LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

Composition of homotopy classes is well defined

Statement

If $f, f^{'} : C_{∙} \to D_{∙}$ and $g, g^{'} : D_{∙} \to E_{∙}$ satisfy $[f] = [f^{'}]$ and $[g] = [g^{'}]$ , then $[g \circ f] = [g^{'} \circ f^{'}] .$ Thus composition on homotopy classes may be defined by $[g] \circ [f] : = [g \circ f] .$

Facts & Assumptions

Given: Chain maps $f, f^{'} : C_{∙} \to D_{∙}$ and $g, g^{'} : D_{∙} \to E_{∙}$ with $[f] = [f^{'}]$ and $[g] = [g^{'}]$ .

[L1]

Equality of classes means that differences are null-homotopic (Homotopy classes of chain maps).

[L2]

Whiskering preserves chain homotopy (Chain homotopy is compatible with addition and composition).

[L3]

Null-homotopic maps form a two-sided additive ideal (Null-homotopic maps form a two-sided additive ideal).

Proof

technique · direct

1.1

By [L1], the maps $f - f^{'}$ and $g - g^{'}$ are null-homotopic. Using [L3], $g f - g^{'} f^{'} = (g - g^{'}) f^{'} + g (f - f^{'})$ is a sum of two null-homotopic maps.

L1L3givenalgebra

2.1

The first summand in step 1.1 is null-homotopic by right whiskering, and the second is null-homotopic by left whiskering; this is exactly [L2] and [L3]. Hence $g f - g^{'} f^{'}$ is null-homotopic, so [L1] gives $[g f] = [g^{'} f^{'}]$ .

L1L2step 1.1algebra∎

Depends on

Used by

The homotopy category of chain complexes Definition

Dependency tree · two levels

7 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)
The Stacks Project, Section 13.8: The homotopy category (standard reference, not scraped)