GOLD STANDARD · TOPOLOGY · HOW X WORKS
Passing every finite obstruction stage does not by itself construct one compatible object at the infinite limit
Obstruction towers solve a problem stage by stage. Every finite truncation may admit a lift or compatible class, yet the choices can fail to assemble into one infinite-stage object. The inverse limit records compatible stage data, while derived-limit terms record a possible defect in commuting homotopy or cohomology with that limit. Convergence therefore needs explicit hypotheses, not only vanishing finite obstructions.
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Build the tower correctly
Separate finite and infinite claims
Control derived-limit defects
Transfer and qualify
Use FAQs, sources and connected routes
CHAPTER 1 OF 12 · BUILD THE TOWER
1. A stage solution is not yet a compatible sequence
Let X0 ← X1 ← X2 ← ··· be a tower of approximations. A solution at stage n is useful only when it maps to the chosen solution at stage n−1. Existence at every stage does not choose a compatible family automatically.
Record the transition maps, basepoints, equivalence relation and whether limits are ordinary, homotopy or derived. These choices determine which convergence statement is meaningful.
CHAPTER 2 OF 12 · BUILD THE TOWER
2. Distinguish inverse limits from homotopy limits
An ordinary inverse limit enforces strict compatibility. A homotopy limit retains coherent paths or higher data witnessing compatibility. Replacing one by the other requires a theorem, often after arranging a tower of fibrations or an equivalent model.
| Object | What it records | Typical risk |
|---|---|---|
| Stage Xn | Finite approximation | May hide later obstruction |
| Compatible sequence | One point in lim Xn | May not capture homotopy-correct limit |
| Homotopy limit | Stages plus coherent compatibility | Needs correct tower model |
| lim πq(Xn) | Compatible stage homotopy classes | May miss a derived-limit kernel |
| lim¹ πq+1(Xn) | Choice or convergence defect | Can survive when every finite stage works |
| Realised object | One global construction | Requires comparison theorem |
Do not use the word limit without specifying the category and comparison map under discussion.
CHAPTER 3 OF 12 · SEPARATE THE CLAIMS
3. Finite obstruction vanishing proves only finite extension
At each stage an obstruction may vanish, allowing a lift to the next approximation. The resulting lift can depend on choices. Later compatibility may require revising earlier representatives, and an infinite revision process need not stabilise.
State the exact finite conclusion: every n-stage problem is solvable, or one coherent chain has been built through stage n. The second is stronger.
CHAPTER 4 OF 12 · SEPARATE THE CLAIMS
4. Build compatibility inductively, not retrospectively
When transition maps have suitable lifting or surjectivity properties, choose the next stage over the already selected previous stage. Without them, independently selected stage solutions can form a nonempty set at every level while the inverse limit is empty.
Track images of later solution sets inside earlier ones. Stabilising images are the practical signal behind Mittag–Leffler-type conditions.
CHAPTER 5 OF 12 · WORK THE TOWER
5. Work a tower with finite solutions but unstable choices
Suppose stage n admits a torsor Sn of lifts. Each Sn is nonempty, but the image of Sn+1 inside S1 shrinks as n grows and never stabilises. A reviewer independently picks a lift at each stage, yet those picks do not form a compatible sequence.
| Stage check | Observed fact | What it proves |
|---|---|---|
| Each Sn nonempty | Every finite problem has a solution | Finite existence only |
| Transition Sn+1→Sn not onto | Some earlier choices cannot continue | Choice matters |
| Images in S1 keep shrinking | No stabilisation shown | Mittag–Leffler condition unproved |
| Independent picks | One lift chosen per stage | Not a compatible family |
| lim Sn | May be empty or smaller | Infinite realisation unresolved |
| Derived-limit term | May record residual choice defect | Needs calculation or vanishing theorem |
The repair is not to declare a tertiary obstruction by name. It is to model the tower, construct compatible choices and test the hypothesis that removes the derived-limit defect.
CHAPTER 6 OF 12 · CONTROL THE DEFECT
6. Use stabilisation or surjectivity where justified
For inverse systems of abelian groups, a Mittag–Leffler condition controls image stabilisation and often forces the first derived limit to vanish under the relevant hypotheses. Surjective transition maps are a common sufficient pattern, not a universal necessity.
Prove the property for the actual groups or solution sets. Similar-looking dimensions or ranks do not establish stabilisation of images.
CHAPTER 7 OF 12 · CONTROL THE DEFECT
7. Read the Milnor-type comparison as a warning label
For suitable towers of fibrations, homotopy groups of the homotopy limit fit between a lim¹ term from one degree higher and the inverse limit of stage homotopy groups. The derived term measures information that stagewise homotopy groups alone do not see.
Check the theorem’s hypotheses and degree conventions. In low degrees the structures may be pointed sets rather than abelian groups, so informal subtraction can be invalid.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Changed-context transfer: Postnikov towers and descent
A Postnikov or Moore–Postnikov tower approximates a space or lifting problem by stages. A descent totalisation approximates coherent local data through increasing cosimplicial depth. Both raise compatibility and convergence questions.
Transfer the finite-versus-infinite checklist, not a claim that the same lim¹ group or convergence theorem applies in every category.
CHAPTER 9 OF 12 · QUALIFY THE RESULT
9. Pair convergence claims with a comparison map
Name the global object, the tower, the map from the global object to the limit or homotopy limit, and the theorem making it an equivalence or identifying its fibre. Without that map, “the tower converges” is only a slogan.
Test whether completion, nilpotence, connectivity, fibrancy or boundedness hypotheses are required. State relative convergence when the global comparison is not established.
CHAPTER 10 OF 12 · QUALIFY THE RESULT
10. Limits and common misconceptions
Vanishing of every listed finite obstruction does not automatically remove an infinite compatibility defect. A zero lim¹ under one tower model does not prove a different tower computes the same object. Stabilising ranks need not stabilise subgroups.
Do not call a compatible family unique when its limit is a torsor. Do not confuse ordinary limits with homotopy limits. Do not infer convergence merely because a spectral sequence has several empty-looking diagonals.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions readers often ask
Can every finite stage be solvable while the infinite problem fails?
Yes. Nonempty solution sets at each stage need not contain one sequence compatible with every transition map, and a derived-limit term can record the remaining defect in suitable algebraic or homotopical settings.
What is the practical role of a Mittag–Leffler condition?
It provides an image-stabilisation criterion that can eliminate a first derived-limit obstruction under the stated algebraic hypotheses. It must be proved for the actual inverse system.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Evidence, connected routes and the next check
The Stacks Project on inverse systems defines the Mittag–Leffler condition and explains why ordinary inverse limits need not preserve exactness. The Stacks Project on derived limits displays the short exact sequence in which an R¹lim term measures a limit defect. A primary research treatment of towers of spaces states the Milnor exact sequence for homotopy limits and cites the classical Bousfield–Kan theorem.
Continue through completeness counterexamples and secondary obstructions, Whitehead towers and obstruction layers, Moore–Postnikov lifting problems, and descent spectral sequences and totalisation. The next check is to construct one compatible sequence rather than one unrelated solution at each finite stage.