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When is Cat(Q) cartesian closed?

Isar Stubbe, Junche Yu

TL;DR

This work provides a complete, elementary criterion for when the category $\text{Cat}(\mathcal{Q})$ of $\mathcal{Q}$-enriched categories is cartesian closed, showing this occurs exactly when $\mathcal{Q}$ is locally localic and a precise four-case formula governs $(g\circ f)\land h$ across object-type triples. The approach unifies prior results (e.g., locales and certain quantales) and yields new examples, while correcting earlier claims about free quantaloids. It also analyzes a range of constructions (coproducts, diagonals, collage-type distributors) to delineate when cartesian closedness is preserved or fails. Overall, the paper clarifies the role of base quantaloid structure in exponentiability within $\text{Cat}(\mathcal{Q})$ and offers concrete guidance for identifying cartesian closed base quantaloids.

Abstract

We give an elementary characterization of those quantaloids Q for which the category Cat(Q) of Q-enriched categories and functors is cartesian closed. We then unify several known cases (previously proven using ad hoc methods) and we give some new examples.

When is Cat(Q) cartesian closed?

TL;DR

This work provides a complete, elementary criterion for when the category of -enriched categories is cartesian closed, showing this occurs exactly when is locally localic and a precise four-case formula governs across object-type triples. The approach unifies prior results (e.g., locales and certain quantales) and yields new examples, while correcting earlier claims about free quantaloids. It also analyzes a range of constructions (coproducts, diagonals, collage-type distributors) to delineate when cartesian closedness is preserved or fails. Overall, the paper clarifies the role of base quantaloid structure in exponentiability within and offers concrete guidance for identifying cartesian closed base quantaloids.

Abstract

We give an elementary characterization of those quantaloids Q for which the category Cat(Q) of Q-enriched categories and functors is cartesian closed. We then unify several known cases (previously proven using ad hoc methods) and we give some new examples.
Paper Structure (3 sections, 7 theorems, 17 equations)

This paper contains 3 sections, 7 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Cartesian closedness of ${\sf Cat}(\mathcal{Q})$
  3. Examples

Key Result

Theorem 1.1

A functor $F\colon\mathbb{A}\hbox{$\xymatrix@1@C=5mm{\ar@{->}[r]&}$}\mathbb{B}$ between $\mathcal{Q}$-enriched categories is exponentiable in ${\sf Cat}(\mathcal{Q})$, i.e. the functor admits a right adjoint, if and only if the following two conditions hold:

Theorems & Definitions (17)

  • Theorem 1.1: CHS09
  • Corollary 1.2: CHS09
  • Example 1.3: CHS09
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Example 3.1: Quantales, locales
  • Example 3.2: A non-locale example
  • ...and 7 more