DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Idempotent and split idempotent

Definition

Let $e : A \to A$ be an endomorphism in a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection).

$e$ is idempotent when $e^{2} = e$ .
An idempotent $e$ splits when there exist morphisms $A \overset{p}{\to} B \overset{i}{\to} A$ such that

$i p = e, p i = 1_{B} .$

Then $B$ is called a splitting object for $e$ .

Depends on

Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Used by

Dependency tree · two levels

3 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

The Stacks Project, Section 12.4, Definition 12.4.1 (standard reference, not scraped)