An objectwise replacement can give every space a convenient model and still fail to define a usable construction on maps. The missing ingredient is functoriality: a controlled way to replace objects and morphisms together so identities and compositions survive at the level being claimed.
This guide continues naturality rectification and homotopy-coherent functors. It explains how functorial cofibrant or fibrant replacements support derived constructions, how natural transformations descend under stated hypotheses, and how to detect when pairwise homotopies do not extend to coherent higher data.
The subject is framework-dependent. Every statement below is tied to a chosen category, weak equivalences and replacement system; there is no category-free machine that turns arbitrary choices into a natural theorem.
A practical orientation
Begin by naming the category and the form of naturality required. A strict replacement functor, a functor on the homotopy category and a homotopy-coherent diagram are related but not interchangeable.
The worked example compares arbitrary CW approximations with the natural map from the geometric realisation of the singular complex.
- Choose the homotopical framework — for fixing the claim.
- Construct derived naturality — for maps and transformations.
- Follow the CW example — for objectwise versus functorial choice.
- Test coherence obstructions — for higher compatibility.
- Transfer to chain complexes — for a changed category.
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 a derived functor from admissible replacements
Work the example
1. Choose the category, weak equivalences and target claim
A replacement system makes sense only after choosing a category and the morphisms that count as weak equivalences. Standard model-category language adds cofibrations, fibrations and functorial factorisations, but the exact structures and hypotheses must be stated.
Also state the target: strict maps in the underlying category, a functor between homotopy categories, or a coherent functor between infinity-categorical localisations. Proving one does not automatically prove the others.
- Underlying category and admissible morphisms.
- Weak equivalences to be inverted.
- Cofibrant or fibrant replacement required.
- Target category or localisation.
- Strict, derived, coherent or invariant-only conclusion.
2. Understand what functorial replacement adds
An objectwise cofibrant replacement chooses QX with a weak equivalence QX → X for each X. A functorial replacement also assigns Qf: QX → QY to every f: X → Y and makes the replacement maps natural. Similar language applies to fibrant replacement X → RX.
Functoriality removes one source of arbitrary comparison maps. It does not remove every higher-choice problem: later constructions, point-set models or transformations may introduce new coherence requirements.
| Construction | Provides | Does not automatically provide |
|---|---|---|
| Objectwise QX | Convenient model for each object | Compatible maps or composition |
| Functorial Q | Objects, maps and natural weak equivalence | All later derived coherence |
| Derived functor | Functor on a homotopy category under hypotheses | Preferred point-set representative |
| Coherent enhancement | Mapping-space level compatibility | Strict equality in the underlying category |
3. Build a derived functor from admissible replacements
For a left Quillen functor, a standard total left derived functor is represented by applying the functor after cofibrant replacement. Dually, a right Quillen functor uses fibrant replacement. The Quillen hypotheses ensure weak equivalences between the relevant admissible objects are handled correctly.
Write the construction and its comparison map explicitly. If the application needs only the induced homotopy-category functor, avoid pretending that a particular point-set composite is canonical beyond the stated replacement choices.
4. Derive natural transformations with the same care
A natural transformation between suitable functors can induce a derived natural transformation, but its definition and invertibility depend on how it behaves on cofibrant or fibrant objects and on the replacement choices. Check the theorem used rather than moving the word “natural” through the derivation by notation alone.
If two point-set transformations agree only after localisation, state derived naturality. If the application needs a coherent transformation on mapping spaces, supply or cite the stronger framework.
| Question | Evidence | If missing |
|---|---|---|
| Is the transformation defined before localisation? | Objectwise maps and natural square | No point-set naturality claim |
| Does it respect the admissible objects? | Behaviour on cofibrant/fibrant inputs | Derived map may be undefined or choice-dependent |
| Is it an isomorphism after localisation? | Weak-equivalence theorem on replacements | Do not call it a derived equivalence |
| Are higher compatibilities needed? | Mapping-space or infinity-categorical data | Restrict to homotopy-category naturality |
5. Use mapping spaces to locate coherence demands
Pairwise homotopies are paths in mapping spaces. Compatibility among those paths asks for higher simplices or fillers. A loop built from two competing compositions may need to be null-homotopic before a chosen 2-dimensional coherence can extend.
