$\mathcal{Q}$-closure spaces

TL;DR

This paper introduces $ ext{Q}$-closure spaces, a generalization of closure spaces in enriched category theory, and establishes an equivalence between the category of these spaces with continuous relations and complete lattices with supremum-preserving maps.

Contribution

It systematically develops the theory of $ ext{Q}$-closure spaces, including their morphisms and examples like fuzzy closure spaces, and proves a fundamental categorical equivalence.

Findings

Category of closure spaces and closed continuous relations is equivalent to complete lattices and $ ext{sup}$-preserving maps.

Fuzzy closure spaces are introduced as a special case of $ ext{Q}$-closure spaces.

The framework generalizes classical closure space theory to enriched categorical contexts.

Abstract

For a small quantaloid $\mathcal{Q}$, a $\mathcal{Q}$-closure space is a small category enriched in $\mathcal{Q}$ equipped with a closure operator on its presheaf category. We investigate $\mathcal{Q}$-closure spaces systematically with specific attention paid to their morphisms and, as preordered fuzzy sets are a special kind of quantaloid-enriched categories, in particular fuzzy closure spaces on fuzzy sets are introduced as an example. By constructing continuous relations that naturally generalize continuous maps, it is shown (in the generality of the $\mathcal{Q}$-version) that the category of closure spaces and closed continuous relations is equivalent to the category of complete lattices and $\sup$-preserving maps.