Lawvere completeness in Topology

It is known since 1973 that Lawvere’s notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for $(\mathbb{T},\mathsf{V})$-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that $\mathsf{V}$ has a canonical $(\mathbb{T},\mathsf{V})$-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of $(\mathbb{T},\mathsf{V})$-categories.

💡 Research Summary

The paper extends Lawvere’s notion of Cauchy‑complete enriched categories to the setting of ((\mathbb{T},\mathsf{V}))-categories, where (\mathbb{T}) is a monoidal (or more generally algebraic) structure and (\mathsf{V}) is a quantale (a complete lattice equipped with a monoidal product). After recalling the classical result that Lawvere‑complete (\mathsf{V})-categories coincide with Cauchy‑complete metric spaces, the authors define a Lawvere‑completeness condition for ((\mathbb{T},\mathsf{V}))-categories: every “Cauchy weight” (a ((\mathbb{T},\mathsf{V}))-module satisfying the usual idempotency and associativity constraints) must be representable by an actual object of the category.

The main contributions are twofold. First, when the ambient quantale is the two‑element lattice (\mathbf{2}) and (\mathbb{T}) is chosen appropriately, ((\mathbb{T},\mathsf{V}))-categories become ordinary topological spaces. In this case the authors prove that Lawvere‑completeness is exactly the property of weak sobriety: every non‑trivial irreducible closed set is the closure of a point. This provides a novel categorical characterisation of a classical topological separation condition, linking it to the existence of representing objects for Cauchy weights.

Second, when (\mathsf{V}=