DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 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.

Shifted chain maps and shifted chain homotopies

Definition

Let $f : C_{∙} \to D_{∙}$ be a chain map and let $k \in Z$ . The shifted chain map $f [k] : C [k]_{∙} \to D [k]_{∙}$ is defined degreewise by $f [k]_{n} : = f_{n - k} .$

If $s : f ≃ g$ is a chain homotopy, its shifted chain homotopy $s [k] : f [k] ≃ g [k]$ is the degree- $1$ family $s [k]_{n} : = (- 1)^{k} s_{n - k} .$

With these conventions, $f [k] - g [k] = d^{D [k]} s [k] + s [k] d^{C [k]} .$

Depends on

Used by

Dependency tree · two levels

6 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 12.14: Homotopy and the shift functor (standard reference, not scraped)