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Exit paths and constructible stacks

David Treumann

arXiv:0708.0659v1math.AT

TL;DR

The paper asks how to extend the sheaf–fundamental-group correspondence to constructible sheaves and stacks of categories on stratified spaces. It introduces S-constructible stacks and the exit-path 2-category, then proves that these stacks are equivalent to Cat-valued 2-functors on that 2-category. The construction depends on a tameness condition for homotopies in the proof of the van Kampen property.

  • Problem

    The paper seeks a framework relating sheaves that are locally constant only along strata and higher-categorical stack-valued generalizations.

    Source: §1 · p. 1, §1.2 · p. 3

  • Method

    It defines S-constructible stacks and the stratified exit-path 2-category EP≤2(X,S), using exit paths and tame homotopies.

    Source: §1 · p. 1, §7.1 · p. 26, §7.2 · p. 27

  • Results

    The 2-category of S-constructible stacks on X is equivalent to the 2-category of Cat-valued 2-functors on EP≤2(X,S).

    Source: §1.3 · p. 3, §7.3 · p. 28

  • Takeaways & Limitations

    Constructible stacks admit a classification by stratified 2-dimensional monodromy data, extending the exit-path description of constructible sheaves.

    Source: §1.1 · p. 2, §7.3 · p. 28

    The proof requires a tameness condition on homotopies, and the authors were unable to obtain a tame-approximation result removing it.

    Source: §7.1 · p. 26

Abstract

from arXiv · show

For a Whitney stratification S of a space X (or more generally a topological stratification in the sense of Goresky and MacPherson) we introduce the notion of an S-constructible stack of categories on X. The motivating example is the stack of S-constructible perverse sheaves. We introduce a 2-category $EP_{\leq 2}(X,S)$, called the exit-path 2-category, which is a natural stratified version of the fundamental 2-groupoid. Our main result is that the 2-category of S-constructible stacks on X is equivalent to the 2-category of 2-functors from $EP_{\leq 2}(X,S)$ to the 2-category of small categories.

1. Introduction

The paper generalizes the correspondence between locally constant sheaves and fundamental-group actions to constructible sheaves and higher-categorical stacks. It introduces exit-path categories and proves that constructible stacks are classified by 2-functors from an exit-path 2-category.

1. Introduction

  • S-constructible sheaves are locally constant on each stratum of a topological stratification, which decomposes the space into suitably compatible manifolds.Topological stratifications are broader than Whitney and Thom–Mather stratifications but narrower than several more general stratified-space notions.Source: §1 · p. 1

1.1. Exit paths and constructible sheaves.

  • The exit-path category EP≤1(X, S) has points of X as objects and homotopy classes of exit paths as morphisms, yielding an equivalence with S-constructible sheaves of sets.Exit paths move through strata without decreasing stratum dimension, and their homotopies satisfy a tameness condition.Source: §1.1 · p. 2
  • For the punctured disk stratification, a constructible sheaf consists of vector spaces V and W, a map a: V →W, and an invertible endomorphism b of W satisfying b ◦a = a.The endomorphism records the loop around the singular origin, while the map records the exit path from the origin.Source: §1.1 · p. 2

1.2. Perverse sheaves.

  • Perverse sheaves form a stack, enabling their local description on cone-shaped charts and motivating an inductive strategy for computing their categories.Concrete descriptions of perverse sheaves often use linear-algebra data, although their objects and especially morphisms can be difficult to handle directly.Source: §1.2 · p. 3

1.3. Constructible stacks.

  • The main theorem identifies S-constructible stacks on X with Cat-valued 2-functors on EP≤2(X, S), whose morphisms are exit paths and whose 2-morphisms are tame homotopy classes between them.For the stratified Riemann sphere, this gives two categories, a functor between them, and a natural automorphism encoding the spherical homotopy.Source: §1.3 · pp. 3–4

1.4. Notation and conventions.

  • The paper uses strict 2-categories, weakly composition-preserving 2-functors, and StS(X) for the 2-category of S-constructible stacks.It distinguishes prestacks, stacks, locally constant stacks, and constructible stacks in its notation.Source: §1.4 · p. 4

2. Locally constant stacks

The section develops locally constant stacks of categories, characterizes them through restriction invariance on contractible neighborhoods, and proves homotopy invariance and base-change properties.

