Properadic coformality of spheres

We define a properad $Y_{\mathrm{\infty }}^{(n)}$ that encodes $n$-pre-Calabi–Yau algebras with vanishing copairing. These algebras include chains on the based loop space of any space $X$ endowed with a fundamental class $[X]$ such that $(X,[X])$ satisfies Poincaré duality of degree $n⩾1$ with local system coefficients, such as an oriented manifold. Extending the notion of coformality of spaces, we define coformality of such a pair $(X,[X])$ in terms of properadic formality of $Y_{\mathrm{\infty }}^{(n)}$-algebra structures on $C_{\ast }(\mathrm{\Omega }X)$. Using a refined version of properadic Kaledin classes, we establish the intrinsic coformality of all spheres in characteristic zero.