Condensed Homotopy Type

• Condensed Homotopy Type is an invariant of schemes that unifies classical étale and pro-étale homotopy types using advanced condensed mathematics.
• It is constructed via hypercoverings on the pro-étale site and Quillen adjunctions, delivering improved descent and computability.
• The framework reveals robust group-theoretic structures through condensed fundamental groups, bridging algebraic geometry, arithmetic topology, and higher homotopy theory.

The condensed homotopy type is an invariant of schemes that refines both the classical étale homotopy type and the pro-étale homotopy type, integrating recent advances from condensed mathematics and higher topos theory. Its construction leverages the machinery of condensed sets in the sense of Clausen and Scholze, provides compatibility with group-theoretic invariants like the fundamental group, and exhibits improved descent and fiber sequence properties compared to prior approaches. Recent works establish rigorous connections among the condensed homotopy type, Galois categories, and the computation of homotopical invariants, positioning this framework as central in the modern intersection of algebraic geometry, arithmetic topology, and higher homotopy theory (Haine et al., 8 Oct 2025, Meffle, 24 Mar 2025).

1. Concept and Definition

The condensed homotopy type of a scheme $X$, denoted $\mathcal{T}_{\text{cond}}(X)$, is constructed as an object in a homotopy category of (pro-)simplicial condensed sets. The construction is deeply informed by the pro-étale topology: for a (quasi-compact and quasi-separated, or "qcqs") scheme, the condensed homotopy type is obtained either as the relative shape of the pro-étale site (after condensation) or, equivalently, as a classifying space $B_{\text{cond}}(\mathrm{Gal}(X))$ associated to the scheme's Galois category in the sense of Barwick–Glasman–Haine (Haine et al., 8 Oct 2025).

This invariant refines previous notions:

• It extends the étale homotopy type of Artin–Mazur and Friedlander by encoding more subtle topological structure.
• It incorporates pro-étale information, especially relevant in non-Noetherian or non-locally connected contexts (Meffle, 24 Mar 2025).
• It packages the profinite or condensed structure into a framework suitable for descent, explicit computation, and comparison with schemes defined over rings of continuous functions.

2. Construction via Sites, Hypercoverings, and Condensed Sets

The condensed homotopy type is defined using hypercoverings on the pro-étale site of a scheme $S$. For a pointed scheme $(S, s)$, one takes all hypercoverings $U_\bullet$ in the pro-étale site and associates to each the "space of components" $\operatorname{comp}(U_\bullet)$. This data is assembled into a pro-object: $$\Pi_{\mathrm{pe}}(S, s) : \operatorname{Ho}(\operatorname{Hy}(S, s)) \to \operatorname{Ho}(\mathbb{S}\mathrm{Topo}_*),\quad (U_\bullet, u) \mapsto \operatorname{comp}(U_\bullet, u)$$ which, after refinement, lives in the homotopy category of pro-simplicial condensed sets (Meffle, 24 Mar 2025).

A key technical step is the use of the Quillen adjunction

$\Pro(\operatorname{comp}_{\operatorname{sh}}) \dashv \Pro(\operatorname{comp}^{\operatorname{sh}}): \Pro(\operatorname{s}\Sh(S_{\mathrm{pe}})) \rightleftarrows \Pro(\sCond),$

where $\operatorname{comp}_{\operatorname{sh}}$ extends the component functor to sheaves, and $\sCond$ denotes the category of simplicial condensed sets (Meffle, 24 Mar 2025, Haine et al., 8 Oct 2025). The refined condensed homotopy type $\widetilde{\Pi}_{\mathrm{pe}}(S) := \mathbb{L}\Pro(\operatorname{comp}_{\operatorname{sh}})(*)$ matches the classical pro-étale type after passage via the forgetful functor.

3. Key Properties: Descent, Fiber Sequences, and Computations

The condensed homotopy type satisfies strong descent and base-change properties:

• Descent: It exhibits hyperdescent along integral (and, more generally, pro-étale) hypercovers, i.e.,

$$\operatorname{colim}_{[n] \in \Delta^{\mathrm{op}}} \mathcal{T}_{\text{cond}}(X_n) \simeq \mathcal{T}_{\text{cond}}(X)$$

for an integral hypercover $X_\bullet \to X$ (Haine et al., 8 Oct 2025).

• Fiber sequences: For morphisms $f: X \to S$ with $S$ "0-dimensional," the fiber of the induced map on condensed homotopy types is, up to homotopy, the condensed homotopy type of the geometric fiber $X_s$.

