基本拓扑
Elementary topoi
摘要：
作为原型范畴，$\mathbf{Set}$具有许多性质，使其在范畴中具有特殊性。从数理逻辑的观点来看，其中一个性质是$\mathbf{Set}$有足够的结构来“适当地”形式化逻辑。然而，我们可以问，在另一个范畴$\mathbf{C}$中形式化逻辑可能意味着什么。一个（基本的）topos的概念提炼了$\mathbf{Set}$的基本特征，这些特征使我们能够这样做。本报告定义了topoi，并描述了topos中一阶逻辑和集合论的发展。
英文摘要：
As the prototypical category, $\mathbf{Set}$ has many properties which make it special amongst categories. From the point of view of mathematical logic, one such property is that $\mathbf{Set}$ has enough structure to "properly" formalise logic. However, we could ask what it might mean to formalise logic in another category $\mathbf{C}$. The notion of an (elementary) topos distills the essential features of $\mathbf{Set}$ which allow us to do this. This expository report defines topoi, and describes the development of first-order logic and set theory within a topos.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
