Isotopy invariance and stratified E₂-structure of the Ran Grassmannian

Let G be a complex reductive group. A folklore result asserts the existence of an E₂-algebra structure on the Ran Grassmannian of G over A¹C, seen as a topological space with the complex-analytic topology. The aim of this paper is to prove this theorem, by establishing a homotopy invariance result: namely, an inclusion of open balls D' ⊂ D in C induces a homotopy equivalence between the respective Beilinson--Drinfeld Grassmannians Gr_G, D'I GrG, DI, for any power I. We use a purely algebraic approach, showing that automorphisms of a complex smooth algebraic curve X can be lifted to automorphisms of the associated Beilinson--Drinfeld Grassmannian. As a consequence, we obtain a stronger version of the usual homotopy invariance result: namely, the homotopies can be promoted to equivariant stratified isotopies, where "equivariant" refers to the action of the arc group L⁺G and "stratified" refers to the stratification induced by the Schubert stratification of GrG and the incidence stratification of CI.