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

Xiao Hu, Lili Shen

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.