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 groups. A notable subcategory is the Temperley-Lieb category.
By the category of non-crossing partitions one denotes the subcategory of the category of all partitions whose underlying set is the set of all non-crossing partitions.
The term “non-crossing” makes sense for partitions of any cyclically ordered set . We say that is non-crossing if we cannot find such that and simultaneously and . In particular, may be infinite under this definition.
Cyclic orders of a finite set are in one-to-one correspondence with permutations of order by defining if and only if there exist with such that and .
In the case of a partition for with , i.e., a partition of , the cyclic order referenced when asking whether is “non-crossing” is the one corresponding to the permutation with for all , with for all , with if and otherwise and with if and 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 which are non-crossing in this sense.
The set of all non-crossing partitions is the underlying set of the subcategory of generated by .