Topological Stratification

• Topological stratification is the decomposition of a space into disjoint, locally closed strata—often manifolds—with closures that satisfy specific frontier conditions.
• It extends classical manifold techniques to singular spaces using intersection cohomology and filtered homotopy invariants.
• Model category structures on stratified spaces provide a robust framework for analyzing singularities and establishing homotopical equivalences.

Topological stratification refers to the decomposition of a topological space into disjoint, locally closed subsets called strata, each typically enjoying additional regularity (e.g., being a manifold) and arranged so that their closures satisfy specific incidence and "frontier" conditions. This framework enables a systematic treatment of spaces with singularities, permitting the extension of manifold-based tools and homotopical invariants to singular and filtered settings.

1. Definition and Canonical Examples

A stratified space is a topological space $X$ together with a partition into locally closed stratum pieces $\{ X_i \}$ such that $X = \coprod_i X_i$, and the closure of each stratum can be written as a union of strata. Notably, each stratum is often a manifold (possibly of varying dimension), and the glueing between strata is subject to additional control, such as the frontier condition: if $X_j \cap \overline{X_i} \neq \varnothing$, then $X_j \subset \overline{X_i}$.

Illustrative examples include:

• The pinched torus, with a 2-manifold part and a singular point.
• Singular algebraic varieties, where smooth strata are glued along singular loci.
• Quotients $M/G$ for group actions with nontrivial stabilizers, stratified by orbit type.
• Contracted submanifolds, where a manifold is collapsed along a subspace to form a singular stratum.

Stratified spaces serve as a foundational language for studying singularities, enabling the use of homological and homotopical machinery on spaces otherwise outside the reach of classical manifold techniques.

2. Intersection Cohomology and Stratified Homotopy

Intersection cohomology $IH^*$ extends singular cohomology to singular (pseudo-manifold) spaces, capturing essential topological information lost if singularities are ignored. Crucially: - $IH^*$ is functorial for stratified maps and invariant under stratified homotopy equivalence. - Its definition incorporates permissible chains that respect the stratification (constraining how chains may intersect lower-dimensional strata). - Homotopy invariance is not absolute: two spaces that are ordinary homotopy equivalent may have different intersection cohomologies unless the equivalence preserves stratification structure.

This suggests intersection cohomology is sensitive to both the topology and the stratified (singularity) data of a space, making stratified homotopy—a homotopy respecting the stratification, typically via filtered or stratified maps—the correct notion of equivalence for such invariants.

3. Filtered Homotopy Groups and Stratified Invariants

Filtered homotopy groups $s\pi_n$ generalize classical homotopy groups to the stratified setting: - For a filtered space $(X, \varphi: X \to P)$, with $P$ a poset of strata, $s\pi_0(X)$ encodes connectivity within and between strata, and higher $s\pi_n$ distinguish how higher connectivity is stratified and how links between strata behave. - These invariants are pivotal: maps inducing isomorphisms on all $s\pi_n$ are precisely the weak equivalences in the stratified (filtered) homotopy theory.

Filtered homotopy groups provide a robust measure of both the internal topology of each stratum and the ways different strata are glued, permitting a Whitehead-type theorem for stratified spaces.

4. Model Category Structures for Stratified Spaces

The development of model categories for stratified spaces enables a homotopical treatment parallel to classical topology: - Filtered simplicial sets ($sSet_P$): Simplicial sets over the nerve $N(P)$ of the poset of strata, with model category structure where weak equivalences are those inducing isomorphisms on filtered homotopy groups and fibrations are right-lifting against admissible filtered horn inclusions. - Filtered spaces ($Top_P$): Topological spaces equipped with a continuous map to $P$, with model structure reflecting that of $sSet_P$ via comparison functors. - Stratified spaces ($Strat$): The category of topological spaces over arbitrary posets, supporting a bifibrant (fibered) model structure. - Stratified simplicial sets ($sStrat$): Simplicial sets over arbitrary posets, likewise supporting global model structures.

Adjunctions between these categories—filtered realization and singular functors—are constructed and shown (often conjecturally, subject to technical points) to be Quillen adjunctions, generalizing the classical Kan–Quillen correspondence to stratified settings.

5. Weak Equivalences, Fibrations, and Whitehead-Type Theorems

A key achievement is the identification of weak equivalences and fibrations adapted to the stratified context: - Weak equivalences: Maps inducing isomorphisms on all filtered homotopy groups for all test objects. - Fibrations: Maps with the right lifting property with respect to filtered horn inclusions—encoding both local and stratification-respecting lifting. - Filtered Whitehead theorem: In the stratified (filtered) model category, a map is a homotopy equivalence iff it induces isomorphisms on filtered homotopy groups, aligning the stratified and classical perspectives for suitable spaces.

Additionally, filtered Serre fibration analogues classify fibrations via right lifting with respect to disks in the filtered sense.

6. Quillen Bifibrations and Global Stratified Homotopy Theory

Employing the notion of Grothendieck/Quillen bifibration, one can lift the model category structure from each fiber (for fixed poset $P$) to the total category of all stratified spaces, ensuring: - Stratified homotopical data is preserved globally. - Cohesive Quillen adjunctions exist between stratified spaces ($Strat$) and stratified simplicial sets ($sStrat$), and likewise for their over-fibers ($Top_P$, $sSet_P$). - Global comparison and transfer of homotopical information—enabling meaningful connections between discrete and topological stratification frameworks.

These structures undergird modern advances in stratified homotopy theory, including the interpretation of the stratified homotopy hypothesis and connections to higher category theory.

7. Classical and Stratified Comparisons

The stratified and filtered viewpoint generalizes classical Whitehead, Serre, and Quillen machinery: - Whitehead theorem for stratified spaces mirrors that for CW complexes, upgraded to filtered homotopy groups. - Kan conditions and Serre fibrations have filtered analogues, ensuring stratification data is preserved under homotopical operations. - Adjunctions—filtered realization and singular functors—behave akin to the classical setting, but retain stratification information at each step.

Table: Classical vs. Stratified Invariants

Notion	Classical	Stratified/Filtered Version
Whitehead Thm.	Isomorphisms on $\pi_n$ (n ≥ 0) ⇒ homotopy equivalence	Isomorphisms on $s\pi_n$ ⇒ filtered homotopy equivalence
Fibrations	Right lifting against disks/horns	Right lifting against filtered horn inclusions
Cohomology	Homotopy invariant	Intersection/stratified homotopy invariant
Model Category	$Top$, $sSet$	$Top_P$, $sSet_P$, $Strat$, $sStrat$
Realization	Kan–Quillen adjunction	Filtered realization/singular functor; Quillen adjunction

8. Significance and Broader Context

The homotopy theory of stratified spaces—developed through model categories, filtered homotopy invariants, and stratified adjunctions—enables: - The classification and analysis of singular spaces in algebraic geometry, topology, and applications such as data analysis. - The extension of foundational theorems to singular and filtered contexts. - The systematic construction of invariants (e.g., intersection cohomology) sensitive to both topological and stratification structure. - Foundations for contemporary developments in higher category theory and factorization homology on singular spaces.

The framework unifies and generalizes previous approaches, providing a precise mathematical infrastructure for the paper of topological stratification, with deep ramifications throughout geometric topology, algebraic geometry, and theoretical data analysis.