Stable Formality Quasi-isomorphisms for Hochschild Cochains

TL;DR

This paper introduces the concept of stable formality quasi-isomorphisms for Hochschild cochains, providing a classification of their homotopy classes using graph complexes, advancing the understanding of algebraic structures in deformation theory.

Contribution

It formalizes stable formality quasi-isomorphisms, defines a homotopy equivalence, and characterizes their classes via the graph complex cohomology.

Findings

Homotopy classes form a torsor for the graph complex cohomology group.

Complete description of stable formality quasi-isomorphisms.

Provides a framework for classifying Hochschild cochain formality.

Abstract

We consider L-infinity quasi-isomorphisms for Hochschild cochains whose structure maps admit "graphical expansion". We introduce the notion of stable formality quasi-isomorphism which formalizes such an L-infinity quasi-isomorphism. We define a homotopy equivalence on the set of stable formality quasi-isomorphisms and prove that the set of homotopy classes of stable formality quasi-isomorphisms form a torsor for the group corresponding to the zeroth cohomology of the full (directed) graph complex. This result may be interpreted as a complete description of homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the "stable setting".