Coefficients for Higher Order Hochschild Cohomology

TL;DR

This paper extends higher order Hochschild cohomology to include multi-module coefficients, broadening its applicability in deformation theory and clarifying the types of modules suitable as coefficients.

Contribution

It introduces an extended definition of higher order Hochschild cohomology that accommodates multi-module coefficients, aligning with existing definitions.

Findings

Extended the definition of Hochschild cohomology to multi-module coefficients.

Identified module types compatible with simplicial set structures.

Maintained consistency with existing symmetric coefficient framework.

Abstract

When studying deformations of an $A$-module $M$, Laudal and Yau showed that one can consider 1-cocycles in the Hochschild cohomology of $A$ with coefficients in the bi-module $End_k(M).$ With this in mind, the use of higher order Hochschild (co)homology, presented by Pirashvili and Anderson, to study deformations seems only natural though the current definition allows only symmetric bi-module coefficients. In this paper we present an extended definition for higher order Hochschild cohomology which allows multi-module coefficients (when the simplicial sets $X_{\bullet}$ are accommodating) which agrees with the current definition. Furthermore we determine the types of modules that can be used as coefficients for the Hochschild cochain complexes based on the simplicial sets they are associated to.