Alphabeta MathHow 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.
Simple object
Definition
An object of an abelian category is simple when and its only subobjects are the zero subobject and (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms).
Equivalently, has no proper nonzero subobject.
Depends on
Used by
- Finite length and finite Hom do not imply a finite category Counterexample
- Absolutely simple objects Definition
- Composition series and composition factors of an object Definition
- Finite k-linear abelian categories Definition
- Semisimple objects and semisimple abelian categories Definition
- Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
- Projective epimorphisms onto the simples generate every finite-length object Lemma
- Finite abelian categories admit finite-dimensional module models Theorem
- Intrinsic finite category hypotheses give a finite projective generator Theorem
- Jordan-Holder theorem in an abelian category Theorem
- Shift-orbit bases for graded simple and projective classes Theorem
- Simple classes freely generate the Grothendieck group of a length category Theorem
- The unit object of a multitensor category is semisimple Theorem
Dependency tree · two levels
4 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)