Hochschild theory of multiplicative sequences of algebras and coalgebra measurings

We study coalgebra measurings between multiplicative sequences of algebras and the maps induced by them on Hochschild homology.

The Hochschild theory of multiplicative sequences is introduced as a functor taking values in graded algebras in the symmetric monoidal category of chain complexes, constructed with the help of the shuffle product.

We develop the universal measuring coalgebra, or Sweedler Hom for multiplicative sequences, as well as study several other Sweedler operations in this context.

In particular, we obtain an enrichment of multiplicative sequences over cocommutative coalgebras.

Using an appropriate theory of bimodules over multiplicative sequences, we study maps induced by comodule measurings on the Hochschild theory with coefficients, as well as the corresponding enriched categories.

Finally, we consider measurings and generalized Sweedler operations between multiplicative sequences induced by comultiplicative sequences of coalgebras, and also the maps in Hochschild theory obtained from them.