Categorical structures enriched in a quantaloid: tensored and cotensored categories

TL;DR

This paper explores the structure of categories enriched in a quantaloid, characterizing cocomplete categories through tensoring and cotensoring properties, and analyzing the relationship between Sup-valued homomorphisms and Q-categories.

Contribution

It provides a new characterization of cocomplete Q-categories via tensor and cotensor structures, expanding the understanding of enriched category theory in quantaloids.

Findings

Cocomplete Q-categories are precisely those that are tensored and conically cocomplete.

Alternatively, cocomplete Q-categories are those that are tensored, cotensored, and order-cocomplete.

Analysis of the relationship between Sup-valued homomorphisms and Q-categories.

Abstract

Our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and order-cocomplete. Bearing this in mind, we analyze how Sup-valued homomorphisms on Q are related to Q-categories. With an appendix on action, representation and variation.