  Several variations on the definition of a Formal Topology exist in the
literature. They differ on how they express convergence, the formal property
corresponding to the fact that open subsets are closed under finite
intersections. We introduce a general notion of convergence of which any
previous definition is a special case. This leads to a predicative presentation
and inductive generation of locales (formal covers), commutative quantales
(convergent covers) and suplattices (basic covers) in a uniform way. Thanks to
our abstract treatment of convergence, we are able to specify categorically the
precise sense according to which our inductively generated structures are free,
thus refining Johnstone's coverage theorem. We also obtain a natural and
predicative version of a fundamental result by Joyal and Tierney: convergent
covers (commutative quantales) correspond to commutative co-semigroups over the
category of basic covers (suplattices).
