A category of enumerated sets (Q1060214)

The objects in the category \({\mathcal N}_ e\) are all the enumerated sets. If \(G_ 0=(S_ 0,\nu_ 0)\) and \(G_ 1=(S_ 1,\nu_ 1)\) are such sets, then a morphism from \(G_ 0\) to \(G_ 1\) is any mapping \(\mu\) : \(S_ 0\to S_ 1\) for which there exists an e-operator \(\phi\) such that (a) \((\forall s\in S_ 0)(\{x: \mu \nu_ 0(x)=\mu (s)\}=\phi (\nu_ 1^{-1}(\mu (s))));\) (b) \(S\not\in \{\mu (s'): s'\in S_ 0\}\Rightarrow \phi (\nu_ 1^{- 1}(s))=\emptyset.\) The object G is called complete, if for every e-subobject \((G_ 0,\mu_ 0)\) of any object \(G_ 1\) (i.e. \(G_ 0=(S_ 0,\nu_ 0)\) is main subobject of \(G_ 1=(S_ 1,\nu_ 1)\) and \(\nu_ 1^{-1}\mu_ 0(S_ 0)\) is a r.e. set) and every morphism \(\mu\) : \(G_ 0\to G\), there exists a morphism \(\mu_ 1: G_ 1\to G\) such that \(\mu =\mu_ 1\mu_ 0\). The central result of the note is: the object \(G=(S,\nu)\), \(| S| \geq 2\), is complete iff there is an \(s\in S\) such that \(N\setminus A\leq_ e\nu^{-1}(s)\) for every r.e. set A \((N=\{0,1,...\})\).