Cogroupoid structures on the circle and the Hodge degeneration

We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$-fold delooping of the filtered loop space $E_2$-groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT19]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$-cogroupoid object in the $\infty$-category of spaces. We relate this cogroupoid structure with the more commonly studied "pinch map" on $S^1$, as well as the Todd class of the Lie algebroid $\mathbb{T}_{X}$; this is an invariant of a smooth and proper scheme $X$ that arises, for example, in the Grothendieck-Riemann Roch theorem. In particular we relate the existence of non-trivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally we record some consequences of this bit of structure at the level of Hochschild cohomology.