More existence theorems for recursion categories Original Research Article

We prove a generalization of Alex Hellerʹs existence theorem for recursion categories; this generalization was suggested by work of Di Paola and Montagna on syntactic P-recursion categories arising from consistent extensions of Peano Arithmetic, and by the examples of recursion categories of coalgebras. Let View the MathML source be a uniformly generated isotypical B#-subcategory of an iteration category View the MathML source, where X is an isotypical object of View the MathML source. We give calculations for the existence of a weak Turing morphism in the Turing completion View the MathML source of View the MathML source when View the MathML source is separated; i.e., when connected domains in View the MathML source are jointly epimorphic. Our proof generalizes as follows. Let View the MathML source be a separated iteration category and let View the MathML source be an iteration functor; i.e., a functor which preserves domains, coproducts, zero morphisms and the iteration operator; it is crucial for the generalization that an iteration functor need not preserve products. If L is faithful, then View the MathML source is a recursion category.