Simplicial structures over the 3-sphere and generalized higher order Hochschild homology

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, ", Ө), which becomes natural after we organize the elements in a convenient manner. We establish these results by way of a barlike 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.