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"We introduce a theory of stratifications of noncommutative stacks (i.e., presentable stable-categories), and we prove a reconstruction theorem that expresses them in terms of their strata and gluing data. This reconstruction theorem is compatible with symmetric monoidal structures, and with more general operadic structures such as En-monoidal structures. We also provide a suite of fundamental operations for constructing new stratifications from old ones: restriction, pullback, quotient, pushforward, and refinement. Moreover, we establish a dual form of reconstruction; this is closely related to Verdier duality and reflection functors, and gives a categorification of M"obius inversion. Our main application is to equivariant stable homotopy theory: for any compact Lie group G, we give a symmetric monoidal stratification of genuine G-spectra. In the case that G is finite, this expresses genuine G-spectra in terms of their geometric fixedpoints (as homotopy-equivariant spectra) and gluing data therebetween (which are given by proper Tate constructions). We also prove an adelic reconstruction theorem; this applies not just to ordinary schemes but in the more general context of tensor-triangular geometry, where we obtain a symmetric monoidal stratification over the Balmer spectrum. We discuss the particular example of chromatic homotopy theory"--

Stratified sets. --- Noncommutative rings. --- Geometry, Algebraic. --- Homotopy theory. --- Reconstruction (Graph theory) --- Algebraic geometry -- Foundations -- Noncommutative algebraic geometry. --- Algebraic topology -- Homotopy theory -- Equivariant homotopy theory. --- Algebraic topology -- Applied homological algebra and category theory -- Duality. --- Manifolds and cell complexes -- Topological manifolds -- Stratifications. --- Algebraic geometry -- Local theory -- Formal neighborhoods. --- Algebraic topology -- Spectral sequences -- General. --- Noncommutative differential geometry.