What internal set theory knows about standard sets

What is the real category of sets?

Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a natural way to describe the relation between mathematics and metamathematics: if metamathematics can be described by using categories (in particular syntactic categories), then the mathematical level is represented by internal categories. Through this two-level interpretation we can clarify the relation between classes and sets in ZF and, in particular, we ca…

单纯同伦类型论不仅仅是单纯的：什么是 $\infty$-范畴？

Simplicial Homotopy Type Theory is not just Simplicial: What are $\infty$-Categories?

本文证明了单纯同伦类型论（sHoTT）存在非单纯对象的模型，揭示了在一般数学基础中 $\infty$-范畴的概念比先前假设的更为普遍。