Partial morphisms in categories of effective objects

AbstractThis paper is divided in two parts. In the first we analyse in great generality data types in relation to partial morphisms. We introduce partial function spaces, partial cartesian closed categories and complete objects, motivate their introduction, and show some of their properties. In the second part we define the (partial) cartesian closed category GEN of generalized numbered sets, prove that it is a good extension of the category of numbered sets, and show how it is related to the recursive topos