  In this paper we investigate the simplicial structure of a chain complex
associated to the higher order Hochschild homology over the $3$-sphere. We also
introduce the tertiary Hochschild homology corresponding to a quintuple
$(A,B,C,\varepsilon,\theta)$, which becomes natural after we organize the
elements in a convenient manner. We establish these results by way of a
bar-like resolution in the context of simplicial modules. Finally, we
generalize the higher order Hochschild homology over a trio of simplicial sets,
which also grants natural geometric realizations.
