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On the tensor product of completely distributive quantale-enriched categories

Adriana Balan

TL;DR

The paper analyzes the monoidal structure of $\mathcal{V}\text{-}\mathbf{Sup}$, the category of separated cocomplete $\mathcal{V}$-categories, proving it is $*$-autonomous and identifying dualizable objects with completely distributive cocomplete $\mathcal{V}$-categories. It develops the free cocompletion 2-monad $\mathbbm D$, proves its KZ property and monoidal compatibility, and uses these tools to construct and analyze the tensor product $\otimes_{\mathcal{V}}$, described as a coreflexive inverter and characterized by a universal bimorphism property. Through a Galois-map perspective, it shows the tensor product objects are weighted colimits of representables and establishes a $2$-categorical equivalence with bimorphisms. The central result is that the tensor product preserves complete distributivity, so completely distributive cocomplete $\mathcal{V}$-categories are precisely the nuclear/dualizable objects in $\mathcal{V}\text{-}\mathbf{Sup}$, with implications for compact closedness and potential generalizations to quantaloid-enriched contexts. The work offers a coherent, algebraic approach to tensor products in the enriched setting, connecting monadic, Galois, and presheaf-theoretic techniques to yield structural insights with broad applicability in enriched category theory.

Abstract

Tensor products are ubiquitous in algebra, topology, logic and category theory. The present paper explores the monoidal structure of the category $\mathcal{V}\hspace{0pt}\mbox{-}\hspace{.5pt}\mathbf{Sup}$ of separated cocomplete enriched categories over a commutative quantale $\mathcal{V}$, the many-valued analogue of complete sup-lattices. We recover the known result that $\mathcal{V}\hspace{0pt}\mbox{-}\hspace{.5pt}\mathbf{Sup}$ is $*$-autonomous and we show that the nuclear/dualizable objects in $\mathcal{V}\hspace{0pt}\mbox{-}\hspace{.5pt}\mathbf{Sup}$ are precisely the completely distributive cocomplete $\mathcal{V}$-categories.

On the tensor product of completely distributive quantale-enriched categories

TL;DR

The paper analyzes the monoidal structure of , the category of separated cocomplete -categories, proving it is -autonomous and identifying dualizable objects with completely distributive cocomplete -categories. It develops the free cocompletion 2-monad , proves its KZ property and monoidal compatibility, and uses these tools to construct and analyze the tensor product , described as a coreflexive inverter and characterized by a universal bimorphism property. Through a Galois-map perspective, it shows the tensor product objects are weighted colimits of representables and establishes a -categorical equivalence with bimorphisms. The central result is that the tensor product preserves complete distributivity, so completely distributive cocomplete -categories are precisely the nuclear/dualizable objects in , with implications for compact closedness and potential generalizations to quantaloid-enriched contexts. The work offers a coherent, algebraic approach to tensor products in the enriched setting, connecting monadic, Galois, and presheaf-theoretic techniques to yield structural insights with broad applicability in enriched category theory.

Abstract

Tensor products are ubiquitous in algebra, topology, logic and category theory. The present paper explores the monoidal structure of the category of separated cocomplete enriched categories over a commutative quantale , the many-valued analogue of complete sup-lattices. We recover the known result that is -autonomous and we show that the nuclear/dualizable objects in are precisely the completely distributive cocomplete -categories.
Paper Structure (22 sections, 19 theorems, 47 equations)

This paper contains 22 sections, 19 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper.
  3. Preliminaires
  4. Quantales.
  5. Quantale enriched categories.
  6. Quantale-enriched distributors.
  7. Limits and colimits.
  8. Cocomplete $\mathcal{V}$-categories.
  9. The free cocompletion monad on $\mathcal{V}\space\hbox{-}\space{\mathbf{Cat}}$
  10. The ($2$-)functor $\mathbbm D$.
  11. The ($2$-)monad $\mathbbm D$.
  12. The Kock-Zöberlein property.
  13. Monoidal structure of the monad $\mathbbm D$.
  14. $\mathbbm D$-algebras.
  15. The tensor product of cocomplete $\mathcal{V}$-categories
  16. ...and 7 more sections

Key Result

Proposition 2.6

Stubbe2005 The following are equivalent for a $\mathcal{V}$-category ${\mathscr{X}}$:

Theorems & Definitions (47)

  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Proposition 2.6
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 4.1
  • ...and 37 more