Completeness of quantaloid-enriched categories up to Morita equivalence
Xiaoye Tang
TL;DR
The article develops a framework for completeness and cocompleteness of quantaloid-enriched categories up to Morita equivalence by constructing presheaf and copresheaf monads on the 2-category of maps between quantaloid distributors. It identifies the Eilenberg–Moore algebras of these monads with skeletal cocomplete and complete $\\mathcal{Q}$-categories, respectively, and introduces notions of $\\mathcal{M}$-cocompleteness, $\\mathcal{M}$-completeness, and their (co)tensored and conical variants. A central result is that $\\mathcal{M}$-cocompleteness is equivalent to being $\\mathcal{M}$-tensored and $\\mathcal{M}$-conically cocomplete (and dually for completeness), aligning Morita-equivalence with Cauchy completion. The paper also provides explicit constructions and examples showing the separation between $\\mathcal{M}$-cocompleteness and ordinary cocompleteness, and clarifies how Morita equivalence allows working within $\\mathbf{Map}(\\mathcal{Q}$$-\\bf Dist)$ for completeness 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.