Categorical structures enriched in a quantaloid: categories, distributors and functors

TL;DR

This paper provides a comprehensive study of categories, functors, and distributors enriched in a quantaloid, extending classical concepts like Kan extensions, limits, and Morita equivalence within this enriched framework.

Contribution

It systematically develops the theory of Q-enriched categories, including adjoints, (co)limits, presheaves, and Morita equivalence, with an appendix on the universality of the quantaloid of enriched categories.

Findings

Unified framework for enriched categories and distributors

Extension of classical categorical concepts to quantaloid enrichment

Universal property of the quantaloid of enriched categories

Abstract

We thoroughly treat several familiar and less familiar definitions and results concerning categories, functors and distributors enriched in a base quantaloid Q. In analogy with V-category theory we discuss such things as adjoint functors, (pointwise) left Kan extensions, weighted (co)limits, presheaves and free (co)completion, Cauchy completion and Morita equivalence. With an appendix on the universality of the quantaloid Dist(Q) of Q-enriched categories and distributors.