Geometries from six-functor formalisms | Higher Topos Theory notes

7.4. Geometries from six-functor formalisms

Six-functor formalisms, developed systematically in [Scholze 2025], provide a categorification of classical cohomology theories. The key insight relevant to our discussion is that every six-functor formalism on a category with an admissibility structure canonically determines a topology, the $$D$$ -topology, and hence gives rise to a geometric site and a fractured topos.

7.4.1. The D-topology

Definition 7.25.

Let $$C$$ be a category with finite limits and let $C^{\ad}\subseteq C$ be an admissibility structure. A presentable six-functor formalism is a lax symmetric monoidal functor

D\colon \Span(C,C^{\ad}) \longrightarrow \Pr.

Definition 7.26.

A family $(X_i\to X)_i$ in $C^{\ad}$ is a $$D$$ -cover if it is a covering family for the canonical topology on $$C$$ and satisfies universal $$!$$ -descent: after every base change $Y\to X$ , the resulting family $(X_i\times_XY\to Y)_i$ satisfies descent for $$D^!$$ . The $$D$$ -topology $\tau_D$ is the topology whose covering families are the $$D$$ -covers.

Proposition 7.27.

The triple $(C,C^{\ad},\tau_D)$ is a geometric site.

Proof

The canonical-cover condition is stable under base change and composition, and universal

\(!\)

-descent has the same stability properties. Since every

\(D\)

-cover consists of admissible morphisms, the resulting topology is compatible with

C^{\ad}

. See [Scholze 2025, Definition 5.13].

Corollary 7.28.

The functor

\Shv_{\tau_D^{\ad}}(C^{\ad})\longrightarrow \Shv_{\tau_D}(C)

exhibits $\Shv_{\tau_D}(C)$ as a complete fractured topos. In particular, it determines a geometric admissibility structure $\Shv_{\tau_D}(C)^{\ad}$ .

7.4.2. Passage to sheaves

Set $\widetilde C:=\Shv_{\tau_D}(C)$ . There are several natural candidates for the class of admissible morphisms in $\widetilde C$ , and they should not be identified without argument. We begin with the class directly controlled by the extension theorem.

Definition 7.29.

A map $f\colon Y\to X$ in $\widetilde C$ is representably admissible if, for every map $h_U\to X$ from a representable sheaf, the pullback $Y\times_Xh_U$ is representable and the induced morphism to $$h_U$$ comes from a morphism in $C^{\ad}$ . We write $\widetilde C^{\ad}_0$ for the resulting admissibility structure.

Proposition 7.30. (Mann–Scholze)

The six-functor formalism $$D$$ extends uniquely from $(C,C^{\ad})$ to $(\widetilde C,\widetilde C^{\ad}_0)$ .

Proof

See [Mann 2022, Proposition A.5.16] and [Scholze 2025, Proposition 5.16].

Remark 7.31.

Marc stated in the lectures that $$D$$ extends to the entire fracture-derived admissibility structure $\Shv_{\tau_D}(C)^{\ad}$ of Corollary 7.28. No proof was given. The cited results establish the extension first for the representably admissible class $\widetilde C_0^{\ad}$ and then for Scholze's minimal local enlargement $\widetilde C_D^{\ad}$ ; they do not identify either class with $\Shv_{\tau_D}(C)^{\ad}$ .

Scholze constructs the further enlargement mentioned in the remark. More precisely, there is a minimal class $\widetilde C_D^{\ad}$ containing $\widetilde C_0^{\ad}$ which is stable under disjoint unions, local on the target, local on the source, and tame. The formalism extends to $(\widetilde C,\widetilde C_D^{\ad})$ ; see [Scholze 2025, Theorem 5.19].

7.4.3. A further congruence localization

The $$D$$ -topology imposes descent along covering families. One may localize further by all morphisms which are universally invisible to $$D^!$$ . This second step need not arise from a Grothendieck topology.

Construction 7.32.

Consider the functor

D^!\colon \Shv_{\tau_D^{\ad}}(C^{\ad})\catop\longrightarrow\Cat.

Let $\Sigma_D$ be the largest congruence contained in the inverse image of the isomorphisms under this functor. More explicitly,

\Sigma_D=\{f\mid \text{every base change of every $\Delta^nf$ is a $D^!$-isomorphism}\}.

Every morphism of $\Sigma_D$ is $\infty$ -connected: the condition for $$n=0$$ implies that every base change is a $$D$$ -cover and hence an effective epimorphism, and the same applies to every diagonal.

Assume that $\Sigma_D$ has small generation, so that it defines an accessible left exact localization. We define $\Shv_{\tau_D}^+(C)$ by the pullback square

where the bottom functor is the fully faithful right adjoint of the localization. The top functor is likewise the fully faithful right adjoint of a left exact localization. The left vertical functor admits a left adjoint, and the pair

\left(\Shv_{\tau_D}^+(C),\Shv_{\tau_D^{\ad}}(C^{\ad})[\Sigma_D^{-1}]\right)

is a fractured topos.

Remark 7.33.

There is a general formalism of fractured localization, in which a compatible localization of the corporeal topos induces a localization of the ambient topos and a new fracture. The preceding construction is an instance of this formalism.

Lemma 7.34.

The six-functor formalism descends further to a functor

D\colon \Span\left(\Shv_{\tau_D}^+(C), \Shv_{\tau_D^{\ad}}(C^{\ad})[\Sigma^{-1}_D]\right) \to \PrL.

Proof

By construction,

\(D^!\)

inverts

\Sigma_D

. The span formalism is self-dual along admissible morphisms: the duality exchanges the pullback and lower-shriek operations. Hence every universal

\(D^!\)

-isomorphism is also a universal

\(D^*\)

-isomorphism. The four operations therefore descend through the two localizations, and the Beck–Chevalley and projection-formula transformations descend with them.

Remark 7.35.

The self-duality used in the proof is also the reason that universal $$!$$ -descent implies universal $$*$$ -descent. It is important here that descent is required after every base change: this makes the class compatible with composition of spans.

References

Peter Scholze. Six-Functor Formalisms. 2025.
Lucas Mann. A $p$-Adic 6-Functor Formalism in Rigid-Analytic Geometry. arXiv preprint arXiv:2206.02022. 2022.