TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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Braided coherence fails in the symmetric form

Statement

There exists a braided monoidal category in which the canonical endomorphisms

$1, c_{X, X}^{2}, c_{X, X}^{4}, \dots$

of $X \otimes X$ are pairwise distinct. Consequently the symmetric slogan "every diagram built from associators and braidings commutes" is false for braided monoidal categories.

Facts & Assumptions

Given: The braid category $B$ .

[L1]

In the braid category, the braiding on $1 \otimes 1 = 2$ is the generator $σ_{1} \in B_{2}$ (The braid category).

[L2]

The group $B_{2}$ is infinite cyclic, so its distinct powers of $σ_{1}$ are distinct morphisms (The two-strand braid group is infinite cyclic).

Proof

technique · direct

1.1

Take $X : = 1$ in the braid category. By [L1], the canonical braiding on $X \otimes X$ is $σ_{1}$ , so its even powers are the endomorphisms $σ_{1}^{2 m}$ of the object $2$ .

givenL1algebra

2.1

By [L2], the elements $1, σ_{1}^{2}, σ_{1}^{4}, \dots$ are pairwise distinct in $B_{2}$ . Therefore the canonical endomorphisms $1, c_{X, X}^{2}, c_{X, X}^{4}, \dots$ are pairwise distinct in this braided monoidal category.

L2step 1.1algebra

3.1

If every formal diagram built from associators and braidings commuted in every braided monoidal category, then the morphisms in step 2.1 would all agree. They do not, so the symmetric-form coherence slogan fails in the braided setting.

step 2.1contradiction∎

Depends on

Used by

FALSE: every diagram built from the associator and the braiding commutes False statement

Dependency tree · two levels

7 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, Remark 8.2.5 (standard reference, not scraped)
Michael Muger, Tensor Categories: A Selective Guided Tour, Section 4 (standard reference, not scraped)