Closure operators and functorial topologies
The basic idea for a closure operator on a category \({\mathcal X}\) is to have, for each object X, an extensive, increasing and idempotent operation [ ]\({}^ X\) on the ordered class of its subobjects \(M\to X\) that belong to a fixed class \({\mathcal M}\) of monomorphisms of \({\mathcal X}\); M is said to be closed if \([M]^ X\) is isomorphic to M. The pairs of the form (X,[ ]\({}^ X)\) and the ``continuous'' morphisms f of \({\mathcal X}\) form a concrete category CS(\({\mathcal X})\) over \({\mathcal X}\). Now, consider a concrete category \({\mathcal A}\) over \({\mathcal X}\). The author sets up a Galois correspondence between the subcategories \({\mathcal C}\) of \({\mathcal A}\) and the faithful functors F: \({\mathcal A}\to CS({\mathcal X})\) (also called global closures over \({\mathcal A})\). The ensuing theory is exploited and applied to the cases where both \({\mathcal A}\) and \({\mathcal X}\) consist of commutative groups or of commutative topological groups.
Galois correspondence between subcategories and faithful functors
scientific article; zbMATH DE number 4077513
scientific article