2.1. Constant stacks.

  • A small category C produces a constant stack C_X; for sets on locally contractible spaces, this recovers the stack of locally constant sheaves.The constant stack is obtained by stackifying the constant C-valued prestack.Source: §2.1 · pp. 4–5
  • Restriction to a point is controlled by contractible neighborhoods: for a contractible open U, the category of sections of a constant stack is equivalent to its stalk category.The associated Hom-sheaves are locally constant, supporting the stalkwise equivalences.Source: §2.1 · p. 5, §2.2 · p. 5

2.2. Locally constant stacks.

  • Locally constant stacks are stacks that become equivalent to constant stacks on an open cover, with restriction equivalences providing a practical characterization.On locally contractible spaces, equivalence of restrictions between suitable contractible open sets is equivalent to local constancy.Source: §2.2 · pp. 5, 7
  • Pullback preserves locally constant stacks, while projection along [0,1] induces inverse equivalences between locally constant stacks on X and on [0,1] × X.The homotopy invariance theorem yields equivalence of the corresponding 2-categories.Source: §2.2 · pp. 5–6
  • Homotopy equivalences induce equivalences of 2-categories of locally constant stacks and equivalences on categories of global sections.This extends interval homotopy invariance from projections to arbitrary homotopy equivalences.Source: §2.2 · p. 7

2.3. Direct images and base change.

  • For Serre fibrations, direct images of locally constant stacks remain locally constant, and the associated base-change map is an equivalence.The base-change statement applies after pullback along any continuous map into the base.Source: §2.3 · pp. 7–8

3. Constructible stacks

The section defines constructible stacks on topologically stratified spaces, establishes their stratified homotopy invariance, and gives restriction criteria and direct-image stability.

3. Constructible stacks

  • An S-constructible stack restricts to a locally constant stack on every stratum, and this notion is preserved by pullback along stratum-preserving maps.The pullback is locally constant on each connected component of every stratum.Source: §3 · p. 8, §3.2 · p. 10

3.1. Topologically stratified spaces.

  • A topological stratification is a filtered decomposition into manifold strata with neighborhoods modeled on open cones times Euclidean spaces.Conical neighborhoods provide the local inductive structure used throughout the section.Source: §3.1 · pp. 8–9

3.2. Constructible stacks.

  • Constructible stacks are invariant under stratified homotopy: projection from (0,1) × X induces an equivalence between the corresponding 2-categories.The proof reduces the claim to equivalences on individual strata and uses base-change for the projection.Source: §3.2 · p. 11
  • Stratified homotopy equivalences induce equivalences of constructible-stack 2-categories and preserve the corresponding categories of global sections.This is the stratified analogue of homotopy invariance for locally constant stacks.Source: §3.2 · pp. 11–12
  • A stack is constructible exactly when restriction along stratified-homotopy-equivalent inclusions is an equivalence; it suffices to test this on conical open subsets.Under these conditions, sections on a conical neighborhood are equivalent to the stalk at its center.Source: §3.2 · p. 12

3.3. Direct images.

  • Direct image along a stratified map sends constructible stacks to constructible stacks.Locally, the inverse image of a conical neighborhood is modeled by a product, so smaller and larger inverse-image neighborhoods are stratified homotopy equivalent.Source: §3.3 · p. 12

4. Example: the stack of perverse sheaves

The section constructs the stack of perverse sheaves from local perversity conditions and proves that it is both a stack and constructible for any topologically stratified space.

4. Example: the stack of perverse sheaves

  • Constructible complexes on conical neighborhoods have vanishing higher cohomology, which makes pullback along (0,1) × X an equivalence for the relevant derived categories.This derived-category result underlies the stack and constructibility arguments for perverse sheaves.Source: §4 · p. 13
  • For each connected stratum, a perversity assigns an integer that bounds the cohomology of restriction and extension-by-zero complexes in opposite directions.These conditions define the S-constructible perverse sheaves of perversity p as a full subcategory of the bounded constructible derived category.Source: §4 · p. 13
  • The prestack assigning perverse sheaves to each open set is a stack.Restriction to smaller open subsets preserves the perverse-sheaf conditions.Source: §4 · p. 13
  • The stack of perverse sheaves is constructible with respect to the topological stratification for any perversity function on connected strata.The proof applies the restriction criterion for constructible stacks together with invariance of the bounded constructible derived category under stratified homotopy equivalence.Source: §4 · pp. 13–14

5. The fundamental 2-groupoid and 2-monodromy

This section constructs the fundamental 2-groupoid and relates 2-monodromy functors to locally constant stacks. It proves the resulting equivalence by decomposing homotopies into elementary pieces and establishing the required descent, full faithfulness, and essential surjectivity.

