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Lax Distributive Laws for Topology, I
Category Theory · Submitted 20 Mar 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $\mathcal{Q}$.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For a quantaloid $\mathcal{Q}$, considered as a bicategory, Walters introduced categories enriched in $\mathcal{Q}$. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. | contribution 0.98 | 0.89 | 0.08 | asserted 1.00 |
| s2 | 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. | contribution 0.51 | 0.90 | 0.13 | asserted 1.00 |
| s3 | 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 $\mathcal{Q}$. | result 0.85 | 0.93 | 0.82 | asserted 1.00 |
| s4 | 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. | contribution 0.53 | 0.85 | 0.11 | asserted 1.00 |
| s5 | 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. | contribution 0.93 | 0.85 | 0.07 | asserted 1.00 |
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The paper
Original 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.
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References
How the literature uses it
6 arXiv papers cite this in 15 sentences
Papers citing it · 8 sentences without a confident label
Builds on it 1 paper
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In this section we establish the non-discrete version of the lax distributive laws considered in [ref] .
Lax distributive laws for topology, II · The non-discrete version of lax distributive laws and their lax algebras · 0.99
Compares with it 1 paper
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Closer to the work in this paper are the lax distributive laws in [ref] , though again these focus on monad-monad interactions.
Lax Liftings and Lax Distributive Laws · Introduction · 0.87
Cites it as background 3 papers
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First we recall the “discrete version" of lax extensions of monads (see [ref] ), by considering a mere functor $T:→$.
Monads on Q-Cat and their lax extensions to Q-Dist · From laxly extended monads on $$ to laxly extended monads on $$ · 0.87
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An interesting connection between topological theories and lax distributive laws is exposed in [ref] .
Duality theory for enriched Priestley spaces · Duality theory for enriched Priestley spaces: concretely · 0.91
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For $=[0,∞]$ one obtains as $$-categories generalized partial (ultra-)metric spaces, as studied by various authors: [ref] .
Lax distributive laws for topology, II · Quantaloid-enriched categories and their distributors · 0.89
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