The category of all non-crossing two-colored partitions is a category of two-colored partitions inducing the co-representation categories of the free symmetric quantum group.
By the category of all non-crossing two-colored partitions one denotes the subcategory of the category of all two-colored partitions whose morphism class is the set of all non-crossing partitions. It was introduced by Tarrago and Weber in [TaWe18], Proposition 5.3 and Theorem 5.4 under the name .
The set of all non-crossing two-colored partitions is the morphism class of the subcategory of generated by the set of two-colored partitions.
The category of all non-crossing two-colored partitions induces the co-representation categories of the free symmetric quantum groups , defined by Wang in [Wang98], Theorem 3.1.