CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Two canonical braided composites agree exactly when their underlying braids agree

Statement

For a fixed object $X$ and a fixed tensor power $X^{\otimes n}$ , two canonical braided endomorphisms of $X^{\otimes n}$ agree in every braided monoidal category if and only if their underlying braids are equal in $B_{n}$ .

Facts & Assumptions

Given: Two canonical braided endomorphisms of the same tensor power $X^{\otimes n}$ .

[L1]

Canonical braided endomorphisms of a fixed tensor power with the same underlying braid are equal (Braided coherence is controlled by underlying braids).

[L2]

The braid category realizes braid-group elements as actual canonical braided morphisms (The braid category).

Proof

technique · direct

1.1

If the two underlying braids are equal, then [L1] says the two canonical composites agree in every braided monoidal category.

L1given

2.1

Conversely, suppose the two canonical composites agree in every braided monoidal category. Apply this to the braid category from [L2]. There the canonical composites are literally the corresponding braids, so equality of the composites forces equality of the underlying braids.

L2step 1.1algebra

3.1

Therefore the two universal statements are equivalent.

step 1.1step 2.1∎

Depends on

Used by

Two canonical maps with different underlying braids do not agree Example

Dependency tree · two levels

8 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)
Michael Muger, Tensor Categories: A Selective Guided Tour, Section 4 (standard reference, not scraped)