CorollaryStatement: AI-generatedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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The free-word formulation implies the canonical-map formulation

Statement

In the one-generator word formalism, the free-word theorem implies the canonical-map form of coherence: the unique arrow of $W$ between two binary words of the same length is carried by evaluation to the unique canonical comparison map between the corresponding parenthesised tensor powers of the chosen object.

Facts & Assumptions

Given: Two binary words $v, w$ of the same length and an object $b$ of a monoidal category.

[L1]

In the binary-word category $W$ , there is exactly one morphism between two words of the same length (The category of binary words).

[L2]

Recursive evaluation at $b$ defines a strong monoidal functor $T_{b} : W \to B$ , and on the unique arrow between equal-length words it uses the canonical comparison isomorphism between the corresponding parenthesised tensor powers of $b$ (The word category is the free monoidal category on one generator).

Proof

technique · direct

1.1

Because $v$ and $w$ have the same length, [L1] gives a unique arrow $u_{v, w} : v \to w$ in $W$ .

givenL1

2.1

By [L2], the image $T_{b} (u_{v, w})$ is exactly the canonical comparison isomorphism between the two parenthesised tensor powers represented by $v$ and $w$ .

L2step 1.1

3.1

Therefore the unique formal arrow of $W$ between $v$ and $w$ is carried by evaluation to the canonical comparison map between the corresponding tensor powers of $b$ . This is precisely the canonical-map formulation in the one-generator setting.

step 1.1step 2.1

4.1

Thus the free-word presentation recovers the canonical-map presentation of coherence for tensor powers of one chosen object.

step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

S. Mac Lane, Categories for the Working Mathematician, Chapter VII.2 (standard reference, not scraped)