A note on measurings and higher order Hochschild homology of algebras

We know that coalgebra measurings behave like generalized maps between algebras. In this note, we show that coalgebra measurings between commutative algebras induce morphisms between higher order Hochschild homology groups of algebras. By higher order Hochschild homology, we mean the the Hochschild homology groups of a commutative algebra with respect to a simplicial set as introduced by Pirashvili.