Quantales : Quantal sets.
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.