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 .

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 .

Let us fix a complex number . Let us denote the vector space of formal linear combination of partitions . That is, is a vector space, whose basis is .

Now, we are going to define some operations on . First, let us define those operations just on partitions.

• The tensor product of two partitions and is the partition obtained by writing the graphical representations of and “side by side”.

• For , we define their composition by putting the graphical representation of below identifying the lower row of with the upper row of . The upper row of now represents the upper row of the composition and the lower row of represents the lower row of the composition. Each extra loop that appears in the middle and is not connected to any of the upper or the lower points, transforms to a multiplicative factor .

• For we define its involution by reversing its graphical representation with respect to the horizontal axis.

Now we can extend the definition of tensor product and composition on the vector spaces linearly. We extend the definition of the involution antilinearly. These operations are called the category operations on partitions.

The set of all natural numbers with zero as a set of objects together with the spaces of linear combinations of partitions as sets of morphisms between and with respect to those operations form a monoidal -category. All objects in the category are self-dual.

A linear category of partitions is any subcategory of the category of all partitions. That is, any collection of linear subspaces containing the identity partition and the pair partition , which is closed under the category operations 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.