  We concentrate on a treatment of a Higgs-Coulomb duality as an absence of
manifest phase transition between ordered and disordered phases of 2d
$\mathcal{N}=(2,2)$ theories. We consider these examples of QFTs in the
Schr\"odinger picture and identify Hilbert spaces of BPS states with morphisms
in triangulated categories of D-brane boundary conditions. As a result of
Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are
equivalent, this resembles an analog of homological mirror symmetry. Following
construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one
is able to introduce interface defects in these theories and associate them to
D-brane parallel transport functors. We concentrate on surveying simple
examples, analytic when possible calculations, numerical estimates and simple
physical picture behind curtains of geometric objects. Categorification of
hypergeometric series analytic continuation is derived as an Atiyah flop of the
conifold. Finally we arrive to an interpretation of the braid group action on
the derived category of coherent sheaves on cotangent bundles to flag varieties
as a categorification of Berry connection on the Fayet-Illiopolous parameter
space of a sigma-model with a quiver variety target space.
