Topological categories, quantaloids and Isbell adjunctions

TL;DR

This paper connects topological categories with quantaloid-enriched categories, expanding Garner's recent results by developing foundational quantaloid theory and demonstrating their relevance to categorical topology and related fields.

Contribution

It introduces a framework unifying topological and enriched category theories using quantaloids, extending Garner's work and broadening applications in categorical topology.

Findings

Unified the notion of topologicity with total cocompleteness in enriched categories.

Developed foundational tools in quantaloid theory for categorical topology.

Linked quantaloid-enriched categories to applications in quantum logic and sheaf theory.

Abstract

In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categorical topology into the context of quantaloid-enriched category theory, a field that previously drew its motivation and applications from other domains, such as quantum logic and sheaf theory.