Alphabeta MathHow statement and proof provenance work
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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
- Modular ribbon fusion category and Hopf-link data Definition
- Negligible morphisms for a pivotal trace Definition
- Pointed ribbon categories from abelian 3-cocycles Definition
- RT state spaces from a modular category Definition
- Tensor and multitensor categories Definition
- The specified rank-one divided-power tilting category Definition
- Tilting objects and negligible morphisms in a root-of-unity category Definition
- Twist and ribbon structure Definition
- Every rigid k-linear abelian monoidal category is a tensor category False statement
- Balancing and the Hopf-link character identity Lemma
- Colored signed twist surgery, circumcision, and paired-handle cancellation Lemma
- Modular Hom state spaces have exact sewing inverses and the dimension formula Lemma
- Multicolored ribbon graphs with arbitrary coupons have a unique evaluation Lemma
- Partial-trace closure of negligible morphisms Lemma
- RT capped operators: geometric Gram factors and exact signature gluing Lemma
- Simple ribbon fusion dimensions are nonzero Lemma
- The complete Kirby-color band-slide identity with open colored graph coupons Lemma
- The corrected finite rank-one R series gives the full tilting ribbon structure Lemma
- The modular Hopf-link matrix squares to D² times duality Lemma
- 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
- The two-braid category is strict rigid monoidal Theorem
- The weighted Reshetikhin–Turaev TQFT and modular-functor factorization 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)