Lax Distributive Laws for Topology, I

Abstract

For a quantaloid $\mathcal{Q}$, considered as a bicategory, Walters introduced categories enriched in $\mathcal{Q}$. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad $\mathbb{T}$ over the discrete presheaf monad of the small quantaloid $\mathcal{Q}$ the primary data of the theory, rather than the lax monad extensions of $\mathbb{T}$ to the category of $\mathcal{Q}$-relations that they equivalently describe. The central piece of the paper establishes a Galois correspondence between such lax distributive laws and lax Eilenberg-Moore $\mathbb{T}$-algebra structures on the set of discrete presheaves over the object set of $\mathcal{Q}$. We give a precise comparison of these structures with the more restrictive notion introduced by Hofmann in the case of a commutative quantale, called natural topological theories here, and describe the lax monad extensions introduced by him as minimal. Throughout the paper, a variety of old and new examples of ordered, metric and topological structures illustrate the theory developed, which includes the consideration of algebraic functors and change-of-base functors in full generality.

32 pages

References in corpus (7)

• Categorical structures enriched in a quantaloid: categories, distributors and functors
• Categorical structures enriched in a quantaloid: tensored and cotensored categories
• Topological categories, quantaloids and Isbell adjunctions
• $\mathcal{Q}$-closure spaces
• Quantale-valued Approach Spaces via Closure and Convergence
• Limits and colimits of quantaloid-enriched categories and their distributors
• Monads on Q-Cat and their lax extensions to Q-Dist

Cited by in corpus (4)

• Completely distributive enriched categories are not always continuous
• A categorical approach to the maximum theorem
• Monads on Q-Cat and their lax extensions to Q-Dist
• Diagonals between $\mathcal{Q}$-distributors