A quotient germ records the local shape seen after symmetry has been collapsed. The preceding article showed that this coarse object can forget which group acted and how. The next useful question is what happens when the quotient is decorated with isotropy and compared through groupoid presentations.
This article starts with finite-group actions and action groupoids. It distinguishes the isotropy group at an orbit, its normal representation and the way these data vary across strata. It then asks which features survive Morita equivalence, the standard notion that different groupoids can present the same orbispace.
The Gold Standard is categorical restraint. State the object being compared, compute a concrete example, transport the invariant through an equivalence, change the presentation and identify what is still missing before making a reconstruction claim.
A practical orientation
Use this guide after learning why a homeomorphism of quotient germs need not recover an action. It adds structured local data while keeping clear that a presentation is not the intrinsic object.
Choose a reading route
Build the decorated germ
Chapters 1–3 define quotient germs, isotropy groups and normal representations.
Compare presentations
Chapters 4–6 introduce action groupoids, essential equivalence and Morita-invariant structure.
Work and transfer examples
Chapters 7–9 compare coarse-equal actions and changed presentations.
State the reconstruction boundary
Chapters 10–11 give invariants, limits and evidence checks.
Chapter contents
- 1. Start with the orbit, not a chosen coordinate picture
- 2. Attach isotropy groups along the quotient strata
- 3. Add the normal isotropy representation
- 4. Package the local action as a groupoid presentation
- 5. Read Morita equivalence as sameness of the presented orbispace
- 6. Choose invariants that descend through presentation change
- 7. Worked comparison: half-lines with different hidden symmetry
- 8. Transfer check: change the presentation without changing the orbispace
- 9. Test effectivity and invisible isotropy
- 10. Use a ladder of reconstruction claims
- 11. Limits, misconceptions and evidence checklist
- 12. FAQs and the next reading route
1. Start with the orbit, not a chosen coordinate picture
Let a compact group act properly on a space or manifold and let x be a point. The quotient germ at the orbit of x is the local equivalence class of neighborhoods in the orbit space. It is coarse: every point on one orbit becomes one point.
A drawing of a cone, half-line or sector can reveal local topology but not necessarily the stabiliser group or its action on normal directions. Record the chosen category—topological, differentiable stratified or groupoid—before using the word equivalent.
The decorated germ adds orbit-type and isotropy information to the quotient neighborhood. Which decoration is sufficient depends on the theorem one hopes to use.
- Orbit-space germ identified.
- Comparison category named.
- Strata and isotropy labels recorded.
- Normal action separated from coarse shape.
2. Attach isotropy groups along the quotient strata
The isotropy group at x consists of elements that fix x. Points in the same orbit have conjugate isotropy groups. Across a proper action, orbit-type strata collect points with the same conjugacy class of stabiliser.
A decoration may assign an isomorphism class of isotropy group to each stratum and compatible specialisation information near its boundary. Labels alone are stronger than the quotient topology but weaker than a full groupoid.
For a trivial action of C2 and a trivial action of C3 on the real line, both quotient germs are an ordinary line. Isotropy labels immediately distinguish them.
| Action | Coarse quotient germ | Isotropy at every point |
|---|---|---|
| Trivial C2 on R | Line | C2 |
| Trivial C3 on R | Line | C3 |
| Trivial group on R | Line | Trivial |
3. Add the normal isotropy representation
At an orbit, tangent directions split into directions along the orbit and normal directions under suitable proper-action hypotheses. The stabiliser acts linearly on the normal space. Its quotient is the local slice model seen by the orbit space.
The group label alone does not determine this representation. C2 can act trivially on a normal line or by reflection. Both have isotropy C2 at the chosen fixed point, but the nearby quotient and orbit types differ.
