CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-01

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The braid category is braided but not symmetric

Statement refuted

Every braided monoidal category is symmetric.

Facts & Assumptions

Given: The braid category $B$ .

[L1]

The braid category is braided monoidal by construction (The braid category).

[L2]

The group $B_{2}$ is infinite cyclic, generated by $σ_{1}$ (The two-strand braid group is infinite cyclic).

Counterexample

technique · direct

1.1givenL1algebra

By [L1], $B$ is a braided monoidal category. Its braiding on the generating object $1 \otimes 1 = 2$ is the crossing braid $σ_{1} \in B_{2}$ .

2.1L2step 1.1algebra

By [L2], $σ_{1}$ has infinite order, so in particular $σ_{1}^{2} \neq 1_{2}$ . Therefore the square of the braiding on $1 \otimes 1$ is not the identity.

3.1step 2.1algebra∎

A symmetric braiding must square to the identity on every tensor product. Step 2.1 shows that this fails in $B$ , so $B$ is braided but not symmetric.

Depends on

Used by

FALSE: every braided monoidal category is equivalent to a strict commutative one False statement

Dependency tree · two levels

6 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

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