Explicit computations illustrate the framework:

• For $T$ a compact Hausdorff space, the condensed homotopy type of $C(T, \mathbb{C})$ as a ring recovers the original space $T$ via its maximal spectrum.
• For w-contractible schemes (e.g. algebraically closed fields), the condensed homotopy type is trivial, given by the constant simplicial object on the (profinite) space of components (Meffle, 24 Mar 2025).
• For $\mathbb{R}$, the condensed homotopy type coincides with the classifying space $B(\mathbb{Z}/2\mathbb{Z})$, reflecting the Galois group of $\mathbb{C}/\mathbb{R}$.

4. Galois Categories, Fundamental Groups, and Classifying Spaces

By identifying the condensed homotopy type with the classifying space of the Galois category, $$\mathcal{T}_{\text{cond}}(X) \simeq B_{\text{cond}}(\mathrm{Gal}(X))$$, one accesses the full representation-theoretic and pro-finite stratified content of a scheme (Haine et al., 8 Oct 2025). This comparison allows:

• Recovery of the classical étale homotopy type after passage to discrete quotients.
• Definition of higher and lower homotopy invariants:
  • The condensed fundamental group $\pi_1^{\mathrm{cond}}(X,x)$ is $\pi_1(\mathcal{T}_{\mathrm{cond}}(X),x)$.
  • After passage to the quasiseparated quotient, $\pi_1^{\mathrm{cond,qs}}(X,x)$ agrees with the classical étale fundamental group in geometric-unibranch settings, and after Noohi completion recovers the pro-étale fundamental group.

Key formulas:

$$\mathcal{T}_{\text{cond}}(X) \simeq B_{\text{cond}}(\mathrm{Gal}(X)),\qquad \pi_1^{\mathrm{cond}}(X,x) := \pi_1(\mathcal{T}_{\text{cond}}(X), x),\qquad (\pi_1^{\mathrm{cond}}(X,x))_{\mathrm{Noohi}} \cong \pi_1^{\mathrm{pro}\text{-}\mathrm{ét}}(X, x)$$

The van Kampen theorem (Theorem 7.51) computes the condensed fundamental group for reducible schemes as a pushout of the étale fundamental groups of the normalizations, with relations arising from the dual graph of components.

5. Comparison with Classical and Pro-étale Homotopy Types

The condensed homotopy type unifies and refines both classical étale and pro-étale homotopy types:

• It aligns with the Artin–Mazur–Friedlander constructions after appropriate completions and discrete projections.
• For qcqs schemes, the pro-étale and condensed homotopy types are both profinite, and the condensed invariant computes étale cohomology with locally constant coefficients (Meffle, 24 Mar 2025, Haine et al., 8 Oct 2025).
• The refinement via condensed sets ensures compatibility with advanced functorial tools, such as Quillen adjunctions and derived functors between sheaf categories and condensed categories, yielding greater functoriality and control.

6. Topological and Group-Theoretic Structures

After passing to the quasiseparated quotient, the condensed fundamental group is a compact Hausdorff group, and for geometrically unibranch schemes it equals the étale fundamental group. This group-theoretic structure is robust under base change and product, as codified in van Kampen and Künneth formulas: $$\pi_1^{\mathrm{cond,qs}}(X \times_k Y, (x, y)) \simeq \pi_1^{\mathrm{cond,qs}}(X, x) \times \pi_1^{\mathrm{cond,qs}}(Y, y)$$

The condensed approach also reveals subtle phenomena: for instance, the condensed fundamental group of $\mathbf{A}^1_{\mathbb{C}}$ can be nontrivial, but its Noohi quotient recovers the classical pro-étale group.

7. Significance and Outlook

The construction of the condensed homotopy type opens new techniques for the paper of schemes and their topological invariants:

• It provides a bridge between algebraic geometry, topos-theoretic descent, and modern topological and homological algebra.
• The approach is compatible with computations using Galois categories and applies uniformly to arithmetic, geometric, and analytic contexts.
• The refinement offered by condensation is particularly significant when handling non-Noetherian schemes, compact Hausdorff spectra, and the paper of functorial or canonical group invariants.
• By leveraging advanced tools from condensed mathematics and higher topos theory, this framework sets the stage for further integration of cohomological invariants, higher stacks, and representation theory into the paper of algebraic and arithmetic schemes.

In summary, the condensed homotopy type synthesizes the topological and arithmetic data of schemes using modern tools from condensed mathematics, pro-étale topology, and higher category theory, enabling a unified approach to descent, computation, and the comparison of fundamental group(oid)s and cohomological invariants (Haine et al., 8 Oct 2025, Meffle, 24 Mar 2025).