Kleisli Monoids Describing Approach Spaces

Abstract

We study the functional ideal monad $\mathbb {I} = (\mathsf {I}, m, e)$ on S e t and show that this monad is power-enriched. This leads us to the category $\mathbb {I}$- M o n of all $\mathbb {I}$-monoids with structure preserving maps. We show that this category is isomorphic to A p p, the category of approach spaces with contractions as morphisms. Through the concrete isomorphism, an $\mathbb {I}$-monoid (X,ν) corresponds to an approach space $(X, \mathfrak {A}),$ described in terms of its bounded local approach system. When I is extended to R e l using the Kleisli extension $\check {\mathsf {I}},$ from the fact that $\mathbb {I}$- M o n and $(\mathbb {I},2)$- C a t are isomorphic, we obtain the result that A p p can be isomorphically described in terms of convergence of functional ideals, based on the two axioms of relational algebras, reflexivity and transitivity. We compare these axioms to the ones put forward in Lowen (2015). Considering the submonad $\mathbb {B}$ of all prime functional ideals, we show that it is both sup-dense and interpolating in $\mathbb {I}$, from which we get that $(\mathbb {I},2)$- C a t and $(\mathbb {B},2)$- C a t are isomorphic. We present some simple axioms describing A p p in terms of prime functional ideal convergence.