Geometry

Topoi provide a notion of generalized geometric object, but a bare topos does not remember the additional structure which distinguishes schemes, analytic spaces, or other geometric objects. The purpose of this final chapter is to explain several formalisms for adding such structure. The recurring pattern is that a geometric object is locally modeled on a chosen category of affine objects, with a distinguished class of admissible morphisms and a topology describing how the local models glue.

We begin in Section 7.1 with the classical example. A scheme can be described as a ringed topos which is locally isomorphic to the Zariski topos of a ring. In this language, classical schemes are the \(0\)-localic Zariski schemes, while classical Deligne–Mumford stacks are the \(1\)-localic étale schemes. Moreover, the category of Zariski schemes and étale morphisms is itself, up to size issues, a topos. These observations isolate the structures that the subsequent theory abstracts.

Lurie's notion of a geometry packages three ingredients: a category \(C\) of affine test objects, a class \(C^{\ad}\) of admissible morphisms, and a topology generated by admissible covers. A structure on a topos is then a left exact functor from \(C\) satisfying descent for these covers. Associated with a geometry are two related sheaf topoi, obtained from all morphisms in \(C\) and from only the admissible morphisms. In Section 7.3, we study the resulting notion of a fractured topos. We show that every geometric site produces a fracture and that, on a fixed topos, fractured structures are equivalent to suitable local admissibility structures. The fracture in turn remembers which structured topoi and morphisms should be regarded as schemes and étale maps.

Six-functor formalisms provide a different source of this structure. A presentable six-functor formalism on a category with admissible morphisms determines the \(D\)-topology, generated by families satisfying universal \(!\)-descent. This produces a geometric site and hence a fractured topos, and the original formalism extends to the resulting categories of sheaves. Thus a cohomological formalism can itself determine the geometry on which it satisfies descent. We describe this construction in Section 7.4.

The final two sections indicate how this perspective enters current geometric frameworks. In Section 7.5, we outline the construction of analytic stacks from analytic rings in the sense of Clausen–Scholze and the six-functor formalism available for them. In Section 7.6, we discuss Scholze–Stefanich's proposed category of Gestalten and the way a six-functor formalism should produce Gestalt-valued invariants satisfying descent.

This is an outlook chapter. The structural results on geometries and fractured topoi are stated precisely, although several proofs are quoted from the literature or omitted. The sections on analytic stacks and especially Gestalten are more schematic: some statements are included only as a roadmap to developing work and are explicitly marked informal where necessary.