This paper provides a preparatory introduction to sheaves and topoi, written as a conceptual continuation of the author's earlier introduction to torsors and as preparatory background for the author's arXiv paper \emph{Grothendieck Topologies and Sheaf-Theoretic Foundations of Cryptographic Security:\ Attacker Models and $Σ$-Protocols as the First Step}~\cite{InoueSecurity}. Rather than attempting an encyclopedic survey of all of topos theory, the exposition develops those parts of the subject that are most relevant for passing from torsor-based local-to-global reasoning to sheaf-theoretic and topos-theoretic reasoning: Grothendieck topologies, sheaves, torsors over a site, descent, sheaf topoi, elementary topoi, Cartesian closed structure, subobject classifiers, and internal logic. The goal is not merely motivational. We try to develop enough genuine topos theory that the reader can understand, not only heuristically but structurally, why the later cryptographic framework of~\cite{InoueSecurity} uses Grothendieck topologies and sheaf-theoretic language. To make the note more self-contained, we also include substantial appendices on basic category theory, Yoneda's lemma, limits and colimits, equalizers and coequalizers, Kan extensions, the relation between internal logic and intuitionistic logic, and exercises with solutions. In the final part, we explain how these ideas prepare the ground for a conceptual understanding of $Σ$-protocols, especially in connection with local consistency, simulability, and the passage from compatible local data to global structure.

翻译：本文提供了关于层与topos的预备性引论，其写作理念延续了作者先前关于torsor的引论，并作为作者arXiv论文《格罗滕迪克拓扑与密码学安全的层论基础：攻击者模型与$Σ$-协议作为第一步》的预备背景。本文并非试图对topos理论进行全面综述，而是重点阐述从基于torsor的局部-整体推理过渡到层论与topos理论推理最相关的部分：格罗滕迪克拓扑、层、site上的torsor、下降、层topos、初等topos、笛卡尔闭结构、子对象分类子与内部逻辑。目标不仅在于提供动机，我们尝试发展足够实质的topos理论，使读者不仅能在启发式层面，更能在结构层面理解为何后续密码学框架采用格罗滕迪克拓扑与层论语言。为增强本文自洽性，我们还包含关于基础范畴论、米田引理、极限与余极限、等值子与余等值子、Kan扩张、内部逻辑与直觉主义逻辑关系以及附解答习题的详实附录。最后部分阐释这些概念如何为理解$Σ$-协议奠定概念基础，特别是关于局部一致性、可模拟性以及从相容局部数据到整体结构的过渡。