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 Spec$(\mathcal{C})\sim \mathcal{X}\sim$(category of injective objects in $\mathcal{C})$, and that $\mathcal{C}$ has natural injective envelopes.