Topological exodromy with coefficients

I will survey ongoing work with Jean-Baptiste Teyssier. Motivated by the study of wild character varieties, we were led to revisit and improve Treumann-Lurie's results on exodromy in the topological setting. In this seminar I will explain how to realize the exodromy equivalence via an explicit push-pull ; as an intermediate step, we establish (folkloristic) formulas for the stalks of the constructible sheaves obtained via exodromy. As a consequence, we manage to drop many assumptions required in the approach described in Higher Algebra. If time permits, I'll sketch a couple of applications to the construction of algebraic moduli spaces.