  In this note we review a construction of a BPS Hilbert space in an effective
supersymmetric quiver theory with 4 supercharges. We argue abstractly that this
space contains elements of an equivariant generalized cohomology theory
$E_G^{*}(-)$ of the quiver representation moduli space giving concretely
Dolbeault cohomology, K-theory or elliptic cohomology depending on the spacial
slice is compactified to a point, a circle or a torus respectively, and
something more amorphous in other cases. Furthermore BPS instantons -- basic
contributors to interface defects or a Berry connection -- induce a BPS algebra
on the BPS Hilbert spaces representing Fourier-Mukai transforms on the quiver
representation moduli spaces descending to an algebra over $E_G^{*}(-)$ as its
representation. In the cases when the quiver describes a toric Calabi-Yau
three-fold (CY${}_3$) the algebra is a respective generalization of the quiver
BPS Yangian algebra discussed in the literature, in more general cases it is
given by an abstract generalized cohomological Hall algebra.
