Graded Lie structure on cohomology of some exact monoidal categories
Yury Volkov, Sarah Witherspoon
TL;DR
This work addresses the Lie structure on the extension algebra ${\mathrm{Ext}}^*_{\mathcal{C}}(\mathbf{1},\mathbf{1})$ in exact monoidal categories by connecting Schwede–Hermann's topological loop definition with an algebraic framework based on homotopy liftings and $A_\infty$-coderivations. The authors extend algebraic techniques from $A_\infty$-coalgebras to exact monoidal categories where the unit $\mathbf{1}$ has a projective power-flat resolution, establishing a canonical Gerstenhaber algebra structure on ${\mathrm{Ext}}^*_{{\mathcal{C}}}(\mathbf{1},\mathbf{1})$ via a bracket defined by homotopy liftings and a coalgebra structure on the shifted resolution $P[-1]$. They relate this bracket to the topological construction of Schwede and Hermann and prove that, under their hypotheses, the two viewpoints yield the same Gerstenhaber bracket up to a sign, thereby answering Hermann's question in this generalized setting. The results bridge topology and algebra in the study of extension algebras, extending Deligne-type and Hochschild-type structures to a broader class of exact monoidal categories, and providing robust tools for computing Lie brackets via $A_\infty$-coderivations and homotopy liftings.
Abstract
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 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.