PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 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 canonical functor from complexes to the homotopy category is additive

Statement

Let $Q : Ch (A) \to K (A)$ be the functor that is identity on objects and sends a chain map $f$ to its homotopy class $[f]$ . Then $Q$ is additive.

Facts & Assumptions

Given: An additive category $A$ .

[L1]

Morphisms in $K (A)$ are homotopy classes of chain maps (The homotopy category of chain complexes).

[L2]

$K (A)$ is additive (The homotopy category is additive).

[L3]

$Ch (A)$ is additive (The category of complexes in an additive category is additive).

Proof

technique · direct

1.1

By [L1], the functor $Q$ sends each chain map to its coset modulo null-homotopy. Therefore for parallel maps $f, g$ , $Q (f + g) = [f + g] = [f] + [g], Q (0) = [0] = 0,$ so $Q$ preserves the additive structure on hom-groups.

L1L2L3givenalgebra

2.1

The zero object and biproduct objects of $Ch (A)$ are sent to the same underlying complexes in $K (A)$ because $Q$ is identity on objects. Since the structural maps are sent to their homotopy classes and still satisfy the biproduct identities, $Q$ preserves finite biproducts. Hence $Q$ is additive.

L1L2step 1.1algebra∎

Depends on

Used by

Homology factors uniquely through the homotopy category Theorem

Dependency tree · two levels

10 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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)