DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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The categorical trace of a morphism into the double dual

Definition

Fix a rigid monoidal category with chosen left duals. For a morphism $a : X \to X^{\lor \lor}$ , the left categorical trace of $a$ is the endomorphism of the unit

${Tr}_{L} (a) : 1 \overset{{coev}_{X}}{\to} X \otimes X^{\lor} \overset{a \otimes 1_{X^{\lor}}}{\to} X^{\lor \lor} \otimes X^{\lor} \overset{{ev}_{X^{\lor}}}{\to} 1.$

For a morphism $b : X^{\lor \lor} \to X$ , the right categorical trace of $b$ is

${Tr}_{R} (b) : 1 \overset{{coev}_{X^{\lor}}}{\to} X^{\lor} \otimes X^{\lor \lor} \overset{1_{X^{\lor}} \otimes b}{\to} X^{\lor} \otimes X \overset{{ev}_{X}}{\to} 1.$

So a bare endomorphism $X \to X$ is not yet an input to categorical trace; one first needs a comparison between $X$ and a double dual.

Depends on

The double dual is a monoidal functor

Used by

A braiding alone does not define a link trace Counterexample
Modular ribbon fusion category and Hopf-link data Definition
Negligible morphisms for a pivotal trace Definition
The dimension of an object relative to a pivotal structure Definition
The ribbon evaluation of an X-colored closed braid 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
Writhe normalization cancels a ribbon kink Example
Balancing and the Hopf-link character identity Lemma
Outside-alcove rank-one tiltings have zero pivotal dimension Lemma
Simple ribbon fusion dimensions are nonzero Lemma
The explicit Fibonacci model has a positive spherical pivotal structure Lemma
The ribbon trace equals the framed-closure evaluation Lemma
What is needed before a trace can be written Remark
Basic properties of the categorical trace Theorem
In a spherical category the left and right traces agree Theorem
The negligible ideal and semisimple trace quotient of root tiltings Theorem

Dependency tree · two levels

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Sources

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definition 4.7.1 (standard reference, not scraped)