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Morphisms and comorphisms of sites II - Distributors of sites
Axel Osmond, Olivia Caramello
https://arxiv.org/abs/2507.20932 https://arxiv.org/pdf/2507.209…

Morphisms and comorphisms of sites II - Distributors of sites
We introduce a notion of distributor of sites, involving suited analogs of flatness and cover-preservation, and show that this notion jointly generalizes those of morphism and comorphism of sites. Given two sites, we exhibit an adjunction between the category of distributors of sites between them and the category of geometric morphisms between the associated sheaf topoi; this adjunction restricts to an equivalence between geometric morphisms and continuous distributors of sites. We finally disc…

Higher Zariski Geometry
Ko Aoki, Tobias Barthel, Anish Chedalavada, Tomer Schlank, Greg Stevenson
https://arxiv.org/abs/2508.11621 https://arxiv.org/pdf/25…

Higher Zariski Geometry
We revisit the classical constructions of tensor-triangular geometry in the setting of stably symmetric monoidal idempotent-complete $\infty$-categories, henceforth referred to as 2-rings. In this setting, we produce a Zariski topology, a Zariski spectrum, a category of locally 2-ringed spaces (more generally $\infty$-topoi), and an affine spectrum-global sections adjunction, based on the framework of ``$\infty$-topoi with geometric structure'' as developed by Lurie in \cite{LurieDAG5}. Using w…