Title: Monodromy & Stratified Homotopy Theory
Speaker: Peter Haine (MIT)
Abstract: The data of a locally constant sheaf of $\mathbb{CC}$-vector spaces on a manifold $X$ is equivalent to the data of its monodromy representation of the fundamental groupoid of $X$. MacPherson generalized this monodromy equivalence in the context of stratified homotopy theory by showing that a constructible sheaf on a (suitably) stratified space $X$ specifies and is specified by its representation of the exit path category of $X$. More recent work of Ayala–Francis–Rozenblyum, Lurie, and others greatly generalizes MacPherson's observation as well as situates it in a more homotopical framework.
In this talk I will give a brief overview of the monodromy equivalence and some of its classical generalizations. I will then give a detailed discussion of stratified spaces and how to extract a homotopical version of MacPherson's exit path category from a stratified space, as well as a new result using this construction to identify the homotopy theory of stratified spaces. This talk precedes a joint paper with Clark Barwick, Saul Glasman, Denis Nardin, and Jay Shah.
Seminar URL: https://people.math.osu.edu/valenzuelavasquez.2/hts/