Topological categories, quantaloids and Isbell adjunctions

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.

23 pages, final version. License updated

References in corpus (3)

• Categorical structures enriched in a quantaloid: categories, distributors and functors
• Categorical structures enriched in a quantaloid: tensored and cotensored categories
• Limits and colimits of quantaloid-enriched categories and their distributors

Cited by in corpus (9)

• Fuzzy Galois connections on fuzzy sets
• Chu connections and back diagonals between $\mathcal{Q}$-distributors
• Lax Distributive Laws for Topology, I
• Fixed points of adjoint functors enriched in a quantaloid
• Limits and colimits of quantaloid-enriched categories and their distributors
• Completeness and injectivity
• Density in categorical topology via quantaloid-enriched categories
• Quantaloidal Completions of Order-enriched Categories and Their Applications
• Diagonals between $\mathcal{Q}$-distributors