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 complete category

Definition

A category is idempotent complete (or Karoubian) when every idempotent endomorphism splits in the sense of Idempotent and split idempotent.

Depends on

Idempotent and split idempotent

Used by

FALSE: every idempotent splits False statement
An additive category with kernels is idempotent complete Theorem
The idempotent completion is idempotent complete and its inclusion is fully faithful and universal Theorem

Dependency tree · two levels

2 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)
Dixy Msapato, The Karoubi envelope and weak idempotent completion of an extriangulated category, Definition 2.1 (standard reference, not scraped)