DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

Composition series and composition factors of an object

Definition

A composition series of an object $A$ in an abelian category is a finite strict chain of subobjects

$0 = A_{0} < A_{1} < \dots < A_{n} = A$

such that every quotient object $A_{i} / A_{i - 1}$ is simple (Simple object, The quotient of an object by a subobject).

The simple quotient objects $A_{i} / A_{i - 1}$ are the composition factors of the series.

Depends on

Used by

Object of finite length Definition
FALSE: every object of an abelian category has a composition series False statement
Jordan-Holder theorem in an abelian category Theorem
Schreier refinement theorem in an abelian category Theorem
Zassenhaus butterfly lemma in an abelian category Theorem

Dependency tree · two levels

7 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

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)