Retraction map categories and their applications to the construction of lambda calculus models

This paper deals with categorical models of the λ-calculus. We generalize the inverse limit method Scott used for his construction of D_∞, and introduce order-enriched ccc's, retraction map categories and ɛ-categories. An order-enriched ccc is a cartesian closed category C equipped with a partial order relation ⩽ on the set of the arrows. A retraction map category of C is R=(R, ⩽, i, j), where ⩽ is a partial order relation on the set |C| of all the objects of C, R is the category of the poset (|C|, ⩽), and i and j are functors from R to C and from R^op to C that satisfy the conditions: (1) j a, b ∘ i a, b ⩾ id_a and (2) i a, b ∘ j a, b ⩽ id_b for every arrow a, b: a → b in R (i.e., a⩽b). The ɛ-category E=E(C, R) of C w.r.t. R is the category whose objects are ideals of (|C|, ⩽) and whose arrows are ideals of (C, ⊑), where ⩽ is the partial order relation in R and ⊑ is the partial order relation defined by f ⊑ g iff dom(f)⩽dom(g), cod(f)⩽cod(g) in R and f⩽j a, b ∘ g ∘ i(a, b in C. We show that every ɛ-category E=E(C, R) is also an order-enriched ccc. Moreover when E and R satisfy a particular condition, E(C, R) has a reflexive object. For example, if there is an ideal U of (|C|, ⩽) satisfying the following conditions, then U is isomorphic to U^U in E and a λ-algebra is constructed from E and U: (1) for every pair of a, b ∈ U, U contains b^a, and (2) for every c ∈ U, there are a, b ∈ U such that c ∈ b^a. We reconstruct P_ω and D_∞ using ɛ-categories.