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.
Pivotal structure
Definition
Let be the monoidal endofunctor of The double dual is a monoidal functor.
A pivotal structure on a rigid monoidal category is an isomorphism of monoidal functors
Equivalently, it is a natural family of isomorphisms such that
after transporting along the monoidal structure of the double-dual functor.
Depends on
Used by
- The dimension of an object relative to a pivotal structure Definition
- The categorical trace of a linear endomorphism is its matrix trace Example
- FALSE: a trace can be defined for an endomorphism in any monoidal category False statement
- Pivotal and spherical structures vary by monoidal automorphisms of the identity Remark
- A twist on a braided rigid category is the same thing as a pivotal structure of Drinfeld type Theorem
- In a spherical category the left and right traces agree 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
- P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definitions 4.7.7-4.7.8 (standard reference, not scraped)