$\mathsf{Q}\text{-}\mathbf{Set}$ is not generally a topos

TL;DR

This paper investigates the conditions under which the category of $ ext{Q}$-sets forms a topos, revealing that this occurs precisely when $ ext{Q}$ is a frame, thus clarifying the structural requirements.

Contribution

It establishes a necessary and sufficient condition for the category of $ ext{Q}$-sets to be a topos, specifically that $ ext{Q}$ must be a frame.

Findings

Category of $ ext{Q}$-sets is a topos iff $ ext{Q}$ is a frame.

Provides a characterization linking algebraic properties of $ ext{Q}$ to categorical structure.

Clarifies the limitations of $ ext{Q}$-sets as a topos based on quantale properties.

Abstract

For a commutative, unital and divisible quantale $\mathsf{Q}$, it is shown that the category of $\mathsf{Q}$-sets is a topos if, and only if, $\mathsf{Q}$ is a frame.