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

The **category of all non-crossing two-colored partitions** is a [[categories of two-colored partitions|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]] $\Pscr^{\circ\bullet}$ whose morphism class is the //set of all non-crossing partitions//. It was introduced by Tarrago and Weber in [(:ref:TaWe18)], Proposition 5.3 and Theorem 5.4 under the name $\mathcal{S}_{\mathrm{glob}}(1)$. 

  * A two-colored partition is said to be **non-crossing** if there exist no blocks $B$ and $B'$ of $p$ with $B\neq B'$ and no legs $i,j\in B$ and $i',j'\in B'$ such that $i\prec i'\prec j$ and $i'\prec j\prec j'$ with respect to the cyclic order of $p$. (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 $\Pscr^{\circ\bullet}$ generated by the set $\{\raisebox{0.125em}{\LPartition{\Pw:1;\Pw:2}{0.6:1,2}}, \raisebox{0.125em}{\LPartition{\Pw:1,3;\Pb:2,4}{0.6:1,2,3,4}}, \LPartition{\Ls:1;\Pw:1}{}\}$ 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 group|free symmetric quantum groups]] $(S_N^+)_{N\in \N}$, defined by Wang in [(:ref:Wang98)], Theorem 3.1.

===== 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 
)]

[( :ref:Wang95 >>
author   : Shuzhou Wang
title    : Free products of compact quantum groups
journal  : Communications in Mathematical Physics
year     : 1995
volume   : 167
number   : 3
pages    : 671--692
url      : http://dx.doi.org/10.1007/BF02101540
)]

[( :ref:Wang98 >>
author   : Shuzhou Wang
title    : Quantum Symmetry Groups of Finite Spaces
journal  : Communications in Mathematical Physics
year     : 1998
volume   : 195
number   : 1
pages    : 195--211
url      : http://dx.doi.org/10.1007/s002200050385
archivePrefix: arXiv
eprint   :0707.3168
)]