TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Unbracketed tensor strings are well defined after coherence

Statement

Let $C$ be a monoidal category and let $A_{1}, \dots, A_{n}$ be objects of $C$ . After coherence, the unbracketed tensor string $A_{1} \otimes \dots \otimes A_{n}$ is a well-defined expression: for $n \geq 2$ , any two parenthesisations of that string are canonically and uniquely identified.

Facts & Assumptions

Given: A monoidal category $C$ and objects $A_{1}, \dots, A_{n}$ of it.

[L1]

Before the coherence page, an unbracketed tensor string of length at least three is not yet defined; only parenthesised words are (Unbracketed tensor strings are not yet defined on this page).

[L2]

For parenthesised tensor words on the same ordered letters, there is a unique canonical natural isomorphism between any two of them (Mac Lane coherence in canonical-map form).

Proof

technique · direct

1.1givenchoose

For $n = 0$ , interpret the empty tensor string as the unit object; for $n = 1$ , as $A_{1}$ ; and for $n \geq 2$ , choose any parenthesised tensor word $w$ on the letters $x_{1}, \dots, x_{n}$ and evaluate it at $(A_{1}, \dots, A_{n})$ .

2.1L2step 1.1

If $w^{'}$ is another such parenthesisation, [L2] gives a unique canonical natural isomorphism $w \Rightarrow w^{'}$ , and evaluating at $(A_{1}, \dots, A_{n})$ yields a unique canonical isomorphism between the two resulting objects of $C$ .

3.1L1step 2.1∎

Therefore suppressing brackets does not change the resulting tensor object except by a unique canonical identification. This discharges the warning of [L1]: after coherence, the notation $A_{1} \otimes \dots \otimes A_{n}$ is well defined.

Depends on

Used by

The monoid-object axioms may be written without associators Theorem

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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Remark 2.9.3 (standard reference, not scraped)