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Associated to any $A_\infty$ $k$-algebra $A$ the Hochschild cochain complex $CH^*(A)$ has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a Gerstenhaber algebra.

If two $A_\infty$ algebras are Morita equivalent, are their Hochschild cochain complexes isomorphic in (i) the category of $k$-dg-algebras and (ii) the category of $k$-dg-Lie algebras, both up to quasi-isomorphism? Are they isomorphic in some category that feels both structures together?

Now suppose that $\mathcal{C}$ is a dg-category over a field $k$. We say that the $k$-dg-algebra $CH^*(\mathcal{C}) = End(id_\mathcal{C})$ is the Hochschild cochain complex. Does $CH^*(\mathcal{C})$ have a bracket that generalizes the known one in the case that $\mathcal{C}$ is a (derived) category of modules? If two dg-categories are quasi-equivalent are their Hochschild cochain complexes quasi-isomorphic?

Is there a point of view that clarifies these issues?

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# Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence

Associated to any $A_\infty$ $k$-algebra $A$ the Hochschild cochain complex $CH^*(A)$ has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a Gerstenhaber algebra.

If two $A_\infty$ algebras are Morita equivalent, are their Hochschild cochain complexes isomorphic in (i) the category of $k$-dg-algebras and (ii) the category of $k$-dg-Lie algebras, both up to quasi-isomorphism? Are they isomorphic in some category that feels both structures together?

Now suppose that $\mathcal{C}$ is a dg-category over a field $k$. We say that the $k$-dg-algebra $CH^*(\mathcal{C}) = End(id_\mathcal{C})$ is the Hochschild cochain complex. Does $CH^*(\mathcal{C})$ have a bracket that generalizes the known one in the case that $\mathcal{C}$ is a (derived) category of modules? If two dg-categories are quasi-equivalent are their Hochschild cochain complexes quasi-isomorphic?

Is there a point of view that clarifies these issues?