Crossed actions of matched pairs of groups on tensor categories

We introduce the notion of $(G, \Gamma)$-crossed action on a tensor category, where $(G, \Gamma)$ is a matched pair of finite groups. A tensor category is called a $(G, \Gamma)$-crossed tensor category if it is endowed with a $(G, \Gamma)$-crossed action. We show that every $(G, \Gamma)$-crossed tensor category $\mathcal C$ gives rise to a tensor category $\mathcal C^{(G, \Gamma)}$ that fits into an exact sequence of tensor categories $\operatorname{Rep G} \to \mathcal C^{(G, \Gamma)} \to \mathcal C$. We also define the notion of a $(G, \Gamma)$-braiding in a $(G, \Gamma)$-crossed tensor category, which is connected with certain set-theoretical solutions of the QYBE. This extends the notion of $G$-crossed braided tensor category due to Turaev. We show that if $\mathcal C$ is a $(G, \Gamma)$-crossed tensor category equipped with a $(G, \Gamma)$-braiding, then the tensor category $\mathcal C^{(G, \Gamma)}$ is a braided tensor category in a canonical way.