TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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In a spherical category the left and right traces agree

Statement

Let $C$ be a spherical tensor category with spherical structure $a$ . For every object $X$ and every endomorphism $x : X \to X$ ,

${Tr}_{L} (a_{X} x) = {Tr}_{R} (x a_{X}^{- 1}) .$

Facts & Assumptions

Given: A spherical tensor category $(C, a)$ , an object $X$ , and an endomorphism $x : X \to X$ .

[F1]

EGNO Theorem 4.7.15 proves exactly the displayed identity for spherical tensor categories.

[L1]

The left and right traces used in the statement are the ones defined in The categorical trace of a morphism into the double dual, and $a_{X} : X \to X^{\lor \lor}$ is the pivotal comparison from Pivotal structure.

Proof

technique · direct

1.1

By [L1], both composites in the statement are well typed in any pivotal category, and the extra spherical hypothesis is exactly the one assumed in [F1].

givenF1L1

1.2

Applying [F1] to the present spherical tensor category gives ${Tr}_{L} (a_{X} x) = {Tr}_{R} (x a_{X}^{- 1}) .$

2.1

Hence in a spherical category the left and right traces agree after inserting the pivotal comparison.

step 1.2∎

Depends on

Used by

FALSE: the left and right traces always agree False statement

Dependency tree · two levels

5 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, Theorem 4.7.15 (standard reference, not scraped)