CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

The uniqueness of the enrichment is an Eckmann-Hilton phenomenon

Statement

The uniqueness of the commutative-monoid enrichment determined by finite biproducts is an instance of the Eckmann-Hilton phenomenon.

Facts & Assumptions

Given: Two candidate bilinear commutative-monoid laws on the hom-sets of a category with finite biproducts.

[L1]

Finite biproducts define a canonical commutative-monoid law on every hom-set (A category with finite biproducts is enriched in commutative monoids).

[L2]

The compatible enrichment is unique (The commutative-monoid enrichment of a category with finite biproducts is unique).

[L3]

Two unital operations satisfying interchange coincide and are commutative (Eckmann–Hilton: two unital operations satisfying interchange coincide and are commutative).

Proof

technique · direct

1.1L1

Fix a hom-set $C (A, B)$ , write $+$ for the canonical law from [L1], and let $⊞$ be any second compatible bilinear law. Let $i_{1}, i_{2} : B \to B \oplus B$ , $p_{1}, p_{2} : B \oplus B \to B$ , and $\nabla_{B} : B \oplus B \to B$ be the standard biproduct maps. Because post- and precomposition are homomorphisms for $⊞$ , the morphism $i_{1} f ⊞ i_{2} g : A \to B \oplus B$ has projections $f$ and $g$ , so it is the pairing $⟨ f, g ⟩$ . By the canonical construction from [L1], this gives $f + g = \nabla_{B} ⟨ f, g ⟩$ .

2.1

Using step 1.1 and bilinearity for $⊞$ ,

$(f + g) ⊞ (h + k) = \nabla_{B} (⟨ f, g ⟩ ⊞ ⟨ h, k ⟩) = \nabla_{B} (i_{1} (f ⊞ h) ⊞ i_{2} (g ⊞ k)) = (f ⊞ h) + (g ⊞ k) .$

Both operations have the same unit $0_{A, B}$ , so $+$ and $⊞$ satisfy the interchange law on $C (A, B)$ . [L1, step 1.1]

3.1L2L3step 2.1

Applying [L3] to $+$ and $⊞$ shows that they coincide and are commutative. This is exactly the uniqueness conclusion of [L2], expressed as an Eckmann-Hilton rigidity statement on each hom-set.

4.1step 3.1∎

Therefore the uniqueness of the enrichment determined by finite biproducts is an Eckmann-Hilton phenomenon.

Depends on

Used by

Additivity can be derived rather than postulated, depending on the axiomatisation Remark

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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.2 (standard reference, not scraped)
nLab, Eckmann-Hilton argument (standard reference, not scraped)