False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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FALSE: a trace can be defined for an endomorphism in any monoidal category

Statement

A trace can be defined for an endomorphism in any monoidal category.

Facts & Assumptions

Given: A monoidal category and an endomorphism $x : X \to X$ .

[L1]

The trace ladder states that rigidity is needed to define categorical trace at all (What is needed before a trace can be written).

[L2]

A pivotal structure is the extra comparison $a_{X} : X \to X^{\lor \lor}$ needed to turn an endomorphism into a traceable morphism (Pivotal structure).

Refutation

technique · direct

1.1

By the hypothesis ladder in What is needed before a trace can be written, categorical trace is first defined on a morphism $a : X \to X^{\lor \lor}$ or $b : X^{\lor \lor} \to X$ , so rigidity is already required before any trace expression is even well typed.

givenL1

2.1

Even in a rigid category, an endomorphism $x : X \to X$ does not itself have the required source and target. One needs a pivotal structure to insert a comparison $a_{X} : X \to X^{\lor \lor}$ or $a_{X}^{- 1} : X^{\lor \lor} \to X$ , as recorded in Pivotal structure.

step 1.1L2

3.1

Therefore a bare monoidal category does not supply a trace for an arbitrary endomorphism. The statement is false because the operation is not even typed without extra duality data.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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, Definition 4.7.1 and Remark 4.7.2 (standard reference, not scraped)