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.
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. Inclusion test: Include chain maps inducing homology isomorphisms in all degrees. Exclusion test: Exclude objectwise isomorphisms with no chain condition, maps matching Euler characteristic only, and homotopy equivalences asserted without maps. Nearest boundary: A chain-homotopy equivalence implies a quasi-isomorphism, but the converse needs conditions. Exit condition: The property ends when one induced homology map is not invertible. Common misclassifications: 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. Nearest named distinctions: Chain isomorphism: Stronger degreewise invertibility. Chain-homotopy equivalence: Usually stronger. Homology isomorphism: The induced criterion. Weak equivalence: A category-dependent broader term.
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.
Relationships to Other Abstractions¶
Current abstraction Quasi-Isomorphism Domain-specific
Parents (1) — more general patterns this builds on
-
Quasi-Isomorphism presupposes Isomorphism Prime
Quasi-Isomorphism presupposes Isomorphism because the morphism is defined by inducing isomorphisms on every homology or cohomology group.
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