Stratified Étale Homotopy Types

• Stratified étale homotopy types are invariants associated with schemes that encapsulate refined homotopical data through étale toposes and profinite stratifications.
• They employ a delocalization technique to reconstruct extended étale homotopy groups using the profinite Galois category, linking categorical structures with classical homotopy invariants.
• This framework unifies topos theory, Galois theory, and homotopy theory, offering concrete computational tools for analyzing schemes and their cohomological properties.

Stratified étale homotopy types are invariants associated with schemes that capture refined homotopical information via the interplay between the étale topos, profinite stratification, and Galois categories. Central to this theory is the assertion that the protruncated shape of a spectral $\infty$-topos is a delocalization of its profinite stratified shape, providing effective methods for reconstructing extended étale homotopy groups from the associated profinite Galois category [1812.11637].

1. Spectral $\infty$-Topoi and Profinite Stratified Spaces

A spectral $\infty$-topos is defined as a bounded coherent $\infty$-topos equipped with a profinite stratification. There exists an equivalence of $\infty$-categories:
[
\widetilde{(-)} : \mathrm{Pro}(\mathrm{Str}\pi) \simeq \mathrm{StrTop}\infty
]
Here, $\mathrm{Pro}(\mathrm{Str}\pi)$ denotes the pro-category of profinite stratified spaces, while $\mathrm{StrTop}\infty$ denotes spectral stratified $\infty$-topoi. For $\mathcal{X} \in \mathrm{StrTop}\infty$, the profinite stratified shape is denoted
[
C = \Pi^{\text{strat}}(\mathcal{X}) \in \mathrm{Pro}(\mathrm{Str}\pi)
]
There is a functor
[
\Pi_\infty : \mathrm{Top}\infty \to \mathrm{Pro}(\mathrm{Spc})
]
assigning to $\mathcal{X}$ its (pro-)shape, formalized as the pro-object $\Gamma!(1_{\mathcal{X}})$, the pro-left-adjoint to the global sections functor applied to the terminal object. The protruncation functor $\tau_{<\infty} : \mathrm{Pro}(\mathrm{Spc}) \to \mathrm{Pro}(\mathrm{Spc}_{<\infty})$ applies the Postnikov truncation tower to pro-objects.

2. Delocalization and Local Equivalences

Given $C \in \mathrm{Pro}(\mathrm{Str}\pi)$ (the profinite stratified shape) and $\mathcal{X} = \widetilde{C}$, a morphism $s$ in $C$ is a local equivalence if its image under the stratified realization $\widetilde{(-)}$ is an equivalence of objects in $\mathcal{X}$. Let $S$ denote the class of all such morphisms. The $\infty$-category $C[S^{-1}]$ is obtained by formally inverting $S$, and its homotopy type is given by
[
H(C[S^{-1}]) = \operatorname{colim}{C[S^{-1}]} 1_{\mathrm{Spc}} \in \mathrm{Pro}(\mathrm{Spc})
]
It is shown that
[
\tau_{<\infty}\Pi_\infty(\mathcal{X}) \simeq H(C[S^{-1}])
]
This identifies the protruncated shape of a spectral $\infty$-topos with the delocalization of its profinite stratified shape (i.e., after inverting local equivalences) [1812.11637].

3. Extended Étale Homotopy Groups via Stratified Shape

For a coherent scheme $X$, its étale $\infty$-topos $X_{\mathrm{\acute{e}t}}$ and $C = \operatorname{Gal}(X)$, the profinite Galois category, serve as foundational objects. The (Artin–Mazur–Friedlander) étale homotopy type is
[
\Pi^{\text{\'{e}t}}_\infty(X) := \Pi_\infty(X_{\mathrm{\acute{e}t}}) \in \mathrm{Pro}(\mathrm{Spc})
]
Choosing a geometric point $x \to X$, the extended étale homotopy pro-groups are
[
\pi_n^{\text{\'{e}t}}(X, x) := \pi_n(\Pi^{\text{\'{e}t}}_\infty(X), x), \quad n \geq 1
]
By the delocalization theorem, there are natural isomorphisms in $\mathrm{Pro}(\mathrm{Grp})$:
[
\pi_n^{\text{\'{e}t}}(X,x) \cong \pi_n(\tau_{<\infty}\Pi_\infty(X_{\mathrm{\acute{e}t}}), x) \cong \pi_n(H(\operatorname{Gal}(X)), x)
]
Therefore, $\pi_n^{\text{\'{e}t}}(X,x)$ can be computed as the homotopy pro-groups of the nerve of $\operatorname{Gal}(X)$ after inverting the specialization morphisms [1812.11637].

4. Construction of the Profinite Galois Category

The profinite Galois category $\operatorname{Gal}(X)$ is constructed as follows:
- Objects: geometric points of $X$
- Morphisms: étale specializations (liftings of one geometric point to the strict localization at another)
- Topology: each Hom-set is endowed with a natural profinite topology, making $\operatorname{Gal}(X)$ an object of $\mathrm{Pro}(\mathrm{FinCat})$

Via the equivalence $\mathrm{Pro}(\mathrm{Str}\pi) \simeq \mathrm{StrTop}\infty$, $\operatorname{Gal}(X)$ is exactly the profinite stratified shape of the étale $\infty$-topos $X_{\mathrm{\acute{e}t}}$ stratified by its Zariski spectrum. The nerve of $\operatorname{Gal}(X)$, after delocalization, recovers the protruncated étale homotopy type, and the extended homotopy groups are computed accordingly [1812.11637].

5. Illustrative Example: The Nodal Cubic

Let $C$ be the nodal cubic over $\mathbb{C}$. The category $\operatorname{Gal}(C)$ admits a continuous functor to the two-point poset ${0 < 1}$, sending the node to $0$ and the generic point to $1$. At the node, there are precisely two lifts of the generic point, so the essential image is the category $D$ with two objects $0,1$ and two distinct maps $0 \rightarrow 1$. Inverting all morphisms in $D$ yields the homotopy type of $S^1$. Thus, the protruncated étale homotopy type of the nodal cubic is
[
\pi_1^{\mathrm{\acute{e}t}}(C) \cong \mathbb{Z}, \quad \pi_n^{\mathrm{\acute{e}t}}(C) = 0 \ \text{for}\ n > 1
]
When $X$ admits a finite Zariski stratification, $\operatorname{Gal}(X)$ sits over the finite poset of strata, and the protruncated étale homotopy type is recovered by taking the nerve of $\operatorname{Gal}(X)$ and inverting all specialization maps; the extended homotopy pro-groups are again those of that nerve [1812.11637].

6. Significance and Context

The ability to reconstruct extended étale homotopy groups from profinite Galois categories via the delocalization of the profinite stratified shape provides concrete computational tools linking topos theory, Galois theory, and homotopy theory. This approach situates the extended étale homotopy type as a homotopical invariant determined by the categorical and stratified structure, offering new insights and computational advantages in the study of schemes and their cohomological invariants. The connection to classical results, as in the case of the nodal cubic and the Riemann existence theorem, exemplifies the utility of this perspective [1812.11637].