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.
Semisimple objects and semisimple abelian categories
Definition
An object of an abelian category is semisimple if it is a finite direct sum of simple objects. The category is semisimple if every object is semisimple. Thus the zero object is semisimple, as the empty direct sum.
Depends on
Used by
- The unit is projective exactly when the tensor category is semisimple Corollary
- Finite semisimple 2-category Definition
- Finite semisimple module category over a fusion category Definition
- Fusion and multifusion categories Definition
- G-graded finite semisimple 2-vector space Definition
- Every finite k-linear abelian category is semisimple False statement
- A separability idempotent constructs the balanced relative tensor product Lemma
- Relative tensor product gives associative composition of separable bimodules Theorem
- The unit object of a multitensor category is semisimple Theorem
Dependency tree · two levels
5 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
- Catherine Meusburger, Tensor Categories, Definition 8.1.4 (standard reference, not scraped)