DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-30

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.

Subcomplex

Definition

Let $C_{∙}$ be a chain complex. A subcomplex of $C$ is a family of subobjects $S_{n} ↪ C_{n}$ such that for every $n$ the differential $d_{n}^{C} : C_{n} \to C_{n - 1}$ restricts to a morphism $d_{n}^{S} : S_{n} \to S_{n - 1}$ making $\begin{array}{ccc} S_{n} & ↪ & C_{n} \\ d_{n}^{S} ↓ & ↓ d_{n}^{C} \\ S_{n - 1} & ↪ & C_{n - 1} \end{array}$ commute.

Because the structure maps of a subobject are monic, each restricted differential is unique when it exists.

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra