Completeness of quantaloid-enriched categories up to Morita equivalence

Abstract

For a small quantaloid $\mathcal{Q}$, we introduce $\mathcal{M}$-(co)complete $\mathcal{Q}$-categories, i.e., (co)complete $\mathcal{Q}$-categories up to Morita equivalence, as Eilenberg--Moore algebras of the presheaf monad on the category of $\mathcal{Q}$-categories and left adjoint $\mathcal{Q}$-distributors, and characterize such $\mathcal{Q}$-categories through $\mathcal{M}$-(co)tensoredness and $\mathcal{M}$-conical (co)completeness.