An Operadic Generalization of the Gerstenhaber-Shack Theorem
Andy Yu
TL;DR
The paper generalizes the Gerstenhaber–Shack theorem by establishing an operadic isomorphism between the simplicial cochain operad of a locally finite poset and the relative Hochschild cochain operad of its incidence algebra. By equipping both cochain theories with operads and multiplications, the authors obtain a homotopy Gerstenhaber algebra and, ultimately, a differential graded algebra isomorphism. This operadic bridge yields a differential graded Lie algebra isomorphism and a computation of the Maurer–Cartan moduli space of formal deformations of $kP$, identifying it with $H^2(P;W)$ where $W$ denotes the big Witt vectors. The results provide a unified operadic perspective on deformation theory for incidence algebras and extend the classical Gerstenhaber–Shack correspondence to an operadic framework. The approach highlights the compatibility of simplicial and Hochschild cohomologies under operadic isomorphisms and informs future deformation-theoretic analyses of posets and categories.
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.