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====== Category of non-crossing partitions ======

The **category of non-crossing partitions** is the free counterpart of the [[category of all partitions]]. It induces the corepresentation category of the [[free symmetric quantum group|free symmetric quantum groups]]. A notable subcategory is the [[temperley_lieb_category|Temperley-Lieb category]].

===== Definition =====

By the **category of non-crossing partitions** one denotes the subcategory $\mr{NC}$ of the [[category of all partitions]] $\Pscr$ whose underlying set is the //set of all non-crossing partitions//.

The term "non-crossing" makes sense for partitions $\pi$ of any [[wp>cyclic order|cyclically ordered set]] $(S,[\cdot,\cdot,\cdot])$. We say that $\pi$ is **non-crossing** if we cannot find $i_1,i_2,j_1,j_2\in S$ such that $i_1\sim_\pi i_2\not\sim_\pi j_1\sim_\pi j_2$ and simultaneously $[i_1,j_1,i_2]$ and $[j_1,i_2,j_2]$.  In particular, $S$ may be infinite under this definition. \\
Cyclic orders of a //finite// set $S$ are in one-to-one correspondence with permutations $\nu:S\to S$ of order $|S|$ by defining $[i,i_1,i_2]$ if and only if there exist $e_1,e_2\in \{1,\ldots,|S|-1\}$ with $e_1<e_2$ such that $\nu^{e_1}(i)=i_1$ and $\nu^{e_2}(i)=i_2$. 

In the case of a partition $p\in \Pscr(k,l)$ for $k,l\in \{0\}\cup \N$ with $k+l>0$, i.e., a partition of $\{1,\ldots,k\}\sqcup \{1,\ldots,l\}\allowbreak\cong \{1_U,\ldots,k_U,1_L,\ldots,l_L\}$, the cyclic order referenced when asking whether $p$ is "non-crossing" is the one corresponding to the permutation with $j_U\mapsto (j-1)_U$ for all $j=2,\ldots,k$, with $i_L\mapsto (i+1)_L$ for all $i=1,\ldots,l-1$, with $1_U\mapsto 1_L$ if $l>0$ and $1_U\mapsto k_U$ otherwise and with $l_L\mapsto k_U$ if $k>0$ and $l_L\mapsto 1_L$ otherwise. (The same notion of being "non-crossing" is obtained by referencing the cyclic order induced by the inverse of this permutation.)

The **set of all non-crossing partitions** is the set of all elements of $\Pscr=\bigcup_{k,l=0}^\infty\Pscr(k,l)$ which are non-crossing in this sense.

The set of all non-crossing partitions is the underlying set of the subcategory of $\Pscr$ generated by $\{\fourpart,\singleton\}$.
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