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Category of two-colored partitions

Categories of two-colored partitions are certain strict monoidal involutive categories, introduced by Freslon and Weber in [FreWeb16], Definition 6.1. Each such category induces the co-representation categories of a sequence of unitary easy quantum groups. Categories of two-colored partitions generalize categories of (uncolored) partitions as defined by Banica and Speicher in [BanSp09].

Definition

The original definition of Freslon and Weber in [FreWeb16], Definition 6.1 was later equivalently reformulated by Tarrago and Weber in [TaWe18], Section 1.3. In this formulation, a category of two-colored partitions is a subset $\Cscr\subseteq \Pscr^{\circ\bullet}$ of the set $\Pscr^{\circ\bullet}$ of all two-colored partitions satisfying the following conditions with respect to the operations for two-colored partitions:

References

[FreWeb16] Freslon, Amaury and Weber, Moritz, 2016. On the representation theory of partition (easy) quantum groups. Journal für die reine und angewandte Mathematik [Crelle's Journal], 2016.
[BanSp09] Banica, Teodor and Speicher, Roland, 2009. Liberation of orthogonal Lie groups. Advances in Mathematics, 222, pp.1461–150.
[TaWe18] Tarrago, Pierre and Weber, Moritz, February 2018. The classification of tensor categories of two-colored non-crossing partitions. Journal of Combinatorial Theory, Series A, 154, pp.464–506.