Measurings of Hopf algebroids and morphisms in cyclic (co)homology theories

TL;DR

This paper studies how measurings between Hopf algebroids induce morphisms on cyclic (co)homology, enriching the understanding of their algebraic and homological structures.

Contribution

It introduces the concept of measurings between Hopf algebroids and SAYD modules, showing they induce compatible morphisms on cyclic (co)homology and enrich the category of modules over operads.

Findings

Measurings induce morphisms on cyclic homology and cohomology.

Enrichment of SAYD modules over comodules via measurings.

Compatibility of morphisms with Hopf-Galois maps.

Abstract

In this paper, we consider measurings between Hopf algebroids and show that they induce morphisms on cyclic homology and cyclic cohomology. We also consider comodule measurings between SAYD modules over Hopf algebroids. These give an enrichment of the global category of SAYD modules over comodules. These measurings also induce morphisms on cyclic (co)homology of Hopf algebroids with SAYD coefficients, which are compatible with Hopf-Galois maps. Finally, we consider non-$\Sigma$ operads with multiplication. We obtain an enrichment of cyclic unital comp modules over non-$\Sigma$ operads, as well as morphisms on cyclic homology induced by measurings of comp modules over operads with multiplication.