Topological exodromy with coefficients

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.