A Picard-Theoretic Brauer Object for Derived Smooth Manifolds

[2607.22668] A Picard-Theoretic Brauer Object for Derived Smooth Manifolds Skip to main content Search Submit Donate Log in Mathematics > General Mathematics arXiv:2607.22668 (math) [Submitted on 5 Jul 2026 ( v1 ), last revised 12 Sep 2026 (this version, v3)] Title: A Picard-Theoretic Brauer Object for Derived Smooth Manifolds Authors: Yimu Mao , Christopher Tropp View PDF HTML (experimental) Abstract: Let $X=(|X|,\mathcal O_X)$ be a derived smooth manifold in the sense of Spivak. After passing from the simplicial $C^\infty$-structure sheaf to a connective spectral structure sheaf $\mathbb O_X$, we construct the intrinsic Picard hypersheaf of invertible $\mathbb O_X$-modules and define its delooping \[ \operatorname{Br}^{\mathrm{Pic}}_X :=B\operatorname{Pic}_{\mathbb O_X}. \] On the ordinary open site of $|X|$, we prove an equivalence of hypersheaves of connected pointed spaces \[ \operatorname{Br}^{\mathrm{Pic}}_X \simeq K(\underline{\mathbb Z},1) \times B^2\operatorname{GL}_1(\mathbb O_X). \] The statement is unconditional at the level of Picard torsors. Its interpretation as a classification of forms of the module category is made under an explicit category-valued open-hyperdescent hypothesis, and representability by an internal $E_1$-algebra is separated further by a global compact-local-generator hypothesis together with internal mapping objects, their base-change equivalences, and relative Morita continuity. For an ordinary paracompact smooth manifold $M$, the real and complex coefficient theories recover, respectively, the pointed-set decompositions \[ H^1_{\mathrm{sing}}(M;\mathbb Z) \times H^2_{\mathrm{sing}}(M;\mathbb Z/2) \quad\text{and}\quad H^1_{\mathrm{sing}}(M;\mathbb Z) \times H^3_{\mathrm{sing}}(M;\mathbb Z). \] Comments: 35 pages, 0 figure, Generative AI assisted with the written exposition; all mathematical ideas, arguments, proofs, and conclusions were developed by the authors Subjects: General Mathematics (math.GM) MSC classes: 18N60, 14A30, 18F20, 55P43, 58A40 Cite as: arXiv:2607.22668 [math.GM] (or arXiv:2607.22668v3 [math.GM] for this version) https://doi.org/10.48550/arXiv.2607.22668 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yimu Mao [ view email ] [v1] Sun, 5 Jul 2026 18:53:42 UTC (12 KB) [v2] Sat, 22 Aug 2026 20:24:58 UTC (34 KB) [v3] Sat, 12 Sep 2026 08:53:01 UTC (34 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.GM < prev | next > new | recent | 2026-07 Change to browse by: math References & Citations SciX 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 Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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