Abstract
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.
Highlights
In 1971, higher order Hochschildhomology was implicitly defined by Anderson in [1]
Higher order Hochschildhomology was even recently generalized over a pair of simplicial sets in [9]
One is a nice mnemonic rule for remembering how to collapse the degree n tensor product to degree n − 1. Another is that in general, higher order Hochschild homology cannot be realized as a functor like the usual can; this simplicial description is the best thing
Summary
In 1971, higher order Hochschild (co)homology was implicitly defined by Anderson in [1]. Higher order Hochschild (co)homology was even recently generalized over a pair of simplicial sets in [9]. One is a nice mnemonic rule for remembering how to collapse the degree n tensor product to degree n − 1 Another is that in general, higher order Hochschild homology cannot be realized as a functor like the usual can (as the Ext functor); this simplicial description is the best thing. ([1], [22]) The higher order Hochschild homology of A with coefficients in M over the simplicial set X is defined to be the homology of the above complex, and is denoted H∗X(A, M ).
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