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.