John Baez and James Dolan, 2023-11-27

Giving 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/...
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 stack is a mere scheme, we get a mere set of points for each commutative ring k, and then we can use this technique to define the Hasse-Weil zeta function of the scheme:
ncatlab.org/jo...
The idea is to define the 'Hasse-Weil species' of the scheme, a certain functor from the groupoid of finite sets to Set, and then decategorify it to get a Dirichlet series.
For a full-fledged algebraic stack we get a groupoid of points for each commutative ring k, and then instead of a Hasse-Weil species we get a Hasse-Weil 'stuff type', an analogously defined pseudofunctor from the groupoid of finite sets to Gpd. We should be able to still define a Dirichlet series from this, but this has not yet been worked out.
Every 2-rig comes from an algebraic stack, but here we consider 2-rigs coming from algebraic stacks of two different kinds: affine schemes (which are not stacky at all), and affine algebraic groups (which can be seen as very stacky algebraic stacks with just one point).
Any affine scheme gives a 2-rig as follows: the affine scheme corresponding to a commutative ring R gives the 2-rig of R-modules. We consider the cases R = ℤ[i] and R = ℤ[x].
Any affine algebraic group gives a 2-rig as follows: the affine algebraic group corresponding to a commutative Hopf algebra H gives the 2-rig of H-comodules. We consider the case where H is the commutative Hopf algebra of ℤ^G of ℤ-valued functions on a finite group G. We focus on the case ℤ/3.
For more on this whole series of conversations, go here:
math.ucr.edu/h...