$\mathscr Q$-Sets and Friends: Categorical Constructions and Categorical Properties

TL;DR

This paper introduces the category of $ ext{ extscr{Q}}$-sets based on a commutative semicartesian quantale, detailing its categorical properties, limits, colimits, and monoidal structures, with implications for change of basis.

Contribution

It defines the category of $ ext{ extscr{Q}}$-sets, analyzes its limits, colimits, and monoidal structures, and explores the impact of quantale morphisms on these structures.

Findings

The category of $ ext{ extscr{Q}}$-sets is complete and cocomplete.

It has a classifier for regular subobjects.

It is $oldsymbol{ ext{ extscr{Q}}^+}$-locally presentable.

Abstract

This work mainly concerns the -- here introduced -- category of $\mathscr Q$-sets and functional morphisms, where $\mathscr Q$ is a commutative semicartesian quantale. We describe, in detail, the limits and colimits of this complete and cocomplete category and prove that it has a classifier for regular subobjects. Moreover, we prove that it is $\kappa^+$-locally presentable category, where $\kappa=max\{|\mathscr Q|, \aleph_0)\}$ and describe a hierarchy of semicartesian monoidal closed structures in this category. Finally, we discuss the issue of 'change of basis' induced by appropriate morphisms between the parametrizing quantales involved in the definition of $\mathscr Q$-sets. In a future work we will address such questions in the full subcategory given by all Scott-complete $\mathscr Q$-sets