Gestalten | Higher Topos Theory notes

7.6. Gestalten

We finish the lecture course with the recent framework of Gestalten, developed by Scholze and Stefanich [Scholze 2026]. Its purpose is to regard all the categorical layers attached to a geometric object as a single ring-like invariant. The construction from a six-functor formalism and its finite-limit and descent properties are precise. The ambient category of all Gestalten, however, is too large to be a topos. The actual topos theorem applies after fixing a base and a finite étaleness level, and after restricting to countable colimits.

Construction 7.45.

There are presentably symmetric monoidal categories $n\Pr$ for all $n\geq 0$ , with

0\Pr = \An, \qquad 1\Pr = \Pr.

The construction is iterated by passing to presentable module categories. Throughout this section the relevant presentability cardinal is $\aleph_1$ , so that “presentable” means compatible with countable colimits. We suppress this cardinal from the notation.

Definition 7.46.

The category of Stefanich rings is

\StRing :=\colim_{1\Pr}\bigl(\CAlg(\An)\longrightarrow\CAlg(1\Pr)\longrightarrow\CAlg(2\Pr)\longrightarrow\cdots\bigr).

Equivalently,

\StRing \simeq\lim_{\CAT}\bigl(\CAlg(\An)\longleftarrow\CAlg(1\Pr)\longleftarrow\CAlg(2\Pr)\longleftarrow\cdots\bigr),

where the transition functor sends a symmetric monoidal category to the endomorphism object of its unit. Thus a Stefanich ring may be written as a sequence $A=(A_0,A_1,A_2,\ldots)$ with $A_n\in\CAlg(n\Pr)$ and compatible isomorphisms

A_n\cong\End_{A_{n+1}}(\unit).

Definition 7.47.

The category of Gestalten is defined as

\Gest := \StRing\catop.

Theorem 7.48. (Scholze–Stefanich)

Let $$A$$ be a Stefanich ring and let $n\geq 0$ . The opposite of the category $\StRing^{n\text{-}\et}_{A/}$ of $$n$$ -étale Stefanich $$A$$ -algebras is a countable topos. More precisely,

\Ind_{\aleph_1}\!\left(\bigl(\StRing^{n\text{-}\et}_{A/}\bigr)\catop\right)

is a topos.

Proof

This is [Scholze 2026, Theorem 7.1].

Thus one should not say that the entire category $\Gest$ is a topos. In applications one works in a countable topos of $$n$$ -étale Gestalten over a fixed base. The restriction to a finite $$n$$ is substantive, although many geometric examples are already $$1$$ - or $$2$$ -étale.

Construction 7.49.

For every $$n$$ , there is a fully faithful functor

\CAlg(n\Pr)\catop \hookrightarrow \Gest.

Indeed, an object $A_n\in\CAlg(n\Pr)$ admits a canonical delooping $A_{n+1}:=\Mod_{A_n}(n\Pr)$ . Iterating this construction supplies the higher terms of a Stefanich ring, while the lower terms are obtained by taking endomorphisms of the unit.

Definition 7.50.

A Gestalt $$A$$ is called $$n$$ -affine if it lies in the image of this fully faithful functor. Equivalently, its corresponding Stefanich ring satisfies $A_{k+1} \cong \Mod_{A_k}(k\Pr)$ for all $k \geq n$ .

Example 7.51.

The $$0$$ -affine Gestalten are precisely the commutative monoids:

\CMon(\An)\catop = \CAlg(\An)\catop \hookrightarrow \Gest.

Since $\Pr_{\st} \subseteq \Pr$ is fully faithful, we similarly get a fully faithful functor

\CAlg(\Sp)\catop \hookrightarrow \Gest.

Remark 7.52.

Objects in the image of $\CAlg(\Sp)$ are $$1$$ -affine, but not $$0$$ -affine in the absolute sense. One may instead work relative to the sphere, in the slice

\Gest_{\S} := \Gest_{/\Gest(\Sp)}.

For every ring spectrum $R\in\CAlg(\Sp)$ , the induced object over $\Gest(\Sp)$ is $$0$$ -affine relative to this base.

7.6.1. Gestalten from six-functor formalisms

Consider a presentable six-functor formalism

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

For every $X\in C$ , restriction to the admissible slice gives a lax symmetric monoidal functor

D_X\colon \Span(C^{\ad}_{/X})\longrightarrow\Pr.

