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.

Let , by a partition of upper and lower points we mean a partition of the set , that is, a decomposition of the set of points into non-empty disjoint subsets, called blocks. The first points are called upper and the last points are called lower. The set of all partitions on upper and lower points is denoted . We denote the union . The number for is called the length of .

We illustrate partitions graphically by putting points in one row and points on another row below and connecting by lines those points that are grouped in one block. All lines are drawn between those two rows.

Below, we give an example of two partitions and defined by their graphical representation. The first set of points is decomposed into three blocks, whereas the second one is into five blocks. In addition, the first one is an example of a non-crossing partition, i.e. a partition that can be drawn in a way that lines connecting different blocks do not intersect (following the rule that all lines are between the two rows of points). On the other hand, the second partition has one crossing.

A block containing a single point is called a singleton. In particular, the partitions containing only one point are called singletons and for clarity denoted by an arrow and .