  Hochschild (co)homology and Pirashvili's higher order Hochschild (co)homology
are useful tools for a variety of applications including deformations of
algebras. When working with higher order Hochschild (co)homology, we can
consider the (co)homology of any commutative algebra with symmetric coefficient
bimodules, however traditional Hochschild (co)homology is able to be computed
for any associative algebra with not necessarily symmetric coefficient
bimodules. In a previous paper, the author generalized higher order Hochschild
cohomology for multimodule coefficients (which need not be symmetric). In the
current paper, we continue to generalize higher order Hochschild (co)homology
to work with associative algebras which need not be commutative and in
particular, show that simplicial sets admit such a generalization if and only
if they are one dimensional.
