Quasi-Isomorphism¶
A morphism of complexes inducing degreewise isomorphisms on homology or cohomology.
Core Idea¶
A quasi-isomorphism is a chain or cochain map whose induced map on homology or cohomology is an isomorphism in every degree.
A map between resolutions induces the same homology although component modules differ. A chain map misses one nonzero homology class.
Structural Signature¶
Sig role-phrases:
- Source complex — Supplies graded objects and differential. It is input. Counterfactual: No complex means no homology.
- Target complex — Receives a chain-compatible map. It is input. Counterfactual: An ungraded target changes the category.
- Chain map — Commutes with differentials. It is morphism. Counterfactual: A graded map alone cannot induce homology maps.
- Induced homology maps — Carry cycles modulo boundaries. It is test. Counterfactual: Objectwise maps are not the criterion.
- Degreewise isomorphism — Defines quasi-isomorphism. It is result. Counterfactual: Failure in one degree fails the property.
What It Is Not¶
- It is not a complex isomorphism.
- It is not equality of Betti numbers alone.
- It is not every chain map.
- It is not always a chain-homotopy equivalence.
- Closest near-miss. A chain-homotopy equivalence implies a quasi-isomorphism, but the converse needs conditions.
Scope of Application¶
- Homological algebra. Localizes complexes.
- Derived categories. Uses quasi-isomorphisms as equivalences.
- Topology. Compares chain models.
- Algebraic geometry. Relates resolutions.
Clarity¶
Include chain maps inducing homology isomorphisms in all degrees. Exclude objectwise isomorphisms with no chain condition, maps matching Euler characteristic only, and homotopy equivalences asserted without maps.
Manages Complexity¶
The map may differ strongly degreewise while preserving homology. Derived equivalence need not supply an ordinary inverse.
Abstract Reasoning¶
- Source complex — Supplies graded objects and differential. No complex means no homology.
- Target complex — Receives a chain-compatible map. An ungraded target changes the category.
- Chain map — Commutes with differentials. A graded map alone cannot induce homology maps.
- Induced homology maps — Carry cycles modulo boundaries. Objectwise maps are not the criterion.
- Degreewise isomorphism — Defines quasi-isomorphism. Failure in one degree fails the property.
Knowledge Transfer¶
Homology-equivalence reasoning transfers across chain models when complexes, grading, and induced maps are preserved; it does not imply a literal isomorphism or chain-homotopy inverse without extra hypotheses.
Examples¶
Applied / In Practice¶
A map between resolutions induces the same homology although component modules differ.
Mapped back: chain → map; homology → isomorphic.
Applied / In Practice¶
A chain map misses one nonzero homology class.
Mapped back: degree → one failure.
Structural Tensions¶
T1 — Chain Detail versus Derived Information. The map may differ strongly degreewise while preserving homology.
Diagnostic: Which level matters?
T2 — Weak Equivalence versus Strict Inverse. Derived equivalence need not supply an ordinary inverse.
Diagnostic: What extra hypotheses hold?
Structural–Framed Character¶
Supplies graded objects and differential. Receives a chain-compatible map. The map may differ strongly degreewise while preserving homology.
Structural Core vs. Domain Accent¶
Carry cycles modulo boundaries. Defines quasi-isomorphism. The property ends when one induced homology map is not invertible.
Instantiates / Related Primes¶
This entry presupposes Isomorphism.
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Approved root. The frozen graph retains quasi-isomorphism without a parent edge.
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Related — Chain isomorphism and Chain-homotopy equivalence. Stronger degreewise invertibility. Usually stronger.
Relationships to Other Abstractions¶
Current abstraction Quasi-Isomorphism Domain-specific
Parents (1) — more general patterns this builds on
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Quasi-Isomorphism presupposes Isomorphism Prime
Quasi-Isomorphism presupposes Isomorphism because the morphism is defined by inducing isomorphisms on every homology or cohomology group.Every reviewed Quasi-Isomorphism instance depends on the parent role: the morphism is defined by inducing isomorphisms on every homology or cohomology group. Removing that role makes the frozen child identity undefined or changes it into a different abstraction. Isomorphism can occur without Quasi-Isomorphism, so the relation is dependency rather than subsumption.
Hierarchy paths (4) — routes to 2 parentless roots
- Quasi-Isomorphism → Isomorphism → Bijectivity → Function (Mapping)
- Quasi-Isomorphism → Isomorphism → Invariance
- Quasi-Isomorphism → Isomorphism → Bijectivity → Injectivity → Function (Mapping)
- Quasi-Isomorphism → Isomorphism → Bijectivity → Surjectivity → Function (Mapping)
Neighborhood in Abstraction Space¶
Quasi-Isomorphism sits in a crowded region of the domain-specific corpus (34th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Category Theory & Higher Structures (18 abstractions)
Nearest neighbors
- Simplicial Localization — 0.90
- Amnestic Functor — 0.90
- Category of Manifolds — 0.89
- Mapping Cylinder — 0.87
- Bijective proof — 0.87
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Chain isomorphism. Tell: Stronger degreewise invertibility.
- Chain-homotopy equivalence. Tell: Usually stronger.
- Homology isomorphism. Tell: The induced criterion.
- Weak equivalence. Tell: A category-dependent broader term.
References¶
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Quasi-isomorphism (revision 1349081515).
- Preserved source candidate: https://flynncoo.github.io/pdfs/papers/On_associative_and_commutative_differential_graded_algebras_in_positive_characteristic%20%281%29.pdf
The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.