An Operadic Generalization of the Gerstenhaber-Shack Theorem

TL;DR

This paper generalizes the Gerstenhaber-Shack theorem by establishing an operadic isomorphism between simplicial cochain complexes and Hochschild cochain complexes of incidence algebras, leading to new insights into deformation theory.

Contribution

It introduces an operadic framework that unifies simplicial and Hochschild cochains, extending the classical Gerstenhaber-Shack theorem to a broader algebraic context.

Findings

Operadic isomorphism between simplicial and Hochschild cochains

Generalization of the Gerstenhaber-Shack theorem

Computation of the moduli space of formal deformations

Abstract

A simplicial cochain complex can be derived from a locally small poset by taking the nerve of the poset viewed as a category. We show that the simplicial cochain complex and a relative Hochschild cochain complex of the incidence algebra of the poset are isomorphic as operads with multiplications. This result implies that the hG-algebras derived from those operads are isomorphic, which is a generalization of the Gerstenhaber-Shack theorem. The isomorphism also induces a differential graded Lie algebra isomorphism, which we use to compute the moduli space of formal deformations of the incidence algebra.