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3 (i) I f P is an I-category, so is ps. (ii) If P is an (w-algebraic) complete 1-category, so is P ' . 2 Standard funetors In dealing with functors between 1-categories, it is natural to require that the functors preserve the class of inclusion morphisms. 4 Let P , P ' be I-categories. (i) An endofunctor F : P --~ P ' is standard if it preserves inclusion morphisms; it is strictness preserving if it preserves strict morphisms. (ii) Suppose P, pI are complete. _, (More,, ~'~) is a continuous (cpo) function.
For f , g E Mot with c o d ( f ) = dom(g), we denote the composition of f and g by f;g (instead of the common notation g o f), which should be read as f followed by g. Given a functor F : P1 --' P2 between two categories P1 and P2 we write Fo : Objp1 --, Objp2 and F,~ : Morpl ~ Morp2 for the induced mappings on objects and morphisms respectively. The identity endofunctor is denoted by ID. We say P is a countable category if Morp is countable. e. PI(A) is the set of all finite subsets of the set A.
In fact, given a morphism f : A --. B we can construct an increasing chain of morphisms between finite sets with lub f as follows. Choose finite sets Ai, Bi,i >_ 0 with A = UiAi, B = LJiBi and for each i > 0 let nl be the least integer with ni ~ i and f(Ai) C B,~,. Define fi : Ai --~ B,, by fl = f [A,. Jifi and B = Ui B~,. Finally in this section, we note that the product P1 x P2 of two I-categories P1 and P2 is an I-category, with A p 1×~ = (AP1, AP2) and the partial order on horn-sets and inclusion morphisms defined co-ordinatewise in the obvious way.