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Category of all non-crossing two-colored partitions
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.
Definition
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 
.
- A two-colored partition is said to be non-crossing if there exist no blocks
and 
of 
with 
and no legs 
and 
such that 
and 
with respect to the cyclic order of . (See also category of all non-crossing partitions in the uncolored case.) - The name set of all non-crossing partitions with neutral blocks is to be taken literally.
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.
Associated unitary easy quantum groups
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.