On Properties of Some Categories with Sets of Nonnegative Integers As Objects

Let $$N$$ be the set of nonnegative integers. The properties of some categories will be researched, including category with computable subsets of $$N$$ as objects and partial computable functions (having computable domain) as arrows (namely Npcomp); category with computably enumerable subsets of $$N$$ as objects and partial computable functions (having computably enumerable domain) as arrows (namely NPCE); category with immune subsets of $$N$$ as objects and functions (having immune sets as domain) as arrows (namely NIm); category with hereditary subsets of $$N$$ as objects and functions (having hereditary sets as domain) as arrows (namely NHer). It is shown in what capacity the notions of “coproduct of two objects,” “product of two objects,” “cone for given diagram,” and “limit of given diagram” are presented in the above-mentioned categories. In particular, it is proved that the coproduct of two objects is the joint of these two objects in Npcomp, NPCE, and NIm categories.