Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation

In this paper, we construct a categorical solution $(\text{huaC},R)$ of the Yang-Baxter equation, i.e. $\text{huaC}$ is a small category and $R:\text{huaC}\times \text{huaC}\text{lon}\text{huaC}\times \text{huaC}$ is an invertible functor satisfying $$(R\times {\text{Id}}_{\text{huaC}})({\text{Id}}_{\text{huaC}}\times R)(R\times {\text{Id}}_{\text{huaC}})=({\text{Id}}_{\text{huaC}}\times R)(R\times {\text{Id}}_{\text{huaC}})({\text{Id}}_{\text{huaC}}\times R),$$ where $\text{huaC}\times \text{huaC}$ is the product category. First, the notion of Rota-Baxter operators on crossed modules of Lie groups is defined and its various properties are established. Then, we use Rota-Baxter operators on crossed modules of Lie groups to construct categorical solutions of the Yang-Baxter equation. We also study the Rota-Baxter operators on crossed modules of Lie algebras which are infinitesimals of Rota-Baxter operators on crossed modules of Lie groups, they can give connections on manifolds. Finally, we study the integration of Rota-Baxter operators on crossed modules of Lie algebras and the differentials of Rota-Baxter operators on crossed modules of Lie groups.