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In previous work, we studied quasi-BPS categories (of symmetric quivers withpotential, of preprojective algebras, of surfaces) and showed they haveproperties analogous to those of BPS invariants/ cohomologies. For example,quasi-BPS categories are used to formulate categorical analogues of the PBWtheorem for cohomological Hall algebras (of Davison-Meinhardt) and of theDonaldson-Thomas/BPS wall-crossing for framed quivers (of Meinhardt-Reineke).  The purpose of this paper is to make the connections between quasi-BPScategories and BPS cohomologies more precise. We compute the topologicalK-theory of quasi-BPS categories for a large class of symmetric quivers withpotential. In particular, we compute the topological K-theory of quasi-BPScategories for a large class of preprojective algebras, which we use (in adifferent paper) to compute the topological K-theory of quasi-BPS categories ofK3 surfaces. A corollary is that there exist quasi-BPS categories withtopological K-theory isomorphic to BPS cohomology.  We also compute the topological K-theory of categories of matrixfactorizations for smooth affine quotient stacks in terms of the monodromyinvariant vanishing cohomology, prove a Grothendieck-Riemann-Roch theorem formatrix factorizations, and check the compatibility between the Koszulequivalence in K-theory and dimensional reduction in cohomology.

Topological K-theory of quasi-BPS categories of symmetric quivers with potential