0-Ideal Monad and Its Applications to Approach Spaces

In this paper, a notion of 0-ideals on a set is proposed and further it is shown that 0-ideals give rise to a power-enriched monad \(\mathbb {Z}\)=\(({\textbf{Z}},m,e)\) on the category of sets, namely a 0-ideal monad. As a first application, a new characterization of approach spaces is given by verifying that the category \({\mathbb {Z}}\)-Mon of \({\mathbb {Z}}\)-monoids is isomorphic to the category App of approach spaces. The second application consists of two components: (i) Based on 0-ideals, a concept of approach 0-convergence spaces is introduced. (ii) By using the Kleisli extension of \({\textbf{Z}}\), the existence of an isomorphism between the category AConv of approach 0-convergence spaces and the category \({(\mathbb {Z},2)}\)-Cat of relational \({\mathbb {Z}}\)-algebras is verified. Then from the fact that \({\mathbb {Z}}\)-Mon and \({(\mathbb {Z},2)}\)-Cat are isomorphic, another new description of approach spaces is obtained by an isomorphism between AConv and App.