5.1. The fundamental 2-groupoid.

  • The fundamental 2-groupoid π≤2(X) has points of X as objects and fundamental groupoids of Moore-path spaces P(x,y) as hom categories, with strictly associative concatenation and strict constant-path units.Its 2-morphisms can be represented by homotopy classes of squares in X with prescribed boundary paths and fixed endpoints.Source: §5.1 · pp. 14–15

5.2. Two-monodromy and locally constant stacks.

  • The section defines 2Mon(X) as Cat-valued 2-functors on π≤2(X), restricts them along open inclusions, and constructs a prestack N from these functors.The unstratified main theorem identifies locally constant stacks with these 2-monodromy functors.Source: §5 · p. 14, §5.2 · p. 15

5.3. A van Kampen theorem for the fundamental 2-groupoid.

  • For every d-cover of a compactly generated Hausdorff space, restriction identifies 2-monodromy functors on X with compatible local 2-monodromy data, giving an equivalence of 2-categories.Local data consist of functors on each Ui, equivalences on inclusions, and coherently commuting higher compatibility maps.Source: §5.3 · pp. 15–16

5.4. The 2-monodromy theorem.

  • For locally contractible X, the prestack associated to a 2-monodromy functor is a locally constant stack, with stalk category equivalent to its value at each point, and N : 2Mon(X) → Stlc(X) is an equivalence.The associated prestack is always a stack; local contractibility yields local constancy and the stated stalk identification.Source: §5.2 · p. 15, §5.4 · p. 16

5.5. The proof of the van Kampen theorem.

  • Every homotopy between Moore paths can be factored, up to homotopy, into finitely many elementary homotopies supported in cover elements, and homotopies between such factorizations admit analogous elementary 3-dimensional decompositions.These local factorizations underpin the descent proof for the van Kampen theorem.Source: §5.5 · pp. 18–19
  • The van Kampen proof establishes that restriction is fully faithful and essentially surjective on morphisms and objects by gluing local transformations and extending local functors along path factorizations.Filtered limits combine compatible local categorical data, while elementary homotopy factorizations verify compatibility for arbitrary paths and homotopies.Source: §5.5 · pp. 19–21
  • The construction defines F(h) for homotopies supported in a single chart Ui as a natural transformation F(α) → F(β).For elementary homotopies, the definition inserts identity transformations along boundary paths and applies the local construction to the chart-supported middle homotopy.Source: §5.5 · p. 21
  • An arbitrary homotopy is assigned a composite of the transformations of its elementary factors, and this assignment is independent of the chosen factorization.The required elementary decomposition follows from proposition 5.9, while factorization independence follows from proposition 5.11.Source: §5.5 · p. 21
  • The local definition extends to all elementary homotopies and remains invariant under elementary homotopies between them, making F well-defined.Any elementary homotopy can be expressed using a chart-supported middle piece, and propositions 5.9 and 5.11 guarantee well-definedness.Source: §5.5 · p. 21
  • The restriction functor is essentially surjective because each comparison Fi(x) → F(x) is an equivalence by construction.Thus the family {Fi} is equivalent to res(F).Source: §5.5 · p. 21

6. Stratified 2-truncations and 2-monodromy

The section axiomatizes stratified 2-truncations and proves that their category-valued 2-functors model constructible stacks. The proof combines cone, homotopy, and van Kampen properties with induction on stratified spaces.

