An invariant can be complete on one subcategory and incomplete on the surrounding world. That is not a weakness when the boundary is stated. Trouble begins when a theorem about finite, connective, nilpotent or otherwise controlled objects is retold as a universal classification claim.
This guide continues the work on transporting power operations through rectification zigzags. It asks what extra evidence is needed to reconstruct maps rather than only object labels, how injectivity, surjectivity and essential image differ, and how to test a classification theorem exactly where its hypotheses stop.
The Gold Standard is a typed completeness statement. Name the source category, target algebraic category, equivalence relation, mapping objects, hypotheses and essential image. Then a failed boundary test improves the theorem’s use instead of being treated as an embarrassment.
A practical orientation
Object invariants answer whether objects can be distinguished. Mapping-space reconstruction asks whether maps, homotopies and composition can also be recovered. A list that separates objects need not encode the full category.
The worked example considers a hypothetical invariant package that is complete for a declared finite connective subcategory. It succeeds on the objects and maps admitted by the theorem, while two objects outside the connectivity hypothesis share the recorded invariants. The correct conclusion is bounded completeness, not global failure or universal classification.
- Type the theorem — for readers stating source, target and equivalence.
- Reconstruct mapping spaces — for full-faithfulness and coherence questions.
- Follow the boundary example — for a concrete completeness audit.
- Test hypotheses and image — for classification-theorem use.
- Transfer the audit — for another prime, model or invariant.
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
Ask for mapping-space reconstruction
Work the example
Worked example: completeness inside a finite connective boundary
1. State invariant completeness as a typed claim
Let C0 be the declared subcategory of topological or homotopical objects and let I send it to an algebraic target A. A completeness claim must say whether I classifies objects up to homeomorphism, homotopy equivalence, weak equivalence or another relation. Those conclusions are different.
Record restrictions such as finiteness, connectivity, nilpotence, localisation, prime, coefficient system, model and chosen enhancement. They are part of the theorem’s type, not small print to be removed from a summary.
- Source category and admitted subcategory.
- Equivalences inverted in source and target.
- Object invariants and operations retained.
- Mapping objects and composition data claimed.
- Hypotheses and essential image.
- Choice or rectification dependence.
2. Separate detection, classification and reconstruction
A detecting invariant proves that a specified difference produces different values. A classifying invariant gives an injective assignment on equivalence classes in its domain. Reconstruction additionally describes which algebraic objects arise and how maps and higher homotopies are recovered.
Surjectivity onto all algebraic packages is usually stronger than necessary. Many theorems identify an essential image cut out by realizability, coherence or finiteness conditions. State that image instead of implying that every formal package comes from a space.
| Claim | What it establishes | What remains |
|---|---|---|
| Detection | A difference is visible | Other objects may still collide |
| Object classification | Objects are separated in C0 | Maps may not be reconstructed |
| Full faithfulness | Mapping spaces are recovered | Not every target object need be realised |
| Equivalence onto image | Objects and maps match essential image | Boundary hypotheses still matter |
3. Ask for mapping-space reconstruction
For X and Y in C0, compare the source mapping space MapC(X,Y) with the derived mapping object between I(X) and I(Y). A bijection on homotopy classes recovers only pi0. Full reconstruction should address higher homotopies and compatibility with composition.
Use derived mapping objects when replacements or resolutions are needed. An underived Hom can lose extensions or coherence. Specify the comparison map and the model in which it is computed rather than saying merely that maps correspond.
| Level | Evidence | Risk if omitted |
|---|---|---|
| Objects | Invariant values | No control of maps |
| Homotopy classes | pi0 of mapping spaces | Higher coherence lost |
| Higher homotopy | All relevant homotopy groups or equivalence | Composition still needs naturality |
| Enriched comparison | Natural mapping-space equivalence | Strongest categorical reconstruction |
4. Check naturality, composition and choice
A family of pairwise mapping equivalences is not yet a functorial reconstruction. Test identities, composition and homotopies between composites. If the construction uses cofibrant replacements, rectification functors or zigzags, record how different choices are compared.
A contractible space of choices can support canonical behaviour up to coherent homotopy. Mere existence of one comparison does not. When coherence is unavailable, restrict the claim to invariant-level consequences that the evidence actually controls.
5. Describe the essential image and realizability tests
List the algebraic relations forced by topology: grading, units, operations, instability or nilpotence conditions, compatibility identities and finiteness. A formal algebraic object outside these relations is not a counterexample to a theorem whose target was always the realizable image.