Record the representation up to equivariant linear isomorphism, not as one arbitrary matrix basis. Basis choices are presentation details.
| C2 normal action | Local quotient | Nearby isotropy |
|---|---|---|
| Trivial on R | Line | C2 everywhere |
| Sign reflection on R | Half-line | Trivial away from origin |
4. Package the local action as a groupoid presentation
An action groupoid has objects given by points and arrows given by group elements carrying one point to another. Orbits, stabilisers and their variation are encoded in one structure rather than appended as unrelated labels.
Different atlases or action descriptions can present the same orbispace. Object spaces and arrow counts may change, so they are not intrinsic invariants.
A useful comparison therefore asks for an equivalence of presentations, commonly expressed through essential equivalences, principal bibundles or Morita equivalence in an appropriate category.
- Objects and arrows specified.
- Orbits recover quotient points.
- Automorphisms recover isotropy.
- Presentation data distinguished from intrinsic data.
5. Read Morita equivalence as sameness of the presented orbispace
Morita-equivalent groupoids can have different object manifolds while describing the same stack-like quotient. The equivalence matches orbit spaces and transports isotropy and local transverse structure in the appropriate sense.
This is stronger than a homeomorphism of coarse quotients. It retains automorphism information that a bare quotient erases. It is also weaker than literal isomorphism of groupoids, which would treat atlas choices as part of the object.
Always state hypotheses. Properness, effectivity, étaleness or other conditions determine which reconstruction and invariance results are available.
| Comparison | Preserves coarse quotient | Preserves presentation literally |
|---|---|---|
| Homeomorphism of quotient | Yes | No |
| Groupoid isomorphism | Yes | Yes |
| Morita equivalence | Yes, suitably | No; preserves presented object |
6. Choose invariants that descend through presentation change
Candidate intrinsic data include the differentiable structure on the orbit space, its canonical stratification, isotropy groups up to transport, normal isotropy representations and sheaf-like constructions defined on the presented orbispace.
A coordinate chart, a selected slice, an object-space dimension in a non-reduced presentation or the number of arrows written in one atlas is not automatically intrinsic. Demonstrate invariance under the chosen equivalence before using it to classify.
The invariant should also be computable. State how data are extracted from a presentation and how an equivalence carries them to another.
| Datum | Likely status | Required check |
|---|---|---|
| Coarse orbit space | Intrinsic but incomplete | Induced quotient map |
| Canonical strata | Morita invariant under hypotheses | Stratum transport |
| Isotropy class | Intrinsic at represented point | Automorphism transport |
| Chosen slice coordinates | Presentation choice | Equivariant change of slice |
7. Worked comparison: half-lines with different hidden symmetry
C2 reflecting the real line has quotient [0,∞). SO2 rotating the plane also has quotient [0,∞), using radius. As topological quotient germs at the origin, both are half-lines.
The isotropy groups at the origins are C2 and SO2. Their normal representations have dimensions one and two. Away from the origin, reflection has trivial stabiliser and rotation also has trivial stabiliser, so the coarse two-stratum pattern still agrees.
The decorated germs distinguish the actions immediately. This does not say every decorated quotient reconstructs every action; it shows exactly which information repaired this coarse collision.
| Presentation | Origin isotropy | Normal representation | Coarse germ |
|---|---|---|---|
| C2 on R by reflection | C2 | One-dimensional sign | Half-line |
| SO2 on R2 by rotation | SO2 | Two-dimensional standard | Half-line |
8. Transfer check: change the presentation without changing the orbispace
Replace a global action chart by overlapping local slices and their transition arrows. The object space and coordinate descriptions change, but orbit points, transported isotropy and canonical strata should agree.
Next refine the atlas by duplicating a chart and adding the necessary transition arrows. Any proposed invariant that doubles merely because the chart was duplicated is presentation-dependent.
