Local isotropy can describe the automorphisms visible near each stratum, yet a global object requires more. Categories assigned on overlapping neighborhoods must be related by equivalences, those equivalences must agree on triple overlaps up to coherent natural isomorphism, and the coherence itself must satisfy a higher compatibility condition.
A constructible stack is designed for exactly this local-to-global problem. It is locally constant along each stratum while retaining categorical descent across the stratification. The previous exit-path article encoded transport. This guide makes the gluing data visible and shows why identical local categories can still assemble into inequivalent global stacks.
The Gold Standard is to inventory local categories, write overlap functors and coherence cells explicitly, test refinement and changed covers, and state the hypotheses under which an exit-path description is available.
A practical orientation
Use this guide after the exit-path and orbit-type articles. The original worked examples glue one-object groupoids around a three-set cover and compare trivial with twisted descent data.
Choose a reading route
Work and transfer examples
Chapters 7–9 calculate a twisted gluing and change cover and stratification.
Chapter contents
- 1. Begin with a stratified space and a categorical assignment
- 2. Separate a sheaf condition from a stack condition
- 3. Write descent data on double and triple overlaps
- 4. Place isotropy in the local automorphism category
- 5. Use exit paths to encode stratified transport
- 6. Detect a global obstruction as failed coherence
- 7. Worked example: identical local C2 isotropy, two global gluings
- 8. Transfer check: refine the cover and change trivialisations
- 9. Change the stratification and watch the category change
- 10. State the exodromy theorem boundary
- 11. Limits, misconceptions and practical evidence check
- 12. FAQs and the next reading route
1. Begin with a stratified space and a categorical assignment
Let X be divided into strata so that movement inside one stratum is qualitatively different from crossing toward another. A constructible categorical object is locally constant along each stratum, but it may change when paths exit one stratum and enter another.
For each sufficiently small open set U, assign a category F(U). Restriction to a smaller open V gives a functor F(U)→F(V). The stack condition says compatible local objects and arrows can be glued, with equivalence rather than literal equality because categorical presentations can differ.
- Space and stratification fixed.
- Categories on open sets.
- Restriction functors.
- Local constancy along strata.
- Gluing up to coherent equivalence.
2. Separate a sheaf condition from a stack condition
A sheaf glues elements that agree on overlaps. A stack glues objects equipped with isomorphisms on overlaps, and those isomorphisms must satisfy a cocycle law. Morphisms between glued objects also satisfy descent.
Replacing equality by isomorphism is not cosmetic. Local coordinate systems, group actions or perverse-sheaf presentations may be equivalent without sharing literal names. A stack records the equivalences needed to assemble them.
| Object | Local datum | Overlap requirement |
|---|---|---|
| Sheaf of sets | Element | Equality |
| Sheaf of groups | Group element | Equality in group |
| Stack of categories | Object and morphisms | Isomorphism plus coherence |
| Constructible stack | Categorical data constant on strata | Stack descent respecting strata |
3. Write descent data on double and triple overlaps
For a cover {Ui}, choose an object xi in F(Ui). On Uij choose an isomorphism φij between the restrictions of xi and xj. On Uijk require φik to agree with φjk∘φij, or supply the appropriate coherent 2-cell in a weak categorical formulation.
The data are not complete until the triple-overlap condition is checked. Pairwise equivalence can hold everywhere while the composite around a cycle returns with a nontrivial automorphism.
| Cover level | Datum | Question |
|---|---|---|
| Single Ui | Local object xi | What exists locally? |
| Double Uij | Equivalence φij | How are presentations compared? |
| Triple Uijk | Coherence cell | Do comparisons compose? |
| Refinement | Transported descent data | Is gluing cover-independent? |
4. Place isotropy in the local automorphism category
At a point or stratum, isotropy appears as automorphisms of the local object. Two neighborhoods can have equivalent local groupoids and the same abstract stabiliser groups while differing in how those automorphisms are identified across overlaps.
Conjugating an overlap functor changes its presentation. A globally meaningful comparison therefore tracks equivalence classes together with coherence, not a preferred generator chosen independently in each chart.
| Local record | What it sees | What it can miss |
|---|---|---|
| Stabiliser group | Automorphisms at one point | Transport between points |
| Orbit-type stratum | Conjugacy class pattern | Overlap coherence |
| Local groupoid | Objects and arrows in a chart | Global twisting |
| Descent stack | Charts, equivalences and coherence | Only what lies outside declared model |
5. Use exit paths to encode stratified transport
Exit paths move from a stratum into one lying in its closure in the permitted direction. Treumann’s exit-path 2-category records points, exit paths and homotopies between them. Under suitable stratification hypotheses, constructible stacks correspond to 2-functors from that 2-category into categories.
The correspondence packages local constancy, crossing functors and higher coherence in one representation. It does not remove hypotheses: the stratification and the class of constructible objects must be suitable for the exodromy statement being used.
- Objects: points or local states.
- 1-morphisms: exit paths.
- 2-morphisms: compatible homotopies.
- 2-functor: categorical transport.
- Hypotheses stated before applying equivalence.
6. Detect a global obstruction as failed coherence
Suppose every Ui carries the same one-object groupoid BG for a group G. Overlap equivalences can act by automorphisms of G. On triple overlaps, their composite may differ from the direct comparison by an inner automorphism represented by an element gij k of G.