6. Stratified 2-truncations and 2-monodromy

  • A stratified 2-truncation assigns a 2-category functorially to each topologically stratified space and satisfies normalization, homotopy invariance, cone, and van Kampen axioms.These axioms ensure that 2-functors into Cat recover constructible stacks.Source: §6 · p. 22

6.1. Cones on 2-categories and axiom (C).

  • For each compact topologically stratified space L, axiom (C) requires an equivalence of 2-categories associated with CL, and the projection-induced functor is an equivalence.Source: §6.1 · pp. 22–23, §6 · p. 22

6.2. Exit 2-monodromy functors and axiom (vK).

  • The van Kampen axiom requires restriction from a stratified space to a d-cover to induce an equivalence on categories of exit 2-monodromy functors.Local functors, equivalences on overlaps, and higher coherence data assemble into global monodromy.Source: §6.2 · p. 23

6.3. The exit 2-monodromy theorem.

  • The proof of Theorem 6.10 uses conical neighborhoods and induction on dimension to prove essential full faithfulness and essential surjectivity.Local models on cones are glued across a d-cover using the van Kampen axiom.Source: §6.3 · pp. 24–25
  • For compact links, Proposition 6.9 identifies constructible stacks on open cones with suitable triples consisting of a category, a constructible stack on the link, and a map from the constant stack.This cone calculation supplies the local step for the global induction.Source: §6.3 · p. 24
  • Theorem 6.10 establishes an equivalence between exit 2-monodromy functors and S-constructible stacks on any topologically stratified space.The associated prestack NF is itself S-constructible.Source: §6.3 · p. 25

7. Exit paths in a stratified space

The section defines exit paths and tame homotopies, assembling them into the exit-path 2-category EP≤2(X,S). It proves that this construction is a stratified 2-truncation and classifies S-constructible stacks by Cat-valued 2-functors.

7. Exit paths in a stratified space

  • EP≤2(X, S) has points of X as objects, Moore paths with the exit property as morphisms, and tame homotopy classes of homotopies between exit paths as 2-morphisms.Source: §7 · p. 25, §7.1 · p. 26

7.1. Tame homotopies.

  • Concatenation of exit paths and tame homotopies defines the strict 2-category EP≤2(X,S), functorial under stratum-preserving maps.The construction therefore gives a functor EP≤2: Strat → 2cat.Source: §7.1 · p. 26, §7.2 · p. 26
  • Tameness is required for the van Kampen proof because it permits homotopies to be subdivided indefinitely; the author conjectures the condition may be removable from related constructions.A tame approximation result needed to compare tame and ordinary fundamental groupoids was not obtained.Source: §7.1 · p. 26

7.2. The exit path 2-category is a stratified 2-truncation.

  • EP≤2 satisfies the stratified 2-truncation axioms, including the homotopy, cone, and van Kampen axioms.The results establish axioms (N) and (H), axiom (C), and axiom (vK).Source: §7.2 · pp. 27–28
  • Every homotopy between exit paths is homotopic to a finite composite of elementary homotopies subordinate to a d-cover.A suitable triangulation and barycentric subdivision produce the required elementary decomposition.Source: §7.2 · p. 28

7.3. Proof of the main theorem.

  • For every topologically stratified space, S-constructible stacks on X are equivalent to Cat-valued 2-functors on EP≤2(X,S).The equivalence follows after proving that EP≤2 is a stratified 2-truncation.Source: §7.3 · pp. 28–29

Appendix A. Stacks

  • Stacks are treated as prestacks of categories satisfying descent over d-covers, with restriction data obeying compatibility conditions.Appendix A formulates prestacks as contravariant 2-functors to Cat and defines stacks through equivalences with 2-limit descent data.Source: Appendix A · p. 29, §A.1 · p. 29, §A.2 · p. 30

A.1. Prestacks.

  • A prestack is a 2-functor Iop → Cat, assigning categories to opens, restriction functors to inclusions, and coherent comparison isomorphisms.The coherence condition is expressed by commutative tetrahedra for chains of inclusions.Source: §A.1 · p. 29

A.2. Stacks.

Stacks are prestacks satisfying a categorical descent condition over d-covers. The appendix establishes locality properties for morphisms, adjoints, equivalences, and stackification.

A.2. Stacks.

  • A d-cover is a cover closed under finite intersections, and a stack is a prestack whose restriction functor to compatible local data is an equivalence.The stack condition is imposed for every open subset and every d-cover.Source: §A.2 · p. 30
  • For stacks, Hom-prestacks and presheaves of morphisms are themselves sheaf-like, while adjoints and equivalences can be checked on stalks.Theorem A.6 also supplies stackification as a right adjoint to the inclusion of stacks into prestacks.Source: §A.2 · p. 30
  • Stackification preserves stalks: the adjunction morphism from a prestack to its stackification induces an equivalence on every stalk.This identifies stackification as a local correction of descent failure.Source: §A.2 · p. 30

A.3. Operations on stacks.

