Phantom covering ideals in categories without enough projective   morphisms
  We give sufficient conditions to ensure that the ideal $\Phi(\mathcal E)$ of $\mathcal E$-phantom maps in a locally $\lambda$-presentable exact category $(\mathcal{A}, \mathcal{E})$ is (special) (pre)covering ideal, where $\mathcal E$ is an exact substructure of $(\mathcal{A}, \mathcal{E})$. As a byproduct, we infer the existence of various covering ideals in categories of sheaves which have a meaningful geometrical motivation. In particular we deal with a Zariski-local notion of phantom maps in categories of sheaves. We would like to point out that our approach is necessarily different from [FGHT13], as the categories involved in most of the examples we are interested in do not have enough projective morphisms. 
Sergio Estrada, Pedro A. Guil Asensio, Sinem Odabasi
math.KT math.AG math.CT