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.
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