A generalisation of Q-sets (and maps between them) called quantal sets, is introduced for an idempotent quantale Q. It turns out that subsets of a quantal set do not correspond bijectively with its subobjects. Quantal sets are shown to form a topos with an internal quantale which classifies subsets. We define appropriate notions of complete quantal set, presheaf and sheaf over Q, and show that the categories of quantal sets and sheaves over Q are equivalent. Based on subsets of quantal sets, a category is constructed in which the properties of the &-operation of Q are reflected. Finally, we construct a bicategory m from Q, and define a concept of quasi-symmetric E-category; it is proved that complete quantal sets may be characterised as quasi-symmetric E-categories satisfying additional properties.