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--- categories_of_two-colored_partitions [2020/03/10 10:11]
amang created
+++ categories_of_two-colored_partitions [2021/11/23 11:56] (current)
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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 [(:ref:FreWeb16)], Definition 6.1. Each such category induces the co-representation categories of a sequence of [[unitary_easy_quantum_group]]. Categories of two-colored partitions generalize categories of (uncolored) partitions as defined by Banica and Speicher in [(:ref:BanSp09)].
+**Categories of two-colored partitions** are certain strict monoidal involutive categories, introduced by Freslon and Weber in [(:ref:FreWeb16)], Definition 6.1. Each such category induces the co-representation categories of a sequence of [[unitary_easy_quantum_group|unitary easy quantum groups]]. Categories of two-colored partitions generalize [[category_of_partitions|categories of (uncolored) partitions]] as defined by Banica and Speicher in [(:ref:BanSp09)].
 ===== Definition =====
-The original definition of Freslon and Weber in [(:ref:FreWeb16)], Definition 6.1 was later equivalently reformulated by Tarrago and Weber in [(:ref: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:
+The original definition of Freslon and Weber in [(:ref:FreWeb16)], Definition 6.1 was later equivalently reformulated by Tarrago and Weber in [(:ref: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 partition|two-colored partitions]] satisfying the following conditions with respect to the [[operations for two-colored partitions]]:
   * $\{\Partition{\Pline (1,0.125) (1,0.875) \Ppoint 0.125 \Pw:1 \Ppoint 0.875 \Pw:1},\Partition{\Pline (1,0.125) (1,0.875) \Ppoint 0.125 \Pb:1 \Ppoint 0.875 \Pb:1},\raisebox{0.125em}{\LPartition{\Pw:1;\Pb:2}{0.6:1,2}},\raisebox{0.125em}{\LPartition{\Pb:1;\Pw:2}{0.6:1,2}}\}\subseteq \Cscr$.
   * $pp'\in \Cscr$ for all $p,p'\in\Cscr$ such that $(p,p')$ is composable.

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