Abstract: We define ``the category of compactifications", which is denoted {\bf{CM}}, and consider its family of coreflections, denoted {\bf{corCM}}. We show that {\bf{corCM}} is a complete lattice with bottom the identity and top an interpretation of the Čech-Stone \beta. A c \in{\bf{corCM}} implies the assignment to each locally compact, noncompact Y a compactification minimum for membership in the ``object-range" of c. We describe the minimum proper compactifications of locally compact, noncompact spaces, show that these generate the atoms in {\bf{corCM}} (thus {\bf{corCM}} is not a set), show that any c \in{\bf{corCM}} not the identity is above an atom, and that \beta is not the supremum of atoms.
Keywords: compactification; coreflection; atom in a lattice
DOI: DOI 10.14712/1213-7243.2022.024
AMS Subject Classification: 54B30 54C10 54D35 06B23 18A40