Dear all, Lili Shen (Sichuan University) will give a talk for this month's PCT seminar (https://pctseminar.github.io/) on Friday, July 10th at 10am JST/11am AED (1am UTC). The zoom link to join the seminar is: https://kyoto-u-edu.zoom.us/j/82401794000?pwd=VjVmzVjmlgxhkdHES4xYeQBTpMzQkS... Title: Towards the categorical foundation of quantale-valued sets Abstract: A classical set may be regarded as a set equipped with an equality relation valued in the two-element Boolean algebra. Fourman and Scott generalized this idea by introducing Ω-sets, where equality is valued in a frame Ω; this construction is closely related to sheaves and topos theory. Höhle and his collaborators later developed the theory of quantale-valued sets, replacing the frame Ω by a quantale Q. This talk provides an overview of the theory of quantale-valued sets and the ongoing efforts to establish its categorical foundation. I will explain why Q-sets form a natural and interesting generalization of Fourman–Scott Ω-sets, and then present two recent results that clarify the extent to which the category of Q-sets behaves like the category of sets or the category of Ω-sets. First, for a commutative and divisible quantale Q, the category of Q-sets is a topos if and only if Q is a frame. Second, for the unit interval [0,1] equipped with a continuous t-norm *, the category of ([0,1], *)-sets is cartesian closed if and only if * is the minimum t-norm. We hope to see you there! Best wishes, Soichiro Fujii, Zeinab Galal, JS Lemay