AI 中文总结
研究通过拉回Toën的范畴导出布劳尔层对导出光滑流形上导出阿祖马亚代数分类，得到范畴布劳尔层同伦型及层化布劳尔类群，揭示拉回范畴布劳尔不变量由导出单位层全同伦型决定。
AI 中文摘要
对于Spivak意义下的导出光滑流形\((X,\mathcal{O}_X)\)，我们沿着从单纯\(C^\infty\)环到连通单纯交换环的遗忘函子拉回Toën的范畴导出布劳尔层，然后在\(X\)的开位点上进行层化。得到的范畴布劳尔层具有同伦型\(\mathfrak{Br}_X \simeq K(\underline{\mathbb Z},1) \times B^2GL_1(\mathcal{O}_X)\)。其层化布劳尔类群为\(dBr(X):=\pi_0\Gamma(X,\mathfrak{Br}_X) \cong H^1(X,\underline{\mathbb Z}) \times \pi_0\Gamma(X,B^2GL_1(\mathcal{O}_X))\)，当结构层离散时可恢复通常的\(H^2(X,\mathcal{O}_X^\times)\)公式。这表明拉回范畴布劳尔不变量由导出单位层的全同伦型决定。
英文摘要
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 \[ H^1_{\mathrm{sing}}(M;\mathbb Z) \times H^2_{\mathrm{sing}}(M;\mathbb Z/2) \quad\text{and}\quad H^1_{\mathrm{sing}}(M;\mathbb Z) \times H^3_{\mathrm{sing}}(M;\mathbb Z). \]
Comments35 pages, 0 figure, Generative AI assisted with the written exposition; all mathematical ideas, arguments, proofs, and conclusions were developed by the authors
