Unusual spectral categories

Abstract

The paper is devoted to a kind of `very non-abelian' spectral categories. Under strong conditions on a category $\mathcal{X}$, we prove, among other things, that, for a given faithful localization $\mathcal{C}\to\mathcal{X}$, we have canonical equivalences $\mathrm{Spec}(\mathcal{C})\sim\mathcal{X}\sim(\mathrm{Category\,\,of\,\,injective\,\,objects\,\,in}\,\, \mathcal{C})$, and that $\mathcal{C}$ has natural injective envelopes.