Exodromy beyond conicality

Abstract We show that compact subanalytic stratified spaces and algebraic stratifications of real varieties have finite exit‐path ‐categories, refining the classical theorems of Lefschetz–Whitehead, Łojasiewicz, and Hironaka on the finiteness of the underlying homotopy types of these spaces. These stratifications are typically not conical; hence, we cannot rely on the currently available exodromy equivalence between constructible sheaves on a stratified space, which requires conicality as a fundamental hypothesis. Building on ideas of Clausen and Jansen, we study the class of exodromic stratified spaces, for which the conclusion of the exodromy theorem holds. We prove two new fundamental properties of this class of stratified spaces: coarsenings of exodromic stratifications are exodromic, and every morphism between exodromic stratified spaces induces a functor between the associated exit path ‐categories. As a consequence, we produce many new examples of exodromic stratified spaces, including: coarsenings of conical stratifications, locally finite subanalytic stratifications of real analytic spaces, and algebraic stratifications of real varieties. Our proofs are at the generality of stratified ‐topoi, hence apply to even more general situations such as stratified topological stacks. In a subsequent paper, we use the previously mentioned finiteness results to construct derived moduli stacks of constructible and perverse sheaves.