Adriana Balan
University Politehnica of Bucharest, Bucharest, Romania

Fifty years ago, Lawvere's observation that ordered sets and metric spaces can be seen as enriched categories over a quantale opened a wide path for the quantitative theory of domains, using and generalising ideas from category theory, algebra, and topology. Among the most useful and pleasing properties of such quantale-enriched categories are undoubtedly (co)completeness with respect to certain classes of (co)limits and commutation of such limits and colimits (with prominent examples featuring the ordered case, like Heyting algebras, continuous or completely distributive lattices).

The present talk considers cocomplete enriched categories over a commutative quantale $\mathcal{V}$. These are known in a variety of guises: as algebras for the $\mathcal{V}$-valued powerset monad on $\mathsf{Set}$, as injective $\mathcal{V}$-categories with respect to fully faithful functors, as cocomplete lattices equipped with an action of $\mathcal{V}$, or even as complete semimodules over $\mathcal{V}$, seen as an idempotent semiring.

There is a symmetric monoidal closed structure on the category $\mathsf{CoCts}(\mathcal{V})$ of cocomplete $\mathcal{V}$-categories and cocontinuous $\mathcal{V}$-functors. The corresponding tensor product $\otimes$ arises naturally, using for example the commutativity of the $\mathcal{V}$-valued powerset monad on $\mathsf{Set}$. Several descriptions of this tensor product will be presented, via Galois connections or the $\mathcal{V}$-valued variant of Raney's $G$-ideals. Nuclear objects/morphisms with respect to this tensor product will turn to be deeply related with the usual projectivity with respect to the class of regular epis in $\mathsf{CoCts}(\mathcal{V})$, hence with completely distributive cocomplete $\mathcal{V}$-categories. In particular, the nuclearity of the Dedekind-MacNeille-Isbell completion will be discussed.