Lawvere completion and separation via closure

TL;DR

This paper introduces a closure-theoretic framework for understanding completeness and separation in enriched categories over quantales and extends it to more general topological theories involving monads and algebra structures.

Contribution

It develops a unified closure-theoretic approach to completeness and separation in $ ext{V}$-categories and generalizes it to $ ext{T}$-categories with monad and algebra structures.

Findings

Provides a closure-theoretic characterization of completeness and separation.

Generalizes the approach to $ ext{T}$-categories with monads and algebra structures.

Lays groundwork for further categorical and topological investigations.

Abstract

For a quantale $\V$, first a closure-theoretic approach to completeness and separation in $\V$-categories is presented. This approach is then generalized to $\Tth$-categories, where $\Tth$ is a topological theory that entails a set monad $\mT$ and a compatible $\mT$-algebra structure on $\V$.