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

Quasi-hereditary algebras, exact Borel subalgebras, A-infinity-categories and boxes