A Picard-Theoretic Brauer Object for Derived Smooth Manifolds

Let $X=(\mathrm{\mid }X\mathrm{\mid },{\mathcal{O}}_X)$ be a derived smooth manifold in the sense of Spivak. After passing from the simplicial $C^{\mathrm{\infty }}$-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 $${\mathrm{Br}}_X^{\mathrm{P}\mathrm{i}\mathrm{c}}:=B{\mathrm{Pic}}_{{\mathbb{O}}_X}\mathrm{.}$$ On the ordinary open site of $\mathrm{\mid }X\mathrm{\mid }$, we prove an equivalence of hypersheaves of connected pointed spaces $${\mathrm{Br}}_X^{\mathrm{P}\mathrm{i}\mathrm{c}}\simeq K(\underset{‾}{\mathbb{Z}},1)\times B^2{\mathrm{GL}}_1({\mathbb{O}}_X)\mathrm{.}$$ 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 $$H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^1(M;\mathbb{Z})\times H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^2(M;\mathbb{Z}\mathrm{/}2)\text{and}{H}_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^1(M;\mathbb{Z})\times H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^3(M;\mathbb{Z})\mathrm{.}$$