The appendix defines pushforward and pullback operations for stacks and records their adjunction relationship. Pullback is obtained through neighborhoodwise limits followed by stackification.

A.3. Operations on stacks.

  • Pushforward along a continuous map is defined directly on opens and preserves stacks, yielding a strict 2-functor on prestacks.The pushforward of a stack remains a stack.Source: §A.3 · p. 30
  • Pullback of a stack is formed from a neighborhoodwise 2-limit and then stackification, rather than by a direct open-set formula.This construction accounts for the categorical descent required of the result.Source: §A.3 · pp. 30–31
  • For a point inclusion, pushforward recovers the stalk category, and for a continuous map f the induced pushforward is right 2-adjoint to pullback.The point case identifies the categorical fiber used by stalk computations.Source: §A.3 · p. 31

A.4. Descent for stacks.

Stacks satisfy descent and can be reconstructed from compatible stacks on a d-cover. The restriction 2-functor from global stacks to such local descent data is an equivalence.

A.4. Descent for stacks.

  • A stack on a d-cover consists of local stacks, equivalences on inclusions, coherent comparison isomorphisms, and a commutative tetrahedron condition.These data encode compatibility across chains of overlapping opens.Source: §A.4 · p. 31
  • The restriction 2-functor St(X) → St({Ui}) is an equivalence of 2-categories.Thus global stacks are completely recoverable from compatible local stack data.Source: §A.4 · p. 31
  • The local descent 2-category may be interpreted as an inverse limit, or inverse 3-limit, of the categories St(Ui).This motivates viewing stacks as forming a 2-stack.Source: §A.4 · p. 31

Appendix B. 2-categories

The appendix fixes conventions for strict 2-categories, their morphisms, and two ways of composing 2-morphisms. It also distinguishes strict, weak, and lax formulations while adopting strict 2-categories for the paper.

Appendix B. 2-categories

  • A strict 2-category has objects, a category of morphisms for each object pair, composition functors, and strictly associative unital composition.Its 1-morphisms are objects of hom categories, while 2-morphisms are morphisms in those hom categories.Source: Appendix B · pp. 31–32
  • A (2,1)-category is a 2-category whose 2-morphisms are all invertible.Source: Appendix B · p. 32
  • The 2-category Cat has categories as objects, functors as 1-morphisms, and natural transformations as 2-morphisms.The notation Cat is distinguished from the 1-category cat.Source: Appendix B · p. 32
  • Weak 2-categories relax strict associativity up to coherent isomorphism, but the paper uses strict models because its relevant 2-categories are strict or can be strictified.Source: Appendix B · p. 32
  • The opposite 2-category keeps the same objects and reverses hom categories, while other possible reversals are not used.Source: Appendix B · p. 32

B.1. Two-dimensional composition in a 2-category.

  • Besides ordinary composition, 2-morphisms admit a second composition ⋆, and the interchange equation allows any compatible composite to be treated as equal.Source: Appendix B · p. 33, §B.1 · p. 33

B.2. Adjoints and equivalences within a 2-category.

This section defines adjoints, equivalences, commutative higher diagrams, and 2-groupoids within a 2-category. These notions organize invertibility, coherence, and the homotopical structure encoded by 2-categorical data.

B.2. Adjoints and equivalences within a 2-category.

  • A right adjoint consists of a reverse 1-morphism and unit and counit 2-morphisms satisfying triangle identities; left adjoints are defined dually.Adjoints are unique up to unique isomorphism compatible with the adjunction data.Source: §B.2 · p. 33
  • A 1-morphism is an equivalence when it has an adjoint whose two adjunction maps are isomorphisms.Source: §B.2 · p. 33

B.3. Commutative diagrams in a 2-category.

  • Commutative triangles, squares, tetrahedra, and prisms encode specified 2-isomorphisms between composites, with higher coherence requiring alternative composites to agree.Source: §B.2 · p. 33, §B.3 · p. 34
  • In a (2,1)-category, commutative higher simplices use invertible 2-morphisms and assemble into the nerve of the 2-category.Source: §B.3 · p. 34

B.4. 2-groupoids.

