ExampleConstruction: AI-adaptedVerification: 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.

Homotopy classes as H-zero of a Hom complex

Example

For the two-term identity complex $0 \to Z \overset{1_{Z}}{\to} Z \to 0,$ the Hom complex from the previous example has $\ker (\partial_{0}) = {(a, a) : a \in Z}, im (\partial_{1}) = {(a, a) : a \in Z},$ so $H_{0} (\underset{}{Hom} (C, C)) = 0.$ Hence every endomorphism of $C$ is zero in $K (A b)$ .

Facts & Assumptions

Given: The Hom complex of the two-term identity complex with itself.

[L1]

The previous example computes $\partial_{1} (m) = (m, m), \partial_{0} (a, b) = b - a$ (The Hom complex of two two-term complexes).

[L2]

Hom in the homotopy category is $H_{0}$ of the Hom complex (Hom in the homotopy category is zero-degree homology of the Hom complex).

Verification

technique · direct

1.1L1givenalgebra

By [L1], a degree- $0$ element $(a, b)$ lies in $\ker (\partial_{0})$ exactly when $b - a = 0$ , so $\ker (\partial_{0}) = {(a, a)}$ . The same formula shows $(a, a) = \partial_{1} (a)$ , hence $\ker (\partial_{0}) = im (\partial_{1}) .$

2.1L2step 1.1algebra∎

Therefore $H_{0} (\underset{}{Hom} (C, C)) = 0$ . By [L2], applied in the abelian category $A b$ , this means ${Hom}_{K (A b)} (C, C) = 0,$ so every endomorphism class of $C$ is zero in the homotopy category.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)