Compact-Open Dualities for Stably Continuous Posets

We organize and generalize several dualities involving continuous posets. The main theorem reads $\mathbf{St}_\alpha\mathbf{Inf}_{\alpha'}\mathbf{Cont}_{\beta'}\mathbf{Sup}_\beta \simeq \mathbf{St}_\beta\mathbf{Inf}_{\beta'}\mathbf{Cont}_{\alpha'}\mathbf{Sup}_\alpha^{\mathrm{op}}$, 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 $\lambda$ stable under dependent sums and quotients, with associated notions of $\lambda$-small infima and ${\lambda}$-filtered suprema. In the second half of the paper, we discuss algebraicity, proximity lattices and perfect maps.