Differential Forms for Smooth Affine Algebras over Operads

The Hochschild--Kostant--Rosenberg theorem is a classical algebro-geometric result: Hochschild homology and cohomology of a smooth commutative algebra can be computed as the space of differential forms and of poly-vector fields on it, respectively. It says, in other words, how to compute the cohomology of a commutative algebra if we pull it back through the canonical projection of operads p : Ass --> Com and consider it merely to be associative. This result was then exploited to study the deformation theory of smooth commutative algebras, and obtain the celebrated Kontsevich formality theorem (cf. D. Tamarkin's proof). In this talk, I will explain how to answer the following natural generalization of this question: given a map of operads f : P --> Q and a smooth Q-algebra A, how can one construct a space Ω*(A) of 'differential forms for A' and when can one produce an isomorphism from the (Hochschild) homology of the pullback P-algebra A to Ω*(A)? In particular, I'll explain how to recover the usual Hochschild--Kostant--Rosenberg theorem using operadic homological algebra. This is joint work with Ricardo Campos (IMAG, Universite Montpellier, CNRS), arXiv:2010.08815.