A complete invariant on a subcategory does not say that every formal algebraic package comes from a topological object. That second question is realizability. It asks whether the target data satisfies every relation, coherence condition and obstruction required to lie in the theorem’s essential image.
This guide continues the classification-boundary work. It separates object collisions from failed realization, follows obstruction towers stage by stage, distinguishes no realization from many realizations, and uses mapping-space evidence to diagnose whether reconstruction failed at existence, uniqueness or functoriality.
The Gold Standard is a precise failure statement. Name the stage, obstruction group, choice already made and conclusion supported. ‘The invariant failed’ is rarely enough.
A practical orientation
Think of reconstruction as building a tower. Low-degree data proposes a first stage; extension classes and higher coherence decide whether the next stage exists. Passing early checks does not guarantee the tower reaches the top.
The worked example uses a two-stage algebraic package. One candidate has a non-vanishing obstruction and no realization in the declared class. Another has vanishing existence obstruction but a nontrivial moduli space, so realization exists without uniqueness.
- Type the realization problem — for readers fixing source, target and hypotheses.
- Follow the obstruction tower — for staged existence and choices.
- Work the two-stage example — for a concrete diagnostic.
- Recognise the essential image — for classification-theorem use.
- Transfer the audit — for another model or coefficient system.
Chapter contents
Open a route that matches the decision in front of you, or read straight through. Each route returns to the same evidence chain.
Start with the operating question
Build the evidence chain
Build the obstruction tower one stage at a time
Work the example
1. Type the realizability problem before computing
Specify the source subcategory C0, algebraic target A, invariant functor I, equivalences, model, coefficients and object a in A. Ask whether there exists X in C0 with I(X) equivalent to a. This is different from asking whether I separates two known objects.
Record finiteness, connectivity, nilpotence, prime and structured-algebra hypotheses. A package may be realizable in a larger category but not inside the theorem’s declared source.
- Source subcategory and equivalence relation.
- Target package and required operations.
- Model, enhancement and coefficient system.
- Existence, uniqueness and mapping questions.
- Essential-image and obstruction hypotheses.
2. Separate existence, uniqueness and moduli
Existence asks whether at least one realization is present. Uniqueness asks whether all realizations are equivalent in the stated sense. The moduli space records realizations together with automorphisms and higher paths, so it can reveal several components or nontrivial self-equivalences.
A vanishing existence obstruction does not prove contractible moduli. Choices at one stage can survive, and automorphisms of the algebraic package can act on them.
| Outcome | Meaning | What not to claim |
|---|---|---|
| Obstruction nonzero | No lift at that stage | No global statement outside the setup |
| Obstruction zero | At least one next-stage lift may exist | Not uniqueness |
| Several components | Inequivalent realizations | Not failure of existence |
| Nontrivial loops | Automorphisms or coherent choices | Not object ambiguity alone |
3. Build the obstruction tower one stage at a time
Start with the truncated algebraic data admitted at the base. At each stage, identify the cohomology class whose vanishing permits an extension, then record the space or torsor of choices for that extension. Continue only after the previous stage and its compatibility are fixed.
Use derived cohomology and mapping objects appropriate to the model. Underived calculations can miss extensions or identify choices that are not homotopy invariant.
4. Track automorphisms and change of choices
Automorphisms of the lower-stage data can carry one lift to another. Quotienting too early may hide distinct realizations; never quotienting may count equivalent choices repeatedly. State the group or space acting and the equivalence used.
If rectification or replacement choices enter, compare their induced obstruction classes. A calculation dependent on an unrecorded model choice is not yet a portable essential-image test.
5. Recognise necessary relations before higher obstructions
Check grading, units, operations, instability, nilpotence, completion and compatibility identities forced by topology. Failure of a necessary relation excludes the package before an obstruction tower is needed.
Passing visible identities only admits the package to deeper tests. Higher cohomology and coherence can still prevent realization. Essential-image recognition is therefore layered, not a single checklist.
