Coxeter gluing for automorphic categories

Coxeter groups are defined by gluing (as a colimit) from finite Coxeter subgroups. We propose a global analogue of this fact in Betti Langlands: for a degeneration family of algebraic curves over the complex numbers, we define a Coxeter gluing functor from the automorphic category of smooth curves to the limit of that of singular curves. We show that this functor is an equivalence if and only if the automorphic gluing functor of Nadler–Yun is an equivalence. In genus 1, we show that the Coxeter gluing functor is an equivalence. This is joint work in progress with David Nadler and Zhiwei Yun.