Finally restrict to an equivalent full transversal where the relevant hypotheses permit it. The computation should move through the equivalence rather than rely on a preferred global group action.
| Presentation change | Should remain | Artefact to reject |
|---|---|---|
| Overlapping slices | Orbispace germ and isotropy transport | Chosen coordinates |
| Duplicated chart | Intrinsic invariants | Raw object or arrow count |
| Equivalent transversal | Transverse structure | Ambient presentation size |
9. Test effectivity and invisible isotropy
A group element can fix every point in a neighborhood. Such ineffective isotropy is invisible to the coarse quotient and to the effective normal action, yet it remains part of the full groupoid or stack.
Compare a trivial C2 action on a line with the trivial group action. Their effective quotient behavior is identical, but the full isotropy differs. A reconstruction theorem for effective orbifolds cannot be applied unchanged to non-effective presentations.
State whether the intended invariant remembers the ineffective kernel. Passing to an effective reduction is a deliberate loss of information, not a harmless simplification.
- Effective and full isotropy separated.
- Kernel of the normal action recorded.
- Reduction step declared.
- Theorem hypotheses checked before reconstruction.
10. Use a ladder of reconstruction claims
The weakest claim is that two presentations have homeomorphic quotient germs. A stronger claim matches stratified differentiable quotient structure. Stronger again is equivalence of isotropy-decorated normal models, and a categorical claim identifies the presented orbispaces up to Morita equivalence.
Do not jump upward without a theorem or constructed equivalence. Matching isotropy groups pointwise may fail to encode how they glue, and matching normal representations may still omit global holonomy or cocycle data.
For local classification, name the neighborhood, morphisms and compatibility required. For global reconstruction, add the descent and gluing evidence that local germs cannot supply.
| Claim level | Evidence | Typical limit |
|---|---|---|
| Coarse germ | Homeomorphism | Forgets isotropy |
| Decorated germ | Strata, isotropy, normal action | May omit gluing |
| Morita class | Equivalence or bibundle | Hypothesis-dependent |
| Literal action | Group and equivariant map | Too presentation-specific |
11. Limits, misconceptions and evidence checklist
Morita equivalence does not mean two groupoids are literally isomorphic. A quotient homeomorphism does not imply Morita equivalence. Isomorphic stabiliser groups do not by themselves determine normal representations or gluing.
Proper-action slice models are local and depend on hypotheses. Infinite-dimensional, non-proper or singular settings may require different tools. The examples here illustrate boundaries; they do not establish a universal reconstruction theorem.
For a comparison, preserve the presentations, quotient maps, strata, isotropy homomorphisms, normal representations, kernels, local equivalences, compatibility diagrams and the precise theorem used. Verify the transported data after changing the atlas.
- Category and hypotheses named.
- Effective kernel recorded.
- Local and global claims separated.
- Transport maps checked.
- Presentation-change test completed.
12. FAQs and the next reading route
Isotropy-decorated quotient germs repair some information lost by coarse quotienting. Groupoids then organise those data and Morita equivalence separates the intrinsic orbispace from one chosen presentation.
Read the Topology root, the quotient-germ equivalence article and this guide in order. The route moves from a reconstruction failure to a structured replacement and a clearer classification boundary.
The parent Gold Standard and How X Works hub connect the idea to a general habit: enrich an abstraction only with information whose role, transport and limits can be explained.
Does the isotropy group determine the quotient germ?
No. The normal representation and its variation also matter, and global gluing can add further information.
Are Morita-equivalent groupoids isomorphic?
Not necessarily. They may have different object spaces or atlases while presenting the same orbispace.
Why record ineffective isotropy?
It can be invisible in the coarse quotient and effective normal action but remains part of the full automorphism structure.
Continue through the verified ecosystem
Sources and scope note
- Orbispaces as differentiable stratified spaces — used for canonical stratification, local quotient structure and Morita-invariant differentiable structure of proper Lie groupoid orbit spaces.
- Geometry of orbit spaces of proper Lie groupoids — used for proper Lie groupoids, orbit-space geometry and the role of isotropy and normal data.
The finite-dimensional action examples and comparison tables are original teaching examples. The sources support the stated groupoid and orbispace framework; this article makes no universal reconstruction claim outside the named hypotheses.