Subobjects and Compactness in Point-Free Convergence

We consider subobjects in the context of point-free convergence (in the sense of Goubault-Larrecq and Mynard), characterizing extremal monomorphisms in the opposite category of that of convergence lattices. It turns out that special ones are needed to capture the notion of subspace. We call them standard and they essentially depend on one element of the convergence lattice. We introduce notions of compactness and closedness for general filters on a convergence lattice, obtaining adequate notions for standard extremal monos by restricting ourselves to principal filters. The classical facts that a closed subset of a compact space is compact and that a compact subspace of a Hausdorff space is closed find generalizations in the point-free setting under the form of general statements about filters. We also give a point-free analog of the classical fact that a continuous bijection from a compact pseudotopology to a Hausdorff pseudotopology is a homeomorphism.