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It is easy to find algebras $\mathbb{T}\in\mathcal{C}$ in a finite tensorcategory $\mathcal{C}$ that naturally come with a lift to a braided commutativealgebra $\mathsf{T}\in Z(\mathcal{C})$ in the Drinfeld center of $\mathcal{C}$.In fact, any finite tensor category has at least two such algebras, namely themonoidal unit $I$ and the canonical end $\int_{X\in\mathcal{C}} X\otimesX^\vee$. Using the theory of braided operads, we prove that for any suchalgebra $\mathbb{T}$ the homotopy invariants, i.e. the derived morphism spacefrom $I$ to $\mathbb{T}$, naturally come with the structure of a differentialgraded $E_2$-algebra. This way, we obtain a rich source of differential graded$E_2$-algebras in the homological algebra of finite tensor categories. We usethis result to prove that Deligne's $E_2$-structure on the Hochschild cochaincomplex of a finite tensor category is induced by the canonical end, itsmultiplication and its non-crossing half braiding. With this new and moreexplicit description of Deligne's $E_2$-structure, we can lift theFarinati-Solotar bracket on the Ext algebra of a finite tensor category to an$E_2$-structure at cochain level. Moreover, we prove that, for a unimodularpivotal finite tensor category, the inclusion of the Ext algebra into theHochschild cochains is a monomorphism of framed $E_2$-algebras, therebyrefining a result of Menichi.

Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories