monoidal categories.

Graded Lie structure on cohomology of some exact monoidal categories