nLab
measuring

In a strict symmetric monoidal category $D$ with symmetry $\tau$, a map $▹:C\otimes A\to A$ of a comonoid $(C,{\Delta }_C)$ on a monoid $(A,{\mu }_A)$ is a measuring, or we say that $C$ measures $A$ if

$$C▹{\mu }_A={\mu }_A\circ (▹\otimes ▹)\circ (C\otimes \tau \otimes A)\circ ({\Delta }_C\otimes A\otimes A):C\otimes A\otimes A\to A.$$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 $c▹(\mathrm{ab})=\sum (c_{(1)}▹a)(c_{(2)}▹b)$. A Hopf action is a special case of measuring which is also an action of a bimonoid where $B=(C,{\mu }_C)$. 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

• S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.

There are also more elaborate versions of measurings, which play role in a Galois theory, see

• T. Brzeziński, On modules associated to coalgebra Galois extensions, J. Algebra, 215, no. 1, 1999, 290-317.