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Consider the relaxation of Eilenberg-MacLane spaces where we allow non-trivial homotopy groups in a band, instead of a single index. We inspect the case where the band stays on the stable range and we prove an equivalence between the (∞−)category of n−connective (2n − 2)−truncated anima and a category of spectral copoints. As a corollary we obtain both an analogous equivalence for arbitrary ∞−topoi and a classification of local systems valued on [n, 2n-2]-spaces, which generalizes the known result for gerbes - the case valued in classifying spaces. To accomplish the proof we use stable methods, left adjointability results, a characterization of under-over stable categories and information coming from the pointed case. To showcase some applications, we use our theorem to re-examine the case of gerbes and slightly improve it, and to shed light on the new case of local systems valued on [n, n + 1]−spaces.