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

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