  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.
