Abstract: Dealing with simply laced Dynkin diagrams, Ingalls and Thomas (Compos. Math. 145, 2009) gave an interpretation of the set of non-crossing partitions in terms of the representation category of a Dynkin quiver: they exhibited, for example, a bijection between the non-crossing partitions and the wide subcategories or also the torsion classes. These results can be reformulated in terms of antichains in additive categories and extended to the non-simply laced cases B_n, C_n, F_4, G_2 and the corresponding hereditary abelian categories. We will show in which way the representation theory approach sheds light on the relationship between crossing and nesting; this relationship is well-known in the case A_n, but seemed to be quite mysterious in the remaining cases.