Phantom covering ideals in categories without enough projective morphisms

TL;DR

This paper establishes conditions under which phantom map ideals in certain exact categories are covering, with applications to sheaf categories and a new approach for categories lacking enough projective morphisms.

Contribution

It introduces new criteria for phantom map ideals to be covering in locally presentable exact categories without relying on enough projective morphisms.

Findings

Identifies conditions for phantom map ideals to be covering

Applies results to categories of sheaves with geometric motivation

Develops a new approach for categories without enough projective morphisms

Abstract

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.