  Our basic objects are partitions of finite sets of points into disjoint
subsets. We investigate sets of partitions which are closed under taking tensor
products, composition and involution, and which contain certain base
partitions. These so called categories of partitions are exactly the tensor
categories being used in the theory of Banica and Speicher's orthogonal easy
quantum groups. In our approach, we additionally allow a coloring of the
points. This serves as the basis for the introduction of unitary easy quantum
groups, which is done in a separate article. The present article however is
purely combinatorial. We find all categories of two-colored noncrossing
partitions. For doing so, we extract certain parameters with values in the
natural numbers specifying the colorization of the categories on a global as
well as on a local level. It turns out that there are ten series of categories,
each indexed by one or two parameters from the natural numbers, plus two
additional categories. This is just the beginning of the classification of
categories of two-colored partitions and we point out open problems at the end
of the article.
