TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

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A twist on a braided rigid category is the same thing as a pivotal structure of Drinfeld type

Statement

Let $C$ be a braided tensor category and let $u_{X} : X \to X^{\lor \lor}$ be its Drinfeld isomorphism. For a natural automorphism $θ$ of the identity functor, put

$ψ_{X} : = u_{X} θ_{X} .$

Then $ψ$ is a pivotal structure if and only if $θ$ is a twist.

Facts & Assumptions

Given: A braided tensor category, its Drinfeld isomorphism $u$ , and a natural automorphism $θ$ of the identity functor.

[F1]

EGNO formula (8.35) and the sentence after it state exactly that $ψ_{X} = u_{X} θ_{X}$ is a tensor isomorphism $X \to X^{* *}$ if and only if $θ$ is a twist.

[L1]

The Drinfeld morphism exists, and in a braided tensor category it is an isomorphism (A braided rigid category has a Drinfeld morphism).

[L2]

A pivotal structure is precisely a monoidal natural isomorphism $id \Rightarrow (-)^{\lor \lor}$ (Pivotal structure).

[L3]

A twist is precisely a natural automorphism satisfying the double-braiding tensor law (Twist and ribbon structure).

Proof

technique · direct

1.1

By [L1], the family $u_{X}$ is invertible, so every natural isomorphism $ψ_{X} : X \to X^{\lor \lor}$ can be written uniquely as $u_{X} θ_{X}$ for a natural automorphism $θ$ of the identity.

givenF1L1construct

2.1

The tensor relation for $u$ from the Drinfeld-morphism theorem inserts exactly one double braiding between $u_{X \otimes Y}$ and $u_{X} \otimes u_{Y}$ . Therefore the condition that $ψ$ be monoidal is equivalent to the condition that $θ$ absorb that double braiding, namely $θ_{X \otimes Y} = (θ_{X} \otimes θ_{Y}) \circ c_{Y, X} \circ c_{X, Y},$ which is the twist law from [L3].

step 1.1L1L2L3

3.1

Hence $ψ$ is a pivotal structure exactly when $θ$ is a twist.

step 2.1L2L3∎

Depends on

Used by

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Dependency tree · two levels

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Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, formula (8.35) and Proposition 8.10.6 (standard reference, not scraped)