Sheaves of categories and the notion of 1-affineness

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We define the notion of 1-affineness for a prestack, and prove an array of results that establish 1-affineness of certain types of prestacks.

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The Galois group of a stable homotopy theory (2014, background): “This phenomenon has been extensively studied (under the name “1-affineness”) in [26].”
On two geometric realizations of an affine Hecke algebra (2012, unknown): “We refer to [32] for the notion of an abelian category over an algebraic stack, and to [33] for a generalization to triangulated (or rather homotopy theoretic) context.”
Affineness and chromatic homotopy theory (2013, background): “We call such derived stacks 0-affine (following [18]).”
Betti Geometric Langlands (2016, background): “Most notably, thanks to Gaitsgory’s fundamental 1-affineness theorem [G5], it is possible [BFN2] to establish a Morita equivalence”