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Canonical static concept record

Quasi-isomorphism

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Summary

A chain map that induces an isomorphism on every homology object.

Record metadata

Canonical name: Quasi-isomorphism
Concept ID: quasi_isomorphism
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Homological algebra
Domains: Homological algebra, Category theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

chain map $f\colon C_\bullet\to D_\bullet$

Axioms / constraints

$H_n(f)$ is an isomorphism for every $n$

Canonically induces

an isomorphism after passage to the derived category

Notes

A quasi-isomorphism need not be an isomorphism or a chain-homotopy equivalence before localization.

Concept sources

Wikipedia — Quasi-isomorphism — Quasi-isomorphism · encyclopedic reference · source ID wp-quasi-isomorphism
nLab — quasi-isomorphism — quasi-isomorphism · mathematical reference · source ID nlab-quasi-isomorphism
Stacks Project — Quasi-isomorphisms — Quasi-isomorphisms · mathematical reference · source ID stacks-quasi-isomorphism

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Chain map Quasi-isomorphism

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: chain_map_to_quasi_isomorphism
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + homology isomorphisms

Authored explanation

Require the induced morphism on every homology object to be an isomorphism.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Quasi-isomorphism — Quasi-isomorphism · encyclopedic reference · source ID wp-quasi-isomorphism
nLab — quasi-isomorphism — quasi-isomorphism · mathematical reference · source ID nlab-quasi-isomorphism
Stacks Project — Quasi-isomorphisms — Quasi-isomorphisms · mathematical reference · source ID stacks-quasi-isomorphism

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.