Take the 2-minute tour ×  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?  share|improve this quest