Categorical Domain Theory: Scott Topology, Powercategories, Coherent Categories

In the present article we continue recent work in the direction of domain theory were certain (accessible) categories are used as generalized domains. We discuss the possibility of using certain presheaf toposes as generalizations of the Scott topology at this level. We show that the toposes associated with Scott complete categories are injective with respect to dense inclusions of toposes. We propose analogues of the upper and lower powerdomain in terms of the Scott topology at the level of categories. We show that the class of finitely accessible categories is closed under this generalized upper powerdomain construction (the respective result about the lower powerdomain construction is essentially known). We also treat the notion of “coherent domain” by introducing two possible notions of coherence for a finitely accessible category (qua generalized domain). The one of them imitates the stability of the compact saturated sets under intersection and the other one imitates the so-called “2/3 SFP” property. We show that the two notions are equivalent. This amounts to characterizing the small categories whose free cocompletion under finite colimits has finite limits.