  Highest weight categories arising in Lie theory are known to be associated
with finite dimensional quasi-hereditary algebras such as Schur algebras or
blocks of category $\mathcal O$. An analogue of the PBW theorem will be shown
to hold for quasi-hereditary algebras: Up to Morita equivalence each such
algebra has an exact Borel subalgebra. The category $\mathcal{F}(\Delta)$ of
modules with standard (Verma, Weyl, \dots) filtration, which is exact, but
rarely abelian, will be shown to be equivalent to the category of
representations of a directed box. This box is constructed as a quotient of a
dg algebra associated with the $A_{\infty}$-structure on $\mathcal{F}(\Delta)$.
Its underlying algebra is an exact Borel subalgebra.
