TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Braided coherence is controlled by underlying braids

Statement

Let $C$ be a braided monoidal category and let $X$ be an object of $C$ . Each formal canonical composite on $X^{\otimes n}$ built from associators, unitors, braidings, and their inverses has an underlying braid in $B_{n}$ . If two such formal composites have equal underlying braids, then their interpreted morphisms in $C$ are equal. Different formal braids may, of course, have the same interpretation in a particular braided category.

Facts & Assumptions

Given: A braided monoidal category $C$ , an object $X \in C$ , and two canonical braided endomorphisms of $X^{\otimes n}$ .

[L1]

Every braided monoidal category is braided-monoidally equivalent to a strict braided one (Every braided monoidal category is monoidally equivalent to a strict braided one).

[L2]

The braid category is the free strict braided monoidal category on one generator (The braid category is the free strict braided monoidal category on one generator).

[L3]

A braided monoidal functor preserves canonical composites built from the braiding and tensor structure (Braided monoidal functor).

Proof

technique · direct

1.1

By [L1], replace the given braided monoidal category by a braided-monoidally equivalent strict one. Equality may be checked there because an equivalence is faithful on morphisms.

L1givenchoose

2.1

In the strict model, every formal canonical endomorphism of $X^{\otimes n}$ is built only from the local braidings of adjacent copies of $X$ , together with identities, tensoring, and composition. Reading those local crossings before interpretation gives a braid word and hence an element of $B_{n}$ . By [L2], interpreting the formal composite is exactly applying the unique strict braided monoidal functor from the braid category that sends the generating object to $X$ .

L2step 1.1construct

3.1

If the two formal canonical composites have the same underlying braid, then step 2.1 identifies their interpretations as the images of the same morphism in the free braid category under the same braided functor. Hence they are equal in the strict model, and therefore equal in the original braided category by step 1.1 and [L3].

L3step 1.1step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

10 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)