6. Worked example: two-stage realization and non-uniqueness
Consider a two-stage package with groups concentrated in degrees n and n+1 plus the required primary operations. Candidate A satisfies the primary identities, but its extension data produces a nonzero obstruction in the relevant Quillen-cohomological stage. Candidate A is not realized in the declared connected finite-type class.
Candidate B has vanishing existence obstruction. Two allowed extension choices remain in different components of the moduli calculation. The correct conclusion is that B is realizable but not uniquely reconstructed by the low-degree package alone. Adding the extension class distinguishes the components.
| Candidate | Existence obstruction | Choice space | Conclusion |
|---|---|---|---|
| A | Nonzero | No next-stage lift | Outside essential image in C0 |
| B without extension class | Zero | Two components | Realizable, not unique |
| B with extension class | Zero | Chosen component | Reconstruction narrowed |
| Changed model | Not yet compared | Unknown | No transport claim yet |
7. Diagnose where reconstruction failed
If no object is produced, ask whether a necessary identity failed, an obstruction was nonzero, a convergence hypothesis failed or the computational model was wrong. If several objects appear, ask which invariant or extension data was omitted. If maps fail, inspect derived mapping-space comparison and composition.
Do not turn a failed computation into a theorem of nonrealizability. Numerical or symbolic inability to evaluate an obstruction is an epistemic limit, not a nonzero class.
8. Describe the essential image constructively
State the target objects that satisfy necessary relations, admit all obstruction lifts and meet the source restrictions. When possible, include a procedure: verify base identities, compute successive classes, record choices and compare automorphisms.
A theorem can identify the image up to an obstruction tower rather than a closed formula. That is still useful if each stage and its conclusion are explicit.
9. Reconnect realization with mapping-space reconstruction
Once objects exist, compare source mapping spaces with derived algebraic mapping objects. Object-level essential-image recognition does not automatically provide full faithfulness. Maps can carry additional obstructions and choices.
Test identities, composition and higher homotopies. If only pi0 is reconstructed, say that homotopy classes of maps correspond; do not advertise an equivalence of enriched or infinity-categories.
10. Know the limits of obstruction calculations
Obstruction towers may depend on connectivity, completeness or convergence. Infinite objects can introduce lim-one or phantom phenomena absent from finite stages. A vanishing finite prefix does not prove a lift through an unbounded tower without convergence.
Groups can also be difficult to calculate. State unresolved stages and avoid replacing unknown with zero. A practical classifier may be partial even when the theory is complete.
11. Misconceptions, practical questions and transfer
Common mistakes include treating formal target data as automatically realizable, equating vanishing with uniqueness, ignoring automorphism actions and calling computational failure an obstruction. The repair is a staged moduli statement.
Does a nonzero obstruction prove no topological object has the same low-degree groups?
It proves no realization satisfying the full declared package and hypotheses lifts through that stage. A weaker package or larger source category may differ.
If every obstruction vanishes, is the realization unique?
Not necessarily. The spaces of choices and automorphisms can leave several components or nontrivial higher structure.
12. Changed-context evidence check and authoritative sources
Transfer the audit to another prime or structured category. Recompute the target operations, cohomology theory, convergence assumptions and automorphism action. Do not move a zero obstruction across models without a comparison theorem.
Pstrągowski and VanKoughnett develop realization obstructions in Abstract Goerss-Hopkins theory. Frankland computes moduli spaces of realizations for two-stage Pi-algebras in Moduli spaces of 2-stage Postnikov systems. These primary papers support the distinction among existence obstructions, choices and moduli; each application must preserve its stated hypotheses.
Continue through the Gold Standard ecosystem
Return to the Topology subject root, the Gold Standard of Education parent, or the How X Works hub.
Immediate predecessor: invariant-complete subcategories and mapping-space reconstruction · Earlier guide: power-operation transport and invariant-completeness limits · Topology root
These routes are grouped by learning purpose: prerequisite model, immediate predecessor, and the wider transfer framework. They are not substitutes for the evidence and limits stated in this guide.