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PreprintarXiv licencearXiv

A Picard-Theoretic Brauer Object for Derived Smooth Manifolds

  • Yimu Mao
  • Christopher Tropp
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Abstract

Let X=(|X|,π’ͺX) be a derived smooth manifold in the sense of Spivak. After passing from the simplicial C∞-structure sheaf to a connective spectral structure sheaf 𝕆X, we construct the intrinsic Picard hypersheaf of invertible 𝕆X-modules and define its delooping BrPicX :=BPic𝕆X. On the ordinary open site of |X|, we prove an equivalence of hypersheaves of connected pointed spaces BrPicX ≃ K(underlineβ„€,1) Γ— B2GL1(𝕆X). 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 E1-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 H1sing(M;β„€) Γ— H2sing(M;β„€/2) and H1sing(M;β„€) Γ— H3sing(M;β„€).

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Cite

BibTeXRIS
BibTeX
@misc{mao2026picardtheoretic,
  title = {{A Picard-Theoretic Brauer Object for Derived Smooth Manifolds}},
  author = {Yimu Mao and Christopher Tropp},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.22668},
  eprint = {2607.22668},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.22668},
}
RIS
TY  - UNPB
TI  - A Picard-Theoretic Brauer Object for Derived Smooth Manifolds
AU  - Yimu Mao
AU  - Christopher Tropp
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2607.22668
UR  - https://arxiv.org/abs/2607.22668
AB  - Let X=(|X|,π’ͺ_X) be a derived smooth manifold in the sense of Spivak. After passing from the simplicial C^∞-structure sheaf to a connective spectral structure sheaf 𝕆_X, we construct the intrinsic Picard hypersheaf of invertible 𝕆_X-modules and define its delooping Br^Pic_X :=BPic_𝕆_X. On the ordinary open site of |X|, we prove an equivalence of hypersheaves of connected pointed spaces Br^Pic_X ≃ K(underlineβ„€,1) Γ— B^2GL_1(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_sing(M;β„€) Γ— H^2_sing(M;β„€/2) and H^1_sing(M;β„€) Γ— H^3_sing(M;β„€).
ER  -