If we give the topos of ℕ-sets its double negation topology, apparently sheaves with respect to this topology are ℤ-sets. It is also interesting to study the double negation topology on the category of Set-valued functors on the category of finite fields, or finite fields of characteristic p:
Olivia Caramello, Topological Galois theory, arxiv.org/abs/1301.0300
Investigating zeta functions arising from 2-rigs. Say a 2-rig is a symmetric monoidal cocomplete category enriched over abelian groups. Then - morally speaking, at least - any 2-rig R comes from an algebraic stack, and the groupoid of 'points' of this stack over a commutative ring k is the groupoid of 2-rig maps from R to kMod. If our...