Separate necessary conditions from sufficient ones. Obstruction groups or higher coherence may prevent realization even when low-degree identities pass. Record whether the theorem proves realization, uniqueness, both or neither.
6. Worked example: completeness inside a finite connective boundary
Suppose I records graded groups, a selected family of power operations and the extension data required by a theorem for finite connective objects in C0. Within that subcategory, the theorem supplies an equivalence between source mapping spaces and derived algebraic mapping objects, and identifies the realizable essential image.
Two nonconnective objects X and Y outside C0 have the same recorded groups and selected operations, but differ in negative-degree extension data omitted by I. Their collision does not refute the theorem on C0. It shows why the connectivity hypothesis and mapping-space comparison must travel with the phrase ‘complete invariant’.
| Case | Hypotheses | Defensible conclusion |
|---|---|---|
| A and B in C0 | Finite, connective, realizable | I separates objects as theorem states |
| Map(A,B) | Derived comparison and coherence pass | Mapping space reconstructed |
| X and Y outside C0 | Connectivity fails | No classification conclusion |
| Formal target package | Realizability not checked | Not yet known to come from topology |
7. Use counterexamples to map the boundary
Vary one hypothesis at a time: remove connectivity, allow infinite type, change the prime, relax nilpotence or forget an operation. Seek pairs with equal recorded invariants and different known structure, or maps that agree algebraically but differ topologically. A boundary counterexample identifies what the hypothesis was doing.
Do not use a counterexample from a different equivalence relation without saying so. Two spaces may be homotopy equivalent but not homeomorphic; an invariant complete for homotopy type was never required to distinguish them as manifolds.
8. Test classification-theorem boundaries systematically
Create a hypothesis matrix. For each row, record theorem support, available comparison, known obstruction and admissible conclusion. Include the essential image and mapping-space level, not only object examples. This prevents a successful object calculation from being upgraded silently into categorical equivalence.
When several hypotheses change together, the test can locate failure but may not identify the responsible assumption. Follow with single-variable cases or state that attribution remains unresolved.
- Finiteness or compactness condition.
- Connectivity and truncation range.
- Nilpotence or completion assumption.
- Prime and coefficient system.
- Model, enhancement and rectification admissibility.
- Realizability and mapping-space comparison.
9. Transport completeness claims through rectification carefully
A Quillen equivalence or rectification theorem can transport homotopy theories under its hypotheses. It does not make every point-set operation literally identical, nor does it erase the need to compare the invariant package along the chosen zigzag.
Check that the source objects are admissible, replacements remain inside the declared subcategory, mapping objects are derived correctly and power operations used by I are preserved with the required coherence. Otherwise keep the classification statement in the model where it was proved.
10. State the limits of finite invariant packages
A finite list can be complete on a tightly controlled class because the class bounds the missing complexity. Outside that class, higher operations, extensions, phantom phenomena or infinite towers may distinguish objects that the list cannot see. Completeness is therefore relational: package, domain and equivalence belong in one sentence.
Computational failure is also not automatically mathematical incompleteness. An invariant may be complete in principle but impractical to calculate. Conversely, easy calculation does not prove that the package captures every relevant map.
11. Misconceptions, practical questions and transfer
Common mistakes include dropping the subcategory from the theorem name, treating pi0 of mapping spaces as full reconstruction, confusing target surjectivity with classification and using rectification as a claim of literal equality. The repair is typed statements and explicit comparison maps.
If an invariant separates every example tested, is it complete?
No. Tests provide evidence on the tested class. Completeness needs a theorem or an argument covering every object in the declared domain and the stated equivalence relation.
Does an object-classification theorem determine all maps?
Not necessarily. Mapping-space reconstruction is a stronger claim that must recover homotopy classes, higher homotopies and composition at the level asserted.
12. Changed-context evidence check and authoritative sources
Transfer the audit to another prime or model category. Recheck admissibility, realizability, operations and derived mapping objects instead of changing only the coefficient label. If the same theorem applies, say why each hypothesis transfers; if not, narrow the claim before calculating examples.
Pavlov and Scholbach give conditions for admissibility and rectification of coloured symmetric operads in Admissibility and rectification of colored symmetric operads. Balderrama develops algebraic theories and obstruction-theoretic tools for power operations in Algebraic theories of power operations. These works support the need for typed hypotheses, derived comparisons and obstruction data; any particular classification theorem must still be read in its own stated domain.
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: power-operation transport and invariant-completeness limits · Earlier guide: rectification-functor choice and power-operation obstructions · 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.