Compact-Open Dualities for Stably Continuous Posets

Abstract

We organize and generalize several dualities involving continuous posets. The main theorem reads $\mathbf{St}_α\mathbf{Inf}_{α'}\mathbf{Cont}_{β'}\mathbf{Sup}_β\simeq (\mathbf{St}_β\mathbf{Inf}_{β'}\mathbf{Cont}_{α'}\mathbf{Sup}_α)^{\mathrm{op}}$, where $\mathbf{St}$, $\mathbf{Inf}$, $\mathbf{Cont}$ and $\mathbf{Sup}$ refer to stability, completeness, continuity and cocompleteness. The indices are "ladders", i.e., classes of sets $λ$ stable under dependent sums and quotients, with associated notions of $λ$-small infima and $λ$-filtered suprema. In the second half of the paper, we discuss algebraicity, proximity lattices and perfect maps.