Completeness of quantaloid-enriched categories up to Morita equivalence

TL;DR

This paper develops a framework for understanding the completeness of quantaloid-enriched categories up to Morita equivalence, using Eilenberg--Moore algebras and characterizations involving (co)tensoredness.

Contribution

It introduces the concept of $ ext{M}$-(co)completeness for $ ext{Q}$-categories and characterizes these categories via new algebraic and categorical conditions.

Findings

Defines $ ext{M}$-(co)complete $ ext{Q}$-categories.

Characterizes such categories through $ ext{M}$-(co)tensoredness.

Provides a framework for Morita equivalence in quantaloid-enriched categories.

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.