  The Kontsevich-Soibelman wall-crossing formula is known to control the
jumping behavior of BPS state counting indices in four-dimensional theories
with $\mathcal{N}=2$ supersymmetry. The formula can take two equivalent forms:
a ``fermionic'' form with nice positivity properties and a ``bosonic'' form
with a clear physical interpretation. In an important class of examples, the
fermionic form of the formula has a mathematical categorification involving PBW
bases for a Cohomological Hall Algebra. The bosonic form lacks an analogous
categorification. We construct an equivalence of chain complexes which
categorifies the simplest example of the bosonic wall-crossing formula: the
bosonic pentagon identity for the quantum dilogarithm. The chain complexes can
be promoted to differential graded algebras which we relate to the PBW bases of
the relevant CoHA by a certain quadratic duality. The equivalence of complexes
then follows from the relation between quadratic duality and Koszul duality. We
argue that this is a special case of a general phenomenon: the bosonic
wall-crossing formulae are categorified to equivalences of $A_\infty$ algebras
which are quadratic dual to PBW presentations of algebras which underlie the
fermionic wall-crossing formulae. We give a partial interpretation of our
differential graded algebras in terms of a holomorphic-topological version of
BPS webs.
