In a strict symmetric monoidal category with unit , symmetry , left and right unit coherences and , a map of a (counital) comonoid on a (unital) monoid is a measuring, or we say that measures if
In Sweedler notation, we also write and . A Hopf action is a special case of measuring which is also an action of a bimonoid where . In other words, is a monoid in the monoidal category of -modules (also called -module monoid or -module algebra). Measurings are used to define the (cocycled) crossed product algebras, see also cleft extension (of an algebra by a bialgebra). For measurings and module algebras see
S. Montgomery, Hopf algebras and their actions on rings, CBMS 82, AMS 1993.
A. Klimyk, K. Schmüdgen, Quantum groups and their representations, Springer, 1997;
and for (co)module (co)algebras and generalizations see also
There are also more elaborate versions of measurings, which play role in a Galois theory, see
If and are -algebras measured by a -coalgebra and is a --bimodule, then we say that is measured by , if there is a -bilinear map such that for all , , , ,
If is in addition a -bialgebra and actions of , we say that a measuring is Hopf action of on the --bimodule . For Hopf actions on bimodules, one can define a bimodule version of a (Hopf) smash product algebra, see Hopf smash product bimodule.