Partition categories also known as linear categories of partitions have been recently heavily studied by researchers from different fields of mathematics and physics such as group theory, compact quantum groups, operator algebras, tensor categories or statistical physics. In the theory of (compact quantum) groups they are used to model the representation theory of a given quantum group.

Note that in the theory of compact matrix quantum groups, the term category of partitions often means an easy category of partitions in the sense of Banica and Speicher.

A linear category of partitions is a rigid monoidal -subcategory of the linear category of all partitions. That is, any collection of linear subspaces containing the identity partition and the pair partition , which is closed under the tensor product, composition and involution is called a linear category of partitions.

For given , we denote by the smallest linear category of partitions containing . We say that generate . Note that the pair partition is contained in the category by definition and hence will not be explicitly listed as a generator. Note that any element in can be obtained from the generators and the pair partition by performing a finite amount of category operations and linear combinations.

Thanks to the fact that any linear category of partitions contains the pair partition, we can introduce additional operations on that stabilize any category .

For , , we define its left rotation as a partition obtained by moving the leftmost point of the upper row on the beginning of the lower row. Similarly, for , , we can define its right rotation by moving the last point of the lower row to the end of the upper row. Both operations are obviously invertible. We extend this operation linearly on .

Proposition. Every category is closed under taking left and right rotations and their inverses.

The most important examples of partition categories are the Category of all partitions, Category of all non-crossing partitions, Category of all pair partitions and Category of all non-crossing pair partitions forming a square of inclusions

Classification of partition categories is available in the case of easy categories of partitions. Another source of examples are the group-theoretical categories of partitions. Outside those two classes, several additional examples are known, but no systematical classification result is available.