A Duality Between Ω-categories and Algebraic Ω-categories

In this paper, we propose a definition of algebraic Ω-categories. Let Ω-POID denote the category of Ω-categories with Ω-functors between them such that inverse image of ideals are also ideals, and let Ω-AlgDomG denote the category of algebraic Ω-categories with Scott continuous functors between them having left Ω-adjoints. We show that Ω-AlgDomG and Ω-POID are dual equivalent to each other.