  We improve the exodromy equivalence of MacPherson, Treumann and Lurie in
several ways: first, we allow stratified spaces that have locally weakly
contractible strata, rather than being locally of singular shape, we remove all
noetherianity assumptions and we consider more general coefficients (e.g.
compactly assembled or stable presentable $\infty$-categories). Furthermore,
our approach shows that the exodromy equivalence is functorial for every
morphism of stratified spaces. As an application, we construct, under suitable
finiteness assumptions, a higher Artin derived stack of hyperconstructible
hypersheaves and prove that every perversity function gives rise to an open
substack of perverse hypersheaves. Using the derived structure, we provide the
construction of a new cohomological Hall algebra, generalizing the ones of
character varieties studied in the past by Schiffmann-Vasserot, Davison and
Mistry.
