Q-modules are Q-suplattices

It is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using enriched category theory and a lemma on KZ doctrines we prove (the generalization of) this fact in the case of ordered sheaves on a small quantaloid. Comparing module-equivalence with sheaf-equivalence for quantaloids and using the notion of centre of a quantaloid, we refine a result of F. Borceux and E. Vitale.

💡 Research Summary

The paper investigates the relationship between modules on a quantaloid and internal suplattices (complete join‑semilattices) in the topos of ordered sheaves on that quantaloid. The classical result that, for a locale L, internal suplattices in the sheaf topos Sh(L) coincide with L‑modules is taken as a starting point. The authors aim to generalize this correspondence to the setting of a small quantaloid 𝒬, where the notion of “ordered sheaf” is interpreted as a 𝒬‑enriched presheaf equipped with an internal order structure.

The technical core rests on enriched category theory and a lemma concerning Kock‑Zöberlein (KZ) doctrines. The authors first show that the 2‑category of ordered sheaves OrdSh(𝒬) carries a KZ‑doctrine that yields a free cocompletion by internal suplattices. Using the KZ lemma, they construct a pair of mutually inverse 𝒬‑enriched functors:

• From an ordered sheaf F they build its suplattice completion Sup(F), which naturally inherits a 𝒬‑module structure.
• Conversely, given a 𝒬‑module M they extract an ordered sheaf U(M) by forgetting the module action while retaining the underlying order‑enriched presheaf.

These constructions establish an equivalence \