The functor category

\Fun\bigl(\Span(C^{\ad}_{/X}),\Pr\bigr)

carries its Day convolution symmetric monoidal structure, with respect to which $$D_X$$ is a commutative algebra object. The category of kernels at $$X$$ is

K_D(X):=\Mod_{D_X}\!\left(\Fun\bigl(\Span(C^{\ad}_{/X}),\Pr\bigr)\right)\qin 2\Pr.

It is functorial in $$X$$ , and its pullback functors admit compatible left and right adjoints. It deloops $$D$$ in the sense that

\End_{K_D(X)}(\unit) \cong D(X)

for all $X\in C$ .

Iterating the kernel construction produces a Stefanich ring

A_{D,X}:=(D(X),K_D(X),K_D^2(X),\ldots).

For the final object $$*$$ of $$C$$ , write $A_D:=A_{D,*}$ . Passing to opposites gives the transmutation functor

[-]_D\colon C\longrightarrow \Gest_{/\Gest(A_D)}, \qquad X\longmapsto [X]_D:=\Gest(A_{D,X}).

Definition 7.53.

We say that $$D$$ satisfies Künneth if, for all $X,Y\in C$ , the canonical tensor comparison

D(X)\otimes_{D(*)}D(Y)\longrightarrow D(X\times Y)

is an isomorphism.

The comparison controls affineness at the first categorified level. In particular, $$D$$ satisfies Künneth if and only if there is an equivalence

K_D(*)\simeq\Mod_{D(*)}(1\Pr).

Analogous relative comparisons in every slice identify $$K_D(X)$$ with $\Mod_{D(X)}(1\Pr)$ and make $$[X]_D$$ $$1$$ -affine. No identification of all higher kernel categories is automatic from the first comparison alone.

Example 7.54.

The six-functor formalism $$D$$ on $\AnRing\catop$ constructed above satisfies Künneth; see [Scholze 2026, Lecture IX].

Proposition 7.55.

The functor

[-]_D\colon C\longrightarrow\Gest_{/\Gest(A_D)}

preserves finite limits. Moreover, every map $[Y]_D\to[X]_D$ induced by a morphism $Y\to X$ in $$C$$ is $$1$$ -étale and $$1$$ -proper.

Proof

See [Scholze 2026, Proposition 9.5].

Proposition 7.56.

If a family $(X_i\to X)_i$ satisfies universal $$!$$ -descent, then the map

\bigsqcup_i[X_i]_D\longrightarrow[X]_D

is an effective epimorphism in the relevant countable topos of Gestalten.

Proof

This is [Scholze 2026, Proposition 9.7]. The converse is not asserted there.

Combining finite-limit preservation with the preceding descent statement gives the following extension. Let $\Shv_D(C)_{\mathrm{ctbl}}$ denote the countable-colimit completion of the representables inside the $$D$$ -sheaf topos. Then $$[-]_D$$ extends to a functor

F\colon \Shv_D(C)_{\mathrm{ctbl}}\longrightarrow \Gest^{1\text{-}\et}_{/\Gest(A_D)}

which preserves countable colimits and finite limits; see [Scholze 2026, Corollary 9.8].

Applying the same construction to the admissible site gives a functor $F^{\ad}$ on the countable part of $\Shv_{\tau_D^{\ad}}(C^{\ad})$ . Its kernel congruence satisfies

(F^{\ad})^{-1}(\Iso)\subseteq\Sigma_D.

This inclusion does not by itself imply that $F^{\ad}$ factors through the localization at $\Sigma_D$ : such a factorization requires the opposite inclusion $\Sigma_D\subseteq(F^{\ad})^{-1}(\Iso)$ .

Remark 7.57.

For analytic rings, Scholze states that the two congruences are equal; see [Scholze 2026, Lecture IX]. Under this additional assertion, the Gestalt-valued functor does factor through the further localization used to define analytic stacks. Marc expects the equality to hold more generally, but the preceding argument proves only the displayed inclusion.

The chapter has therefore passed through three increasingly global encodings of geometry. A geometry specifies affine tests and admissible covers; a fractured topos records their local behavior intrinsically; a six-functor formalism can generate such a geometry and, after transmutation, a Gestalt-valued invariant. The last step is promising precisely because it remembers every categorical level at once, but its strongest descent and comparison statements remain part of an active theory.

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References

Peter Scholze. Geometry and Higher Category Theory. 2026.