Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: the standard unit-constraint identities must all be imposed as independent axioms

Statement

False claim: besides the pentagon and triangle, the tensor-product formulas for the left and right unitors and the equality on the unit object must all be imposed as independent axioms.

Facts & Assumptions

Given: The unit-constraint formulas already proved on this page.

[L1]

The left unitor satisfies λX⊗Y=(λX⊗1Y)∘α1,X,Y (The left unitor of a tensor product is determined by the associator).

[L2]

The right unitor satisfies ρX⊗Y∘αX,Y,1=1X⊗ρY (The right unitor of a tensor product is determined by the associator).

[L3]

The two unitors agree on the tensor unit (The two unitors agree on the tensor unit).

Refutation

technique · direct
1.1

By [L1] and [L2], the two tensor-product formulas for the unitors follow from the monoidal axioms and are not independent axioms.

L1L2
1.2

By [L3], even the remaining equality on the unit object is a theorem rather than an extra axiom.

L3
2.1

Therefore the claim is false.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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