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The powerset monad on quantale-valued sets

Lili Shen, Xiaojuan Zhao

TL;DR

The paper develops a $\mathcal{Q}$-enriched generalization of the powerset monad by working with small involutive quantaloids $\mathcal{Q}$ and the associated categories of symmetric $\mathcal{Q}$-categories. It constructs the presheaf-based powerset functor $\mathsf{P}_{\mathsf{s}}$ via the adjunction between $\mathcal{Q}$-SymCat and $\mathcal{Q}$-Cat, with unit given by the symmetrized Yoneda embedding and multiplication by a symmetrized supremum, and proves Beck-style monadicity to show strict monadicity of $\mathfrak{U}_{\mathsf{s}}$; this yields a concrete $\mathsf{Q}$-powerset monad on $\mathsf{Q}$-Set. The main result identifies the $\mathsf{Q}$-powerset of a $\mathsf{Q}$-set $X$ as the image of $X$ under $\mathfrak{U}_{\mathsf{s}}\mathsf{P}_{\mathsf{s}}$, i.e., the symmetrization of the presheaf $\mathbf{D}_*(\mathsf{Q})$-category of $X$, providing an explicit algebraic description that generalizes the classical powerset monad and recovers it when $\mathsf{Q}= \mathbf{2}$. These findings give a rigorous framework for quantale-valued set constructions and clarify how symmetrization interacts with enrichment to produce the $\mathsf{Q}$-powerset.

Abstract

For a small involutive quantaloid $\mathcal{Q}$, it is shown that the category of separated complete $\mathcal{Q}$-categories and left adjoint $\mathcal{Q}$-functors is strictly monadic over the category of symmetric $\mathcal{Q}$-categories. In particular, the (covariant) powerset monad on the category of quantale-valued sets is precisely formulated.

The powerset monad on quantale-valued sets

TL;DR

The paper develops a -enriched generalization of the powerset monad by working with small involutive quantaloids and the associated categories of symmetric -categories. It constructs the presheaf-based powerset functor via the adjunction between -SymCat and -Cat, with unit given by the symmetrized Yoneda embedding and multiplication by a symmetrized supremum, and proves Beck-style monadicity to show strict monadicity of ; this yields a concrete -powerset monad on -Set. The main result identifies the -powerset of a -set as the image of under , i.e., the symmetrization of the presheaf -category of , providing an explicit algebraic description that generalizes the classical powerset monad and recovers it when . These findings give a rigorous framework for quantale-valued set constructions and clarify how symmetrization interacts with enrichment to produce the -powerset.

Abstract

For a small involutive quantaloid , it is shown that the category of separated complete -categories and left adjoint -functors is strictly monadic over the category of symmetric -categories. In particular, the (covariant) powerset monad on the category of quantale-valued sets is precisely formulated.
Paper Structure (4 sections, 9 theorems, 91 equations)

This paper contains 4 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Categories and symmetric categories enriched in a quantaloid
  3. Complete categories enriched in a quantaloid
  4. The powerset monad on quantale-valued sets

Key Result

Lemma 2.1

Let $(X,\alpha)$, $(Y,\beta)$ be $\mathcal{Q}$-categories. If $(X,\alpha)$ is symmetric, then $f\colon(X,\alpha)\to(Y,\beta)$ is a $\mathcal{Q}$-functor if, and only if, $f\colon(X,\alpha)\to(Y,\beta_{\mathsf{s}})$ is a $\mathcal{Q}$-functor.

Theorems & Definitions (22)

  • Lemma 2.1
  • Remark 3.1
  • Example 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • Proposition 3.6
  • Remark 3.7
  • Proposition 3.8
  • ...and 12 more