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

In this paper, we consider coalgebra measurings between Hopf algebroids and show that they induce morphisms on cyclic homology and cyclic cohomology. We also consider comodule measurings between stable anti-Yetter Drinfeld (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-symmetric operads with multiplication and modules over them which have both cyclic and Gerstenhaber type structures, known as cyclic unital comp modules. We obtain an enrichment of cyclic unital comp modules over comodules, as well as morphisms on cyclic homology induced by comodule measurings of comp modules over operads with multiplication.