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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
- Frobenius–Schur sign in a normalized anyonic vacuum channel Definition
- Modular ribbon fusion category and Hopf-link data Definition
- Negligible morphisms for a pivotal trace Definition
- Pointed ribbon categories from abelian 3-cocycles Definition
- The dimension of an object relative to a pivotal structure Definition
- Tilting objects and negligible morphisms in a root-of-unity category 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
- All-label unitary Ising normal data and ribbon coherence Lemma
- Modular Hom state spaces have exact sewing inverses and the dimension formula Lemma
- Outside-alcove rank-one tiltings have zero pivotal dimension Lemma
- Partial-trace closure of negligible morphisms Lemma
- The corrected finite rank-one R series gives the full tilting ribbon structure Lemma
- The explicit Fibonacci model has a positive spherical pivotal structure Lemma
- Objectwise double-dual isomorphisms do not supply a pivotal structure Remark
- 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
- The full toral-weight root category is braided, rigid and spherical 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)