  • A 2-groupoid is a 2-category whose morphisms are equivalences and whose 2-morphisms are isomorphisms.Its nerve models spaces with homotopy groups vanishing above dimension 2.Source: §B.4 · p. 35
  • A one-object 2-groupoid is a 2-group, and its algebraic structure can encode fundamental and second homotopy groups together with a Postnikov invariant.Source: §B.4 · p. 35

B.5. 2-functors between 2-categories.

The appendix defines 2-functors with object, hom-functor, and coherence data, then explains strictness, laxness, composition, and the resulting 2-category of 2-functors.

B.5. 2-functors between 2-categories.

  • A 2-functor assigns objects, hom-category functors, and natural isomorphisms comparing the two composites, while preserving identities and satisfying coherence.Source: §B.5 · p. 35
  • A strict 2-functor has identity coherence maps, whereas the paper uses 2-functor for the broader non-strict notion.Terminology varies across authors, who may call these broader maps pseudofunctors.Source: §B.5 · p. 35
  • The paper excludes lax 2-functors, whose coherence maps need not be invertible, because non-lax notions are taken to be appropriate for stacks of categories.Source: §B.5 · p. 35
  • Every 2-functor preserves left and right adjoints and sends equivalences to equivalences.Source: §B.5 · p. 36
  • 2-functors compose, with coherence data for the composite, and strict hom 2-functors provide standard examples into Cat.Source: §B.6 · p. 36, §B.5 · p. 36

B.6. Composite 2-functors.

  • 2-natural transformations and modifications between 2-functors form hom categories, yielding a 2-category 2Funct(C, D).Source: §B.7 · p. 36

B.7. The 2-category 2Funct(C, D).

The 2-category 2Funct(C, D) organizes 2-functors, 2-natural transformations, and modifications. Its adjunctions and equivalences can be checked objectwise in D.

B.7. The 2-category 2Funct(C, D).

  • A 2-natural transformation assigns each object a 1-morphism and each source 1-morphism an invertible compatibility 2-morphism, subject to prism coherence.Source: §B.7 · p. 36
  • A transformation in 2Funct(C, D) has an adjoint exactly when each component transformation in D has the corresponding adjoint.Source: §B.7 · p. 36
  • A transformation is an equivalence in 2Funct(C, D) exactly when every component is an equivalence in D.Source: §B.7 · p. 36
  • The framework therefore reduces equivalence questions for 2-functor transformations to objectwise equivalence checks in the target 2-category.Source: §B.7 · p. 36

B.8. Adjoint 2-functors between 2-categories.

This section defines 2-adjunctions and equivalences between 2-categories, characterizing equivalences through essential full faithfulness and essential surjectivity. It then specifies inverse and filtered direct 2-limits of categories-valued 2-functors.

B.8. Adjoint 2-functors between 2-categories.

  • A 2-adjunction consists of 2-functors F and G together with a 2-natural equivalence between Hom(F−,−) and Hom(−,G−).The equivalence is taken in the 2-category 2Funct(Cop × D, Cat).Source: §B.8 · p. 36
  • An equivalence requires adjoint 2-functors whose induced transformations 1C → GF and FG → 1D are equivalences.These transformations arise by applying the adjunction to identity 2-functors.Source: §B.8 · p. 37
  • A 2-functor is part of an equivalence exactly when it is essentially fully faithful and essentially surjective.Essential full faithfulness requires equivalences on hom-categories, while essential surjectivity requires every target object to be equivalent to an image.Source: §B.8 · p. 37

B.9. Direct and inverse 2-limits of 1-categories.

  • The inverse 2-limit category assigns objects in each F(i), compatible isomorphisms along morphisms, and morphisms commuting with those structure maps.Compatibility is imposed for each commutative triangle in the indexing 2-category.Source: §B.9 · p. 37
  • For a filtered poset I, the direct 2-limit is formed from objects across the categories F(i), with morphisms represented by a limit after restriction to common stages.For ℓ≤k, x|k denotes the image of x under the functor induced by the unique morphism ℓ→k.Source: §B.9 · p. 37, §B.8 · p. 37
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