Peter Freyd claims that any partial recursive function arises as Hom(1,f) for some endomorphism from N to N .
From his comment I could not see why. If one assumes the solution
D = N + [N->N] then one may define the recursor Y in the usual lambda calculus style. BUT how can one transform the exponentiation functor in the ambient ccc to a functor on the reflective cat of strict maps ? I guess one needs a little bit more assumptions : e.g. that the category of predomains contains the reflective subcat of domains and strict maps which in turn contains the category of total strict maps (by the lift functor) which is equivalent to the category of predomains . Can one axiomatize this situation in a sufficiently strong way, e.g. that one can transfer the exponentiation functor from the category of predomains to the category of domains and strict maps ? Thomas Streicher =========================================================================