Pro-étale Homotopy Type in Schemes

• Pro-étale homotopy type is an invariant defined by the shape of the pro-étale topos.
• It leverages w-contractible affines and the pro-étale site to ensure local contractibility and robust cohomology.
• The theory integrates model categories, ∞-categories, and profinite methods to refine classical étale invariants.

The pro-étale homotopy type is an invariant of schemes, formulated as a refinement of the Artin–Mazur étale homotopy type, designed to capture additional information, particularly suited to $\ell$-adic and infinite topological constructions. Central to its definition is the pro-étale site and topos as developed by Bhatt and Scholze, which allow for the construction of a locally contractible Grothendieck topology on schemes, and the associated shape (homotopy type) as a pro-object in spaces or condensed anima. Recent advances further refine this invariant in the frameworks of model categories, $\infty$-categories, profinite spaces, and condensed mathematics.

1. The Pro-Étale Site and Topos

The pro-étale site $X_{\proet}$ of a scheme $X$ is defined as the category of all weakly étale $X$-schemes, i.e., morphisms $f:Y\to X$ that are flat and whose diagonal $\Delta_f:Y\to Y\times_X Y$ is flat. The coverings in this site are families $\{Y_i\to Y\}$ that constitute fpqc coverings (Bhatt et al., 2013).

The associated topos $\Shv(X_{\proet})$ hosts sheaves whose descent and cohomology properties are governed by pullback–pushforward along weakly étale maps. An essential property—and a motivator for applications to infinite constructions in $\ell$-adic cohomology—is that every quasi-compact and quasi-separated $X$ is covered by affines that are w-contractible. An affine is w-contractible if every faithfully flat weakly étale cover admits a section; equivalently, its coordinate ring is w-strictly local, and the space of connected components is extremally disconnected.

This local contractibility endows $\Shv(X_{\proet})$ with features where stalk functors are exact and commute with limits and colimits, countable products are exact, $\R\!\lim=\lim$ for surjective inverse systems, and Postnikov towers converge (Bhatt et al., 2013).

2. Formal Constructions of the Pro-Étale Homotopy Type

Given the pro-étale topos, the pro-étale homotopy type is defined as the shape of the associated $\infty$-topos:

$\Shape(X_{\proet}) := \Shape(\Shv(X_{\proet})) \in \Pro(\Spaces),$

where "shape" refers to the pro-object in spaces determined by the $\infty$-topos (Bhatt et al., 2013, Barnea et al., 2015). Concretely, due to local weak contractibility, this shape can be computed as the limit (over pro-étale hypercovers by w-contractible affines) of their simplicial nerves:

$\Shape(X_{\proet}) = \lim_{U_\bullet\to X} N(U_\bullet),$

where $N(-)$ denotes the simplicial nerve.

In the refinement of Artin–Mazur's étale homotopy type, the pro-étale variant adapts the classical approach to the subtler topology provided by $X_{\proet}$. For schemes $X$ that are quasi-compact and quasi-separated, the pro-étale shape is a pro-object in profinite spaces, often determined by a single split affine w-contractible hypercovering (Meffle, 24 Mar 2025).

3. Relations to Profinite Homotopy Theory and Model Categories

In "Profinite homotopy theory" (0803.4082), a cofibrantly generated model structure is established on simplicial profinite sets, extending the Grothendieck–Galois correspondence to higher homotopy groups and continuous cohomology. The classical étale homotopy type functor of Artin–Mazur and Friedlander,

$$\mathrm{Et}: (\text{loc. noetherian schemes}) \rightarrow \mathrm{pro}\text{-}\mathbf{sSet},$$

after rigid profinite completion, provides a canonical comparison:

$$\widehat{\mathrm{Et}}(X) := (\mathrm{Et} X)^{\wedge} \in \mathbf{sProf}.$$

This construction yields profinite homotopy groups and continuous cohomology that agree with the corresponding (pro-)étale invariants. The pro-étale shape is strictly compatible with the model-categorical approach, and Quillen equivalences connect these settings to $\infty$-category theory (Barnea et al., 2015).

Barnea and Schlank (Barnea et al., 2011) introduce the projective model structure on pro-simplicial sheaves, showing how their derived functor construction of the absolute (and relative) étale homotopy type lifts the classical Artin–Mazur theory and naturally extends to maps of topoi, providing robust groundwork for homotopical obstructions and descent theory.

