In a strict symmetric monoidal category with symmetry , a map of a comonoid on a monoid is a measuring, or we say that measures if
C\triangleright \mu_A = \mu_A\circ(\triangleright \otimes \triangleright)\circ (C\otimes \tau\otimes A)\circ (\Delta_C\otimes A\otimes A) : C\otimes A\otimes A\to A.
In Sweedler notation, we also write . A Hopf action is a special case of measuring which is also an action of a bimonoid where . Measurings are used e.g. do define the (cocycled) crossed product algebras, see also cleft extension. 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