  Together with Koenig and Ovsienko, the first author showed that every
quasi-hereditary algebra can be obtained as the (left or right) dual of a
directed bocs. In this monograph, we prove that if one additionally assumes
that the bocs is basic, a notion we define, then this bocs is unique up to
isomorphism. This should be seen as a generalisation of the statement that the
basic algebra of an arbitrary associative algebra is unique up to isomorphism.
The proof associates to a given presentation of the bocs an
$A_\infty$-structure on the $\operatorname{Ext}$-algebra of the standard
modules of the corresponding quasi-hereditary algebra. Uniqueness then follows
from an application of Kadeishvili's theorem.
