# nLab measuring

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

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