The precise obstruction depends on the diagram shape and framework. The practical audit is stable: list the composites that should agree, the homotopies supplied, and the next boundary that those homotopies form.
6. Worked example: arbitrary and functorial CW approximations
Suppose each space X is assigned an arbitrary CW approximation qX: CX → X. For a map f: X → Y, a lift CX → CY may exist up to homotopy, but an independently chosen lift for g: Y → Z need not compose to the chosen lift for gf. The objectwise approximations support invariants, yet they do not by themselves define a strict functor C.
In a standard convenient category of spaces, the construction QX = |Sing X| gives a functorial CW-type replacement with a natural evaluation map QX → X and a weak homotopy equivalence under the usual model structure. A map f induces |Sing f|, so identities and compositions are inherited from the functors. This solves the map-choice problem for this replacement, not every later coherence problem.
| Step | Arbitrary CW choices | Functorial singular-realisation choice |
|---|---|---|
| Object | CX with weak equivalence to X | |Sing X| with natural evaluation |
| Map f | Choose a lift up to homotopy | Use |Sing f| |
| Composition | Chosen lift of gf may differ | Functor composition holds |
| Invariant use | Often sufficient for homotopy invariant | Supports derived functorial constructions |
| Limit | Comparison choices remain | Framework assumptions still required |
7. Test whether the replacement solves the actual problem
After introducing Q or R, repeat the naturality audit for the full construction. A filtration, truncation, localisation or chosen representative applied afterward may break strict functoriality again.
Use universal properties where possible. A construction characterised in the homotopy category or an infinity-category may have cleaner naturality than one defined through many stagewise choices.
8. Run coherence-obstruction tests dimension by dimension
For a composable pair, compare the assigned map for gf with the composite of assigned maps. For triples, compare the two ways of combining the composition homotopies. Continue only as far as the claimed structure requires.
A failed filler does not always invalidate the invariant theorem. It may show that the construction cannot carry the stronger coherent structure with the chosen data. Record the obstruction and use a strict replacement, a different enhancement or an invariant-only fallback.
- Dimension 0: objects exist in the declared category.
- Dimension 1: every admissible map receives a map.
- Dimension 2: composition comparisons have homotopies.
- Dimension 3: competing homotopy composites admit compatible fillers.
- Higher dimensions: the selected enhancement supplies the required coherences.
9. Choose the honest fallback when coherence is unavailable
If a functor on the homotopy category is proven, state that and stop there. If only an invariant property survives all choices, state the property without inventing maps. If naturality holds on a restricted subcategory, declare the restriction.
These are mathematically useful results. The right strength is the strongest conclusion whose objects, maps and compatibilities are actually controlled.
10. Respect replacement and size limitations
Functorial factorisations may require model-category hypotheses, cofibrant generation or other smallness conditions, depending on the construction. A convenient category of spaces is not identical to the category of all topological spaces with every point-set pathology.
Derived constructions can also forget point-set features. Equality after localisation does not mean the original maps are equal, and a weak equivalence need not be a homeomorphism.
11. Misconceptions and practical questions
Common mistakes include calling objectwise choices functorial, assuming a functorial replacement makes every later choice natural, treating equality in a homotopy category as strict equality and naming a coherence obstruction without specifying the diagram or mapping space.
Is every CW approximation automatically functorial?
No. Objectwise approximations can compute homotopy invariants, but compatible map choices and composition require additional structure. Use a known functorial construction or state the weaker invariant result.
Does a functor on the homotopy category give a homotopy-coherent functor?
Not by itself. The homotopy category records morphism classes and forgets higher mapping-space information needed for coherence.
12. Changed-context transfer, evidence check and sources
Transfer the audit to chain complexes. Choose projective or injective replacements under the relevant hypotheses, construct the derived functor, check natural transformations on admissible replacements and state whether the conclusion lives in the derived category or a stronger enhancement.
Mark Hovey’s Model Categories develops model structures, functorial factorisations and derived functors. Jacob Lurie’s Higher Topos Theory develops infinity-categorical tools for coherent diagrams and localisation. The exact replacement and coherence claims in any application still require their 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: naturality rectification and coherent fallbacks · Earlier guide: coherent comparison systems and naturality 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.