Analytic stacks | Higher Topos Theory notes

7.5. Analytic stacks

In this section, we discuss analytic stacks in the sense of Clausen–Scholze as an example of the preceding formalism.

A condensed anima is a hypercomplete sheaf on the site $\CHaus$ of compact Hausdorff spaces equipped with the topology in which a finite collection of maps $\{X_i \to X\}_i$ is declared to be a covering if and only if the map $\bigsqcup_i X_i \twoheadrightarrow X$ is surjective. Equivalently, it is a hypercomplete sheaf on the site $\ProFin$ of profinite sets, equipped with the topology inherited from $\CHaus$ .

In practice, we will work with light condensed animae: hypercomplete sheaves on the site of countably profinite sets (i.e. profinite sets presented by countable cofiltered diagrams). We will be denoting this by

\CondAn := \Shv(\{\text{Countably profinite sets}\})^{\hyp}.

In this setting, we take “ring” to mean “light condensed animated commutative ring”.

Definition 7.37.

An analytic ring is a pair $A = (A^{\triangleright}, \Mod_A)$ consisting of a light condensed animated commutative ring $A^{\triangleright}$ together with a reflective subcategory $\iota\colon \Mod_A \hookrightarrow \Mod_{A^{\triangleright}}$ , with left adjoint $L\colon \Mod_{A^{\triangleright}} \to \Mod_A$ , satisfying the following conditions:

$\Mod_A$ is closed under colimits (also limits, but this is automatic).
$\Mod_A$ is closed under the internal hom $\iHom(M,-)$ for every $M \in \Mod_{A^{\triangleright}}$ .
The functor $\iota L\colon \Mod_{A^{\triangleright}} \to \Mod_{A^{\triangleright}}$ preserves connective objects with respect to the canonical t-structure.
We have $A^{\triangleright} \in \Mod_A$ .

Example 7.38.

The basic examples arise from complete topological rings, in particular from Huber pairs. The analytic ring structure remembers which condensed modules are complete for the chosen topology; it is therefore strictly more information than the underlying condensed ring $A^{\triangleright}$ . See [Scholze 2025, Lecture IX] for the constructions used below.

Remark 7.39.

In the original definition of analytic rings, this was defined in the “non-light” setting. In this case, there is an additional condition one needs to put: the Frobenius map $A \to A/p$ is a map of analytic rings for every prime $$p$$ . In the light setting, this condition turns out to be automatic. (It is not known whether this condition is automatic in the “non-light” setting.)

Definition 7.40.

A map $f\colon A \to B$ of analytic rings is a map $A^{\triangleright} \to B^{\triangleright}$ satisfying the condition that the restriction-of-scalars functor $\Mod_{B^{\triangleright}} \to \Mod_{A^{\triangleright}}$ restricts to a functor

f_*\colon \Mod_B \to \Mod_A.

We denote its left adjoint by

f^*\colon \Mod_A \to \Mod_B,

which is given by first including into $\Mod_{A^{\triangleright}}$ , then applying base change to $\Mod_{B^{\triangleright}}$ , and then reflecting back into $\Mod_B$ . We denote the resulting category of analytic rings by $\AnRing$ .

Definition 7.41.

A map $f\colon A \to B$ of analytic rings is said to be:

Proper if the functor $$f_*$$ induces an equivalence
$\Mod_B \iso \Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A.$
An open immersion if the functor $$f^*$$ admits a further left adjoint
$f_!\colon \Mod_B \to \Mod_A$
which is fully faithful and $\Mod_A$ -linear. Equivalently, it is fully faithful and satisfies the projection formula
$f_!(f^*(M) \otimes_B N) \cong M \otimes_A f_!(N).$
$$!$$ -able if the canonical map of analytic rings
$(B^{\triangleright}, \Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A) \longrightarrow (B,\Mod_B)$
is an open immersion.

Lemma 7.42.

A map $$f$$ is $$!$$ -able if and only if it factors as a proper map followed by an open immersion.

Proof

For a map

A\to B

, equip

B^{\triangleright}

first with the analytic structure induced from

\(A\)

. The map from

\(A\)

to this induced analytic ring is proper by definition, while the comparison from the induced analytic ring to

\(B\)

is an open immersion precisely when

A\to B

\(!\)

-able. Conversely, any factorization of this form satisfies the defining condition. See [Scholze 2025, Definition 9.16].

Proposition 7.43.

There is a presentable six-functor formalism

D\colon \Span(\AnRing\catop, \AnRing^{!,\op}) \to \Pr, \qquad A \mapsto \Mod_A.

Proof

The proper and open classes above satisfy the compactification, base-change, and projection-formula hypotheses, and every

\(!\)

-able map has the factorization of Lemma 7.42. The resulting analytic six-functor formalism is constructed in [Scholze 2025, Lecture IX]. The formal assembly of the coherent span-valued functor may also be expressed using [Cnossen et al. 2025].

Definition 7.44.

The category of analytic stacks is defined to be

\AnStck := \Shv^+_{\tau_D}(\AnRing\catop).

Here the superscript $$+$$ denotes the further congruence localization of Construction 7.32, not merely sheafification for the $$D$$ -topology. In the light setting all sites and colimit constructions are understood with the corresponding countability restrictions.

References

Peter Scholze. Six-Functor Formalisms. 2025.
Bastiaan Cnossen, Tobias Lenz, Sil Linskens. Universality of span 2-categories and the construction of 6-functor formalisms. 2025.