If these elements can be removed by changing local trivialisations, the descent is equivalent to the untwisted one. If a coherent residual class remains, the same local isotropy has produced a different global stack. The obstruction lives in the gluing, not in any single chart.
| Observation | Possible conclusion | Not yet justified |
|---|---|---|
| All local categories equivalent | Same local type | Same global stack |
| All pair overlaps equivalent | Pairwise compatibility | Triple coherence |
| Cocycle removable by re-trivialisation | Equivalent descent presentation | Canonical trivialisation |
| Residual coherent twist | Nontrivial global gluing | Different underlying topological space |
7. Worked example: identical local C2 isotropy, two global gluings
Cover a circular stratified neighborhood by three opens U1, U2 and U3 arranged cyclically. Put the one-object groupoid BC2 on each open. In the first descent datum, every overlap functor is the identity and the triple coherence is trivial. A local object returns unchanged after one circuit.
In the second datum, use the same local categories and pairwise identity equivalences but choose a nontrivial coherent automorphism around the cycle in a setting where it cannot be removed by the allowed local re-trivialisations. Every point still sees C2 isotropy, yet global parallel transport remembers the twist.
A coarse orbit-type inventory declares the two constructions identical. The descent record separates them by the composite around the cover. This is the precise sense in which local isotropy fails to globalise by itself.
| Feature | Untwisted gluing | Twisted gluing |
|---|---|---|
| Local category on Ui | BC2 | BC2 |
| Pairwise equivalence | Identity presentation | Identity presentation |
| Cycle composite | Trivial | Nontrivial coherent automorphism |
| Coarse isotropy inventory | C2 everywhere | C2 everywhere |
| Global stack | Untwisted | Twisted |
8. Transfer check: refine the cover and change trivialisations
Split one open into two and pull the descent data to the refinement. A genuine global distinction should persist, although the cocycle is represented by more overlap maps. If the twist vanishes only because the refined calculation dropped a triple-overlap condition, the comparison is incomplete.
Next conjugate the local C2 presentations or rename their unique object. The global class should not change. This distinguishes a coordinate artifact from a descent obstruction.
| Changed context | Expected invariant | Failure exposed |
|---|---|---|
| Cover refinement | Equivalent global stack | Cover-dependent bookkeeping |
| Local re-trivialisation | Same descent class | Presentation dependence |
| Base point moved along stratum | Equivalent local transport | Broken constructibility |
| Stratification coarsened | Requires new theorem check | Overextended exodromy claim |
9. Change the stratification and watch the category change
Constructibility is relative to a stratification. Refining the stratification can permit new jumps; coarsening can require formerly distinct local behavior to become constant. The same underlying space can therefore support different constructible categories.
Haine, Porta and Teyssier study exodromic stratifications beyond the standard conical setting and prove stability properties under stated conditions. Their result is a reason to check hypotheses carefully, not permission to assume every coarsening preserves every constructible-stack model.
- Stratification recorded with the object.
- Local constancy checked on new strata.
- Exit-path category recomputed or compared.
- Coarsening theorem hypotheses verified.
- No automatic transfer from one model to another.
10. State the exodromy theorem boundary
Treumann’s result concerns constructible stacks on suitable topologically stratified spaces and an exit-path 2-category. Later work extends exit-path descriptions to constructible hypersheaves under specified niceness and chain conditions, and beyond conicality through exodromic stratified spaces.
A classroom calculation should therefore name the category of values, regularity of stratification, finiteness or local conditions and the exact equivalence invoked. Without them, an exit-path diagram is a useful model, not automatically a classification theorem.
| Claim level | Needed statement | Safe conclusion |
|---|---|---|
| Local example | Explicit cover and coherence | This descent datum glues |
| Exit-path model | Constructibility and stratification hypotheses | Transport encodes objects |
| Classification equivalence | Exact theorem assumptions | All objects recovered up to equivalence |
| Changed stratification | Comparison theorem | Transfer is justified |
11. Limits, misconceptions and practical evidence check
Identical stabiliser groups do not determine a global stack. Pairwise equivalences do not replace triple coherence. A nontrivial cocycle may depend on the coefficient 2-group and allowed equivalences, so the obstruction must be stated in the correct category.
Record the stratification, cover, local categories, restriction functors, overlap equivalences, coherence cells, refinement maps, re-trivialisations and theorem hypotheses. Check the cycle composite and reproduce it on a refinement.
- No global classification from local groups alone.
- Triple and higher coherence retained.
- Equivalence category explicit.
- Refinement and re-trivialisation tested.
- Theorem hypotheses cited precisely.
12. FAQs and the next reading route
Constructible stacks show how local categorical information becomes global only through descent. They turn “the same isotropy everywhere” from a conclusion into the beginning of a gluing calculation.
Read the Topology root, the orbit-type sheaves article and the exit-path categories article before this guide. The Gold Standard parent and How X Works hub connect the same discipline across subjects: retain the comparison maps, test changed contexts and state the theorem boundary.
Why is pairwise agreement not enough?
Because three comparison maps can compose around a triple overlap or cycle with a nontrivial automorphism. Coherence decides whether the local objects truly glue.
Does a constructible stack have to be constant?
It is locally constant along each chosen stratum, but it can change across strata and can carry nontrivial global gluing.
Can every constructible stack be read from exit paths?
Only under the hypotheses of an appropriate exodromy theorem. The space, stratification, value category and constructibility conditions must be checked.
Continue through the verified ecosystem
Sources and scope note
- Treumann — Exit Paths and Constructible Stacks — used for constructible stacks and the exit-path 2-category description.
- Lejay — Constructible Hypersheaves via Exit Paths — used for the scope and hypotheses of an extended exit-path representation for hypersheaves.
- Haine, Porta and Teyssier — Exodromy Beyond Conicality — used for exodromic stratified spaces, coarsenings and theorem-boundary cautions.
The three-open BC2 examples and tables are original teaching constructions. The sources support exit-path and exodromy results; the article does not claim a classification outside their stated hypotheses.