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Abelian groups are monoidal under the tensor product

Statement

The category $A b$ of abelian groups is monoidal with tensor product $\otimes_{Z}$ and unit object $Z$ . The associator and unitors are the canonical tensor-product isomorphisms over the commutative ring $Z$ , and a morphism $A \otimes_{Z} B \to C$ is exactly the same