Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31‡ rests on unproved material (inherited)
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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Abelian groups are monoidal under the tensor product

Statement

The category Ab of abelian groups is monoidal with tensor product ⊗Z and unit object Z. The associator and unitors are the canonical tensor-product isomorphisms over the commutative ring Z, and a morphism A⊗ZB→C is exactly the same data as a bilinear map A×B→C.

Facts & Assumptions

Given: Abelian groups A,B,C,D.

[L1]

Abelian groups and Z-modules have the same objects and morphisms (Abelian groups and Z-modules have the same objects and morphisms).

[L2]

The category of abelian groups already exists as an abelian category (Abelian groups form an abelian category).

[L3]

Tensor products of modules are associative and have the regular module as tensor unit (Associativity of tensor products for compatible bimodules, The regular module is a tensor unit: R⊗RN≅N and M⊗RR≅M).

[L4]

Over a commutative ring there are natural symmetry and associativity isomorphisms for tensor products (Symmetry and associativity isomorphisms for tensor products over a commutative ring).

[L5]

For modules over a unital ring, homomorphisms out of A⊗RB are in bijection with balanced maps out of A×B (Universal property of the tensor product for balanced maps into abelian groups).

Proof

technique · direct
1.1

By [L1], every abelian group is a Z-module and every group homomorphism is Z-linear. Thus the tensor product over Z of two abelian groups is again an abelian group, and the tensor-product maps are morphisms in Ab.

givenL1
1.2

The associativity isomorphism ((A⊗ZB)⊗ZC)≅A⊗Z(B⊗ZC) and the unit isomorphisms Z⊗ZA≅A and A⊗ZZ≅A are exactly the maps supplied by [L3] and [L4] for the commutative ring Z.

L3L4
2.1

By [L5], homomorphisms A⊗ZB→C are the same as balanced maps A×B→C. By step 1.1 and [L1], balanced over Z is exactly bilinear for abelian groups, so the same universal property identifies homomorphisms A⊗ZB→C with bilinear maps A×B→C. Together with step 1.2 this is the monoidal structure on Ab.

step 1.1step 1.2L1L5
3.1

Therefore Ab is monoidal under ⊗Z with unit object Z.

step 1.1step 1.2step 2.1L2L3L4L5∎

Depends on

Used by

Dependency tree · two levels

37 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