False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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FALSE: a monoidal category carries at most one symmetry

Statement

A monoidal category carries at most one symmetry.

Facts & Assumptions

Given: The monoidal category of $Z / 2$ -graded vector spaces over a field of characteristic different from $2$ .

[L1]

A symmetry is a braiding $c_{X, Y}$ with $c_{Y, X} c_{X, Y} = 1$ (Symmetric monoidal category).

Refutation

technique · direct

1.1L1given

On $Z / 2$ -graded vector spaces, the ordinary swap $x \otimes y \mapsto y \otimes x$ and the Koszul swap $x \otimes y \mapsto (- 1)^{∣ x ∣ ∣ y ∣} y \otimes x$ are two natural symmetries of the same tensor product. Because the characteristic is not $2$ , these two maps differ on odd-degree simple tensors.

2.1step 1.1∎

Since these two symmetries differ on odd-degree simple tensors, the monoidal category has more than one symmetry. Therefore the statement is false.

Depends on

Symmetric monoidal category

Used by

Nothing in the library uses this result yet.

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

G. M. Kelly, Basic Concepts of Enriched Category Theory, Section 1.4 (standard reference, not scraped)