There are many situations in algebra, geometry, and topology where a category of sheaves that one would like to understand is expressible in terms of representations of a more fundamental combinatorial object. For example, the oo-category of local systems on a nice topological space T is equivalent to functors out of the underlying oo-groupoid of T. More generally, the oo-category of constructible sheaves on a nice stratified topological space T -> P is equivalent to functors out of the exit-path oo-category Exit(T) of T. Because of monodromy and exit-paths, we refer to such equivalences as exodromy equivalences.
We'll begin by surveying known exodromy results in topology and algebraic geometry. We'll then explain a program initiated by Nadler to bring exodromy to the setting of microlocal sheaf theory.