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Category of non-crossing two-colored pair partitions
The category of non-crossing two-colored pair partitions is a category of two-colored partitions inducing the co-representation categories of the free orthogonal quantum group.
Definition
By the category of non-crossing two-colored pair partitions one denotes the subcategory of the category of all two-colored partitions 
whose morphism class is the set of all non-crossing two-colored pair partitions. It was introduced by Tarrago and Weber in [TaWe18], Proposition 3.3 (b) under the name 
.
- A two-colored partition
is called a pair partition (see category of all pair partitions in the uncolored case), if every block 
of 
satisfies 
. - It 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 two-colored pair partitions is to be taken literally.
The category of non-crossing two-colored pair partitions is the subcategory of generated by the two-colored partition 
.
Associated unitary easy quantum groups
The category of non-crossing two-colored pair partitions induces the co-representation categories of the free orthogonal quantum groups 
, defined by Wang in [Wang95], Example 4.2.
References
