====== Category of all two-colored partitions ======

The **category of all two-colored partitions** is a [[categories of two-colored partitions|category of two-colored partitions]] inducing the co-representation categories of the [[symmetric group|symmetric groups]].

===== Definition =====

By the **category of all two-colored partitions** one denotes the category of two-colored partitions $\Pscr^{\circ\bullet}$ whose morphism class is the //set of all two-colored partitions//. It was introduced by Tarrago and Weber in [(:ref:TaWe18)], Theorem 8.3 under the name $\mathcal{S}_{\mathrm{grp},\mathrm{glob}}(1)$. 

A canonical generator of $\Pscr^{\circ\bullet}$ is the set $\{\Partition{\Pline (1,0) (2,1) \Pline (2,0) (1,1) \Ppoint 0 \Pw:1,2 \Ppoint 1 \Pw:1,2}, \raisebox{0.125em}{\LPartition{\Pw:1;\Pw:2}{0.6:1,2}}, \raisebox{0.125em}{\LPartition{\Pw:1,3;\Pb:2,4}{0.6:1,2,3,4}}, \LPartition{\Ls:1;\Pw:1}{}\}$. 

===== Associated unitary easy quantum groups =====

The category of two-colored pair partitions with neutral blocks induces the co-representation categories of the [[symmetric group|symmetric groups]] $(S_N)_{N\in \N}$.

===== References =====


[( :ref:TaWe18 >>
author:  Tarrago, Pierre and Weber, Moritz
title:   The classification of tensor categories of two-colored non-crossing partitions
year:    2018
journal: Journal of Combinatorial Theory, Series A
volume:  154
month:   February
pages:   464--506
url:     https://doi.org/10.1016/j.jcta.2017.09.003
archivePrefix: arXiv
eprint   :1509.00988 
)]