DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The homotopy category of chain complexes

Definition

Let $A$ be an additive category. The homotopy category of chain complexes on $A$ is the category $K (A) .$ Its objects are the chain complexes in $A$ , and for chain complexes $C_{∙}, D_{∙}$ its morphisms are the homotopy classes ${Hom}_{K (A)} (C, D) : = [C, D]_{K} .$

Composition is induced from composition of representatives and is well defined by Composition of homotopy classes is well defined.

Depends on

Used by

FALSE: chain-homotopic maps are equal as chain maps False statement
FALSE: the homotopy category is obtained by identifying quasi-isomorphisms with identities False statement
The canonical functor from complexes to the homotopy category is additive Proposition
Zero homology does not make an object zero in the homotopy category Proposition
Hom in the homotopy category is zero-degree homology of the Hom complex Theorem
Homology factors uniquely through the homotopy category Theorem
Shift is an additive autoequivalence of the complex and homotopy categories Theorem
The homotopy category is additive Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Section 13.8: The homotopy category (standard reference, not scraped)
Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)