4. Properties, Examples, and Cohomological Applications

Properties

• Local contractibility: Any object in $\Shv(X_{\proet})$ admits a (hyper)cover by w-contractible affines, ensuring exactness and strong cohomological properties (Bhatt et al., 2013).
• Profinite nature: For qcqs $X$, the pro-étale homotopy type is a pro-object in profinite spaces (Meffle, 24 Mar 2025).
• Cohomological realization: Cohomology of $\comp$-sheaves (i.e., those depending only on spaces of components) is computed as colimits over w-contractible hypercovers, recovering pro-étale cohomology (Meffle, 24 Mar 2025).
• Descent and fiber sequences: The condensed version of the pro-étale homotopy type satisfies descent along integral morphisms, and base-change fiber sequences for smooth proper morphisms (up to suitable completions) (Haine et al., 8 Oct 2025).

Examples

• If $k$ is separably closed, $\Shape(\Spec k_{\proet}) \simeq *$.
• For $C$ a smooth affine curve over an algebraically closed field, $\Shape(C_{\proet}) \simeq B(\pi_1^{\proet}(C))$, where $\pi_1^{\proet}(C)$ matches the profinite completion of the topological fundamental group (Bhatt et al., 2013).
• Over $\RR$, the unique nontrivial Galois cover $\Spec(\CC) \to \Spec(\RR)$ yields the classifying space $B(\ZZ/2)$ as the pro-étale homotopy type (Meffle, 24 Mar 2025).

Table: Homotopy types in special cases

$X$	Pro-étale homotopy type	Fundamental group
$\Spec k$, $k$ separably closed	$*$	$1$
Smooth curve $C$	$B(\pi_1^{\proet}(C))$	$\pi_1^{\proet}(C)$
$\Spec(\RR)$	$B(\mathbb{Z}/2)$	$\mathbb{Z}/2$

5. Refined Invariants: Fundamental Groups and Condensed Homotopy

The refined pro-étale fundamental group $\pi_1^{\proet}(X, \bar{x})$ is defined as the group of automorphisms of the fiber functor on the category of lisse $\widehat{\mathbb{Z}}_\ell$-sheaves, topologized as a Noohi group. Its profinite completion is the classical étale $\pi_1$, while its pro-discrete quotient is the SGA 3 fundamental group (Bhatt et al., 2013).

The condensed homotopy type, as studied in (Haine et al., 8 Oct 2025), refines the pro-étale and étale homotopy types via a colimit-preserving assignment over the category of schemes, and its fundamental group $\pi_1^{\cond}(X, x)$, as a condensed group, recovers Bhatt–Scholze’s $\pi_1^{\proet}(X,x)$ upon Noohi completion. In geometrically unibranch and qcqs settings, the quasiseparated quotient of $\pi_1^{\cond}$ recovers the classical profinite étale fundamental group (Haine et al., 8 Oct 2025).

Explicit computations demonstrate new phenomena: for example, $\mathbf{A}^1_{\mathbf{C}}$ possesses a nontrivial condensed fundamental group, but its Noohi completion is trivial, recovering the classical answer.

6. Comparison with Classical and Profinite Theories

The pro-étale homotopy type specializes to the Artin–Mazur–Friedlander étale homotopy type after protruncation or profinite completion (Haine et al., 8 Oct 2025). Rigid profinite completion functors as studied by Quick (0803.4082) provide a model-categorical underpinning, ensuring agreement with classical topological invariants in the profinite regime. The formalism through model structures and derived functors ensures compatibility between strict model category and $\infty$-categorical approaches (Barnea et al., 2015, Barnea et al., 2011).

Barwick–Glasman–Haine’s Galois category construction encodes the condensed homotopy type as the classifying anima of a Galois category, providing a higher-categorical generalization of the classical monodromy–exodromy programs (Haine et al., 8 Oct 2025).

7. Impact, Open Problems, and Future Directions

The pro-étale homotopy type, through its various models, unifies $\ell$-adic and profinite perspectives, provides a robust invariant for arithmetic and motivic problems, and supplies a framework that accommodates condensed mathematics and $\infty$-topoi. Open questions include the explicit comparison of pro-étale homotopy groups as condensed groups with classical invariants, full understanding of the theory over arbitrary fields, and exploration of phenomenology in the condensed and non-geometrically unibranch settings (Meffle, 24 Mar 2025, Haine et al., 8 Oct 2025).

Ongoing research explores refinements for relative settings, higher Grothendieck obstructions, the global Galois category, and applications to rational points and van Kampen formulas (Barnea et al., 2011, Haine et al., 8 Oct 2025).

---

References:

• "The pro-étale topology for schemes" (Bhatt et al., 2013)
• "Profinite homotopy theory" (0803.4082)
• "A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type" (Barnea et al., 2011)
• "Pro-categories in homotopy theory" (Barnea et al., 2015)
• "The Pro-Étale Homotopy Type" (Meffle, 24 Mar 2025)
• "The condensed homotopy type of a scheme" (Haine et al., 8 Oct 2025)