nLab
Hopf action

In a strict symmetric monoidal category $C$ with symmetry $\tau$, a map $▹:B\otimes A\to A$, where $(B,{\Delta }_B)$ is a comonoid and $(A,{\mu }_A)$ a monoid, is a measuring if

$$B▹{\mu }_A={\mu }_A\circ (▹\otimes ▹)\circ (B\otimes \tau \otimes A)\circ ({\Delta }_B\otimes A\otimes A):B\otimes A\otimes A\to A$$B\triangleright \mu_A = \mu_A\circ(\triangleright \otimes \triangleright)\circ (B\otimes \tau\otimes A)\circ (\Delta_B\otimes A\otimes A) : B\otimes A\otimes A\to A

where we wrote $B={\mathrm{id}}_B$ etc. If $B$ is in fact a bimonoid and if the measuring $▹:B\otimes A\to A$ is an action, then $▹$ is said to be a Hopf action. In the $k$-linear case, a $k$-algebra $(A,{\mu }_A)$ equipped with a Hopf action is called also a left $B$-module algebra; it is the same as a monoid (=algebra) in the monoidal category of left $B$-modules, where the monoidal structure is induced by the coaction ${\Delta }_B$. It is straightforward to modify the condition above to the case of non-strict symmetric monoidal categories. A dual concept is a Hopf coaction or equivalently, a notion of a $B$-comodule algebra.