Lawvere completion and separation via closure

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 $\V$.

💡 Research Summary

The paper develops a unified closure‑theoretic framework for describing completeness and separation in categories enriched over a quantale 𝕍, and then extends this framework to the more general setting of 𝕋‑categories, where 𝕋 = (𝕋, 𝕍, ξ) is a topological theory consisting of a set monad 𝕋, the same quantale 𝕍, and a compatible 𝕋‑algebra structure ξ on 𝕍.

1. Closure in 𝕍‑categories.
Starting from a quantale 𝕍, the authors define a closure operator
 cl : ℘|X| → ℘|X|
on the underlying set of any 𝕍‑category 𝔛 = (X, a). For a subset A⊆X, cl(A) consists of all points x for which the enriched hom‑value a(x, y) is “≤ k” (the unit of 𝕍) for some y∈A. This operator is monotone, extensive, and idempotent, exactly mirroring the usual topological closure when 𝕍 =