Compact-Open Dualities for Stably Continuous Posets

Abstract

We organize and generalize several dualities involving continuous posets. The main theorem reads ${\mathbf{St}}_{\alpha }{\mathbf{Inf}}_{{\alpha }^{\prime }}{\mathbf{Cont}}_{{\beta }^{\prime }}{\mathbf{Sup}}_{\beta }\simeq ({\mathbf{St}}_{\beta }{\mathbf{Inf}}_{{\beta }^{\prime }}{\mathbf{Cont}}_{{\alpha }^{\prime }}{\mathbf{Sup}}_{\alpha }{)}^{\mathrm{op}}$, 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.

23 pages; v2: mainly added some references