A Picard-Theoretic Brauer Object for Derived Smooth Manifolds

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 \[ H^1_{\mathrm{sing}}(M;\mathbb Z) \times H^2_{\mathrm{sing}}(M;\mathbb Z/2) \quad\text{and}\quad H^1_{\mathrm{sing}}(M;\mathbb Z) \times H^3_{\mathrm{sing}}(M;\mathbb Z). \]