An analytic mono is a coflat complement of a strong mono. A mono is disjunctable if it has a coflat complement. An integral object is a reduced primary object. A prime of an object is an integral strong subobject. A non-initial object is irreducible if it is not the join of two proper strong subobjects.
Any simple object is integral.
If A is locally disjunctable then an object is integral iff it is reduced and irreducible. One can show that the class of affine subsets is closed under intersection. For instance, if A is the category of affine schemes then Spec coincides with the classical Zariski topology.