Convergence-theoretic characterizations of compactness

Fundamental variants of compactness are characterized in terms of concretely reflective convergence subcategories: topologies, pretopologies, paratopologies, hypotopologies and pseudotopologies. Hyperquotient maps (perfect, quasi-perfect, adherent and closed) and quotient maps (quotient, hereditarily quotient, countably biquotient, biquotient, and almost open) are characterized in terms of various degrees of compactness of their fiber relations, and of sundry relaxations of inverse continuity.