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We define and study a sheaf-theoretic cohomological Hall algebra for suitablygeometric Abelian categories $\mathcal{A}$ of homological dimension at mosttwo, and a sheaf-theoretic BPS algebra under the conditions that $\mathcal{A}$is 2-Calabi-Yau and has a good moduli space. We show that the BPS algebra forthe preprojective algebra $\Pi_Q$ of a totally negative quiver is the freealgebra generated by the intersection cohomology of the closure of the locusparameterising simple $\Pi_Q$-modules in the coarse moduli space.  We define and study the BPS Lie algebra of arbitrary 2-Calabi-Yau categories$\mathcal{A}$ for which the Euler form is negative on all pairs of non-zeroobjects, which recovers the BPS algebra as its universal enveloping algebra forsuch "totally negative" 2CY categories. We show that for totally negative 2CYcategories the BPS algebra is freely generated by intersection complexes ofcertain coarse moduli spaces as above, and the Borel-Moore homology of thestack of objects in such $\mathcal{A}$ satisfies a Yangian-type PBW theorem forthe BPS Lie algebra. In this way we prove the cohomological integrality theoremfor these categories.  We use our results to prove that for $C$ a smooth projective curve, and for$r$ and $d$ not necessarily coprime, there is a nonabelian Hodge isomorphismbetween the Borel-Moore homologies of the stack of rank $r$ and degree $d$Higgs bundles, and the appropriate stack of twisted representations of thefundamental group of $C$. In addition we prove the Bozec-Schiffmann positivityconjecture for totally negative quivers; we prove that their polynomialscounting cuspidal functions in the constructible Hall algebra for $Q$ havepositive coefficients, strengthening the positivity theorem for the Kacpolynomials of such quivers.

BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks