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

Coefficients for Higher Order Hochschild Cohomology