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.
Rigid object and rigid monoidal category
Definition
An object of a monoidal category is rigid if it has both a left dual and a right dual in the sense of Left dual and right dual object.
A monoidal category is rigid if every object in it is rigid.
Muger's notes also use the word autonomous for the same condition, and Joyal-Street use compact in the non-symmetric setting.
Depends on
Used by
- Not every monoidal category is rigid Counterexample
- Twist and ribbon structure Definition
- Rigidity alone does not make a tensor category Remark
- A braided rigid category has a Drinfeld morphism Theorem
- Finite-dimensional vector spaces are rigid Theorem
- In a rigid category every morphism of monoidal functors is an isomorphism 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, Definition 2.10.11 (standard reference, not scraped)
- Michael Muger, Tensor Categories: A Selective Guided Tour, Section 1.5 (standard reference, not scraped)