  In previous work, we studied quasi-BPS categories (of symmetric quivers with
potential, of preprojective algebras, of surfaces) and showed they have
properties analogous to those of BPS invariants/ cohomologies. For example,
quasi-BPS categories are used to formulate categorical analogues of the PBW
theorem for cohomological Hall algebras (of Davison-Meinhardt) and of the
Donaldson-Thomas/BPS wall-crossing for framed quivers (of Meinhardt-Reineke).
  The purpose of this paper is to make the connections between quasi-BPS
categories and BPS cohomologies more precise. We compute the topological
K-theory of quasi-BPS categories for a large class of symmetric quivers with
potential. In particular, we compute the topological K-theory of quasi-BPS
categories for a large class of preprojective algebras, which we use (in a
different paper) to compute the topological K-theory of quasi-BPS categories of
K3 surfaces. A corollary is that there exist quasi-BPS categories with
topological K-theory isomorphic to BPS cohomology.
  We also compute the topological K-theory of categories of matrix
factorizations for smooth affine quotient stacks in terms of the monodromy
invariant vanishing cohomology, prove a Grothendieck-Riemann-Roch theorem for
matrix factorizations, and check the compatibility between the Koszul
equivalence in K-theory and dimensional reduction in cohomology.
