Existence of enough injectives for contraherent cosheaves

Determine whether the categories of contraherent cosheaves and locally contraherent cosheaves on schemes have enough injective objects.

Background

Contraherent and locally contraherent cosheaves are presented as dual analogues of quasi-coherent sheaves. Unlike quasi-coherent sheaves, whose ambient categories are Grothendieck abelian categories with enough injectives, the relevant contraherent categories are exact rather than abelian and are not expected to possess enough injective objects. The paper explicitly notes that no specific counterexamples are known, leaving the existence question unresolved.

References

Dual-analogously to the categories of quasi-coherent sheaves not having enough projective objects, one does not expect the categories of contraherent or locally contraherent cosheaves to have enough injective objects (though we are not aware of any specific counterexamples).

— The contraherent version of the theorem of Slavik and Stovicek  (2609.08491 - Positselski, 8 Sep 2026) in Section 1, Introduction