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, ɛ, θ), 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.