Two model or infinity-categorical enhancements can present the same homotopy theory while using different objects, fibrant-cofibrant replacements and mapping constructions. Agreement of homotopy categories is an important checkpoint, but it does not by itself recover mapping spaces, higher composition or monoidal coherence.
This guide continues replacement-choice comparison. It explains what an enhancement-comparison functor must preserve, how mapping-space equivalence strengthens objectwise derived equivalence and where tensor or higher-coherence claims need additional structure.
The goal is to prevent both extremes: demanding strict point-set equality when equivalence is enough, and claiming coherent monoidal sameness from evidence that only lives in the homotopy category.
A practical orientation
Separate essential surjectivity on derived objects, equivalence on mapping spaces and compatibility with extra structure. These are related but not interchangeable.
The worked example compares two presentations of a homotopy theory through derived mapping spaces and then tests a tensor product claim.
- State the enhancements — for hypotheses and scope.
- Test mapping spaces — for stronger equivalence.
- Follow the worked example — for a concrete comparison.
- Audit monoidal coherence — for tensor claims.
- Mark the limits — for honest conclusions.
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
Work the example
Stress-test the decision
Audit monoidal comparison data
1. State what each enhancement presents
Name the model category, simplicial category, dg category or infinity-category, its weak equivalences and the localisation or underlying homotopy theory intended for comparison.
State whether the desired conclusion concerns homotopy categories, mapping spaces, limits and colimits, or a monoidal structure. The evidence threshold depends on the claim.
- Objects and weak equivalences.
- Fibrant, cofibrant or derived mapping construction.
- Comparison functor and domain of definition.
- Underlying localisation.
- Extra structures to be preserved.
2. Check essential surjectivity on derived objects
Every object of the target homotopy theory should be equivalent to the image of an object from the source, within the declared scope. This tests whether the comparison misses whole derived object types.
Essential surjectivity is weaker than an equivalence. It says objects are represented, not that all higher maps between them are preserved.
| Level | Test | What remains unknown |
|---|---|---|
| Objects | Essential surjectivity | Mapping information |
| Homotopy classes | Full faithfulness on Ho | Higher homotopies |
| Mapping spaces | Equivalence for every pair | Extra structure |
| Monoidal enhancement | Coherent tensor compatibility | Other chosen structures |
3. Compare derived mapping spaces, not only Hom sets
For suitable replacements X and Y, compare the derived mapping space Map(X,Y) with the target mapping space between their images. Require a weak equivalence natural in both variables for a fully faithful infinity-categorical comparison.
The set of connected components recovers morphisms in the homotopy category, but higher homotopy groups encode homotopies and their coherences. Equal Hom sets can hide different mapping spaces.
4. Track composition and units
Mapping-space equivalences must be compatible with composition and identity maps at the relevant coherent level. Pairwise equivalences chosen independently do not automatically assemble into a functor.
Use a comparison construction whose enrichment or localisation supplies composition, or record the required higher homotopies explicitly.
| Check | Evidence | Failure if omitted |
|---|---|---|
| Objects | FX and equivalence classes | Missing targets |
| Maps | Map-space equivalence | Lost higher morphisms |
| Composition | Coherent comparison of composites | Pairwise data does not assemble |
| Units | Identity compatibility | Not a functorial equivalence |
5. Separate presentation choices from invariant conclusions
Fibrant-cofibrant replacements, hammock localisations, simplicial mapping objects or nerves may look different while presenting equivalent mapping spaces. State the comparison route rather than treating notation as identity.
Use the weakest sufficient invariant conclusion. A derived calculation may need only an equivalence after localisation, while structured transport needs the enhancement.
6. Worked example: two presentations and a tensor claim
Suppose model categories M and N present the same underlying homotopy theory through a Quillen-style comparison. Derived objects correspond, and for cofibrant-fibrant replacements the comparison induces weakly equivalent mapping spaces. This supports equivalence of the associated infinity-categorical homotopy theories under the stated hypotheses.
Now suppose both categories carry tensor products. Agreement of objects and mapping spaces does not yet prove the derived tensors correspond. One also needs comparison maps F(X⊗Y)→FX⊗FY, unit compatibility and coherent associativity and symmetry data.
| Claim | Available evidence | Conclusion |
|---|---|---|
| Same derived objects | Essential surjectivity | Supported |
| Same higher mapping theory | Natural mapping-space equivalences | Supported if composition compatible |
| Same tensor on objects | Objectwise tensor comparison | Partial |
| Monoidal equivalence | Unit, associativity, symmetry and higher coherence | Requires stronger proof |
7. Use localisation to express the invariant core
An infinity-categorical localisation inverts weak equivalences while retaining higher mapping information. A comparison that induces an equivalence of localisations captures more than an equivalence of ordinary homotopy categories.
State which localisation theorem or construction is used and retain size, enrichment and replacement hypotheses.
8. Audit monoidal comparison data
A lax, oplax or strong monoidal comparison carries different information. Name the direction and whether the structure maps become equivalences on derived objects.
Check the unit, associativity, symmetry or braiding diagrams and their higher coherences. Objectwise equivalence of tensor outputs is not enough if choices fail to compose coherently.
- Tensor comparison maps and direction.
- Derived equivalence of structure maps.
- Unit compatibility.
- Associativity and symmetry coherence.
- Compatibility with replacement and localisation.
9. Know when homotopy-category agreement is sufficient
If the task asks only for an invariant group, existence of a derived object or a morphism class, an equivalence of homotopy categories may be enough. Do not build higher structure without a need.
If the task involves mapping objects, natural transformations between transformations, descent, operads or tensor-compatible constructions, the forgotten higher information matters.
10. Recognise comparison failures without overclaiming
A comparison can be essentially surjective but not fully faithful, fully faithful on Hom sets but not mapping spaces, or an infinity-categorical equivalence that fails to preserve the chosen tensor. Diagnose the exact level.
Failure of a strong monoidal comparison does not imply the underlying homotopy theories differ. It may identify a structure-specific obstruction.
11. Misconceptions and practical questions
The homotopy category is not the entire homotopy theory, natural objectwise equivalence is not automatically coherent and “same derived category” can be ambiguous without naming the enhancement.
Does an equivalence of homotopy categories imply equivalent mapping spaces?
No. It identifies objects and homotopy classes of maps but can forget higher homotopies. Additional enriched or infinity-categorical evidence is needed.
Does an equivalence of infinity-categories preserve tensor products?
Only if the tensor is part of the structure being compared and the equivalence carries suitable coherent monoidal data.
12. Changed-context transfer, evidence check and sources
Transfer the audit to two dg enhancements of a triangulated category. First compare their localisations and mapping objects; only then ask whether tensor, t-structure or other chosen data is preserved.
Hovey’s Model Categories develops model structures and derived comparisons. Lurie’s Higher Topos Theory develops infinity-categorical localisation and higher mapping information. The exact equivalence and monoidal hypotheses depend on the selected enhancements.
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: replacement-choice comparison and enhancement limits · Earlier guide: functorial replacement systems and coherence 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.