ExampleConstruction: AI-adaptedVerification: 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.
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Two canonical maps with different underlying braids do not agree

Example

In the braid category, the two canonical endomorphisms of $1 \otimes 1 \otimes 1$ represented by $σ_{1} σ_{2}$ and $σ_{2} σ_{1}$ are different.

Facts & Assumptions

Given: Braided coherence and the braid category.

[L1]

Two canonical braided composites agree in every braided monoidal category exactly when their underlying braids agree (Two canonical braided composites agree exactly when their underlying braids agree).

[L2]

The braid category realizes the braid group on three strands as the endomorphisms of the object $3 = 1 \otimes 1 \otimes 1$ (The braid category).

Verification

technique · direct

1.1givenL2algebra

The two displayed composites have underlying braids $σ_{1} σ_{2}$ and $σ_{2} σ_{1}$ in $B_{3}$ .

2.1step 1.1algebra

These braids are not equal: their images in $S_{3}$ are $(1 2 3)$ and $(1 3 2)$ , which are distinct permutations.

3.1L1L2step 2.1∎

Therefore [L1] implies that the two canonical composites do not agree. In particular, they are distinct endomorphisms of the object $3$ in the braid category from [L2].

Depends on

Used by

Nothing in the library uses this result yet.

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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Exercise 8.2.7 (standard reference, not scraped)