From Torsors to Topoi: An Introduction with a View Toward $\Sigma$-Protocols in Cryptography

Mathematics > Category Theory [Submitted on 17 Mar 2026] From Torsors to Topoi: An Introduction with a View Toward Σ-Protocols in Cryptography Takao Inoué 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 \Sigma-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 \Sigma-protocols, especially in connection with local consistency, simulability, and the passage from compatible local data to global structure. Comments: 27 pages. Introductory but substantial note on torsors, sheaves, topoi, and internal logic, written as preparatory background for the author's sheaf-theoretic approach to cryptographic security and Σ-protocols. Includes appendices on category theory, Yoneda lemma, limits and colimits, Kan extensions, and exercises with solutions Subjects: Category Theory (math.CT); Cryptography and Security (cs.CR) MSC classes: 18F10, 18F20, 18N10, 03G30, 94A60 Cite as: arXiv:2603.16274 [math.CT]   (or arXiv:2603.16274v1 [math.CT] for this version)   https://doi.org/10.48550/arXiv.2603.16274 Focus to learn more Submission history From: Takao Inoue [view email] [v1] Tue, 17 Mar 2026 09:05:13 UTC (23 KB) Access Paper: HTML (experimental) view license Current browse context: math.CT < prev   |   next > new | recent | 2026-03 Change to browse by: cs cs.CR math References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)