  Calabi-Yau algebras are particularly symmetric differential graded algebras.
There is a construction called `Calabi-Yau completion' which produces a
canonical Calabi-Yau algebra from any homologically smooth dg algebra.
Homologically smooth dg algebras also form a 2-category to which the
construction of `equivariant completion' can be applied. In this theory two
objects are called `orbifold equivalent' if there is a 1-morphism $X$ with
invertible quantum dimensions between them. Any such relation entails a whole
family of equivalences between categories. We show that an orbifold equivalence
between two homologically smooth and proper dg algebras lifts to an orbifold
equivalence between their Calabi-Yau completions under certain conditions on
$X$.
