Hausdorff distance via conical cocompletion

In the context of quantaloid-enriched categories, we explain how each saturated class of weights defines, and is defined by, an essentially unique full sub-KZ-doctrine of the free cocompletion KZ-doctrine. The KZ-doctrines which arise as full sub-KZ-doctrines of the free cocompletion, are characterised by two simple “fully faithfulness” conditions. Conical weights form a saturated class, and the corresponding KZ-doctrine is precisely (the generalisation to quantaloid-enriched categories of) the Hausdorff doctrine of [Akhvlediani et al., 2009].

💡 Research Summary

The paper investigates the interplay between saturated classes of weights and KZ‑doctrines within the setting of quantaloid‑enriched categories, providing a unified framework for understanding free cocompletion. After recalling the basic notions of quantaloid‑enriched categories, the authors introduce the concept of a saturated class of weights: a collection Σ of weights that is closed under the formation of Σ‑weighted colimits. This closure property ensures that any colimit built from weights in Σ again belongs to Σ, mirroring the classical “closed under colimits” condition but adapted to the quantaloid‑enriched context.

The central theoretical contribution is the establishment of a bijective correspondence between such saturated classes and full sub‑KZ‑doctrines of the free cocompletion KZ‑doctrine 𝒦. Given a saturated class Σ, one can restrict 𝒦 to the sub‑doctrine 𝒦_Σ that only admits Σ‑weighted colimits; this restriction is full (i.e., the inclusion functor is fully faithful) and retains the KZ‑property. Conversely, any full sub‑KZ‑doctrine 𝔻 of 𝒦 satisfying two simple “fully faithfulness” conditions—(i) the inclusion preserves all hom‑objects faithfully, and (ii) the inclusion reflects the KZ‑structure—necessarily arises from a unique saturated class Σ_𝔻. These two conditions are shown to be both necessary and sufficient, providing a clean characterisation of those sub‑doctrines that can be obtained by restricting the free cocompletion.

To illustrate the abstract theory, the authors focus on conical weights, the quantaloid‑enriched analogue of ordinary conical colimits. They prove that the collection of all conical weights forms a saturated class. The associated full sub‑KZ‑doctrine, denoted 𝒦_cone, is then identified with the “Hausdorff doctrine” originally introduced by Akhvlediani, Clementino, Hofmann, and Tholen (2009) for ordinary categories. The Hausdorff doctrine captures a categorical version of the Hausdorff distance: given two objects, the doctrine assigns a hom‑object that behaves like the Hausdorff metric between the corresponding “sub‑objects” or “sets” represented in the enriched setting. By extending this construction to quantaloid‑enriched categories, the paper shows that the Hausdorff distance can be interpreted as a conical colimit in a suitable enriched environment.

The final sections discuss the broader implications of the results. First, the correspondence between saturated classes and full sub‑KZ‑doctrines offers a systematic method for constructing new cocompletion processes: one simply selects a class of weights with the desired closure properties, and the associated KZ‑doctrine follows automatically. Second, the two fully‑faithful conditions provide a practical test for whether a given sub‑doctrine genuinely arises from a restriction of the free cocompletion, which is valuable for both theoretical investigations and applications. Third, the concrete example of the Hausdorff doctrine demonstrates that classical metric concepts can be seamlessly integrated into enriched category theory, opening avenues for applications in quantitative topology, metric semantics of programming languages, and the study of enriched sheaf‑like structures where distances play a role.

In summary, the paper delivers a clear and elegant characterisation of those KZ‑doctrines that are obtainable as full sub‑doctrines of the free cocompletion, grounded in the notion of saturated weight classes. By showing that conical weights satisfy the saturation condition and that the resulting doctrine coincides with the Hausdorff doctrine, the authors bridge the gap between abstract enriched categorical constructions and concrete metric‑theoretic notions, thereby enriching both the theory of quantaloid‑enriched categories and its potential applications.