Category of Δ-functors

Abstract

We define the category Func_Δ with functors F:D_F→ Scott (D_FεCPO) as objects and pairs $(f:D_F \to D_G ,\eta :F\mathop \to \limits^ \cdot G \circ f)$ as morphisms (η is a natural transformation). We show that this category is closed under the common domain theoretical operations +,X,⊥ and →. The category Func_Δ is an O-category and all the operations we define on it are continuous functors, so we will be able to solve recursive equations in Func_Δ. We also show that if we restrict Func_Δ to functors that preserve directed colimits then the category is not closed under the → operation. The category Func_Δ is a basis for a model of second-order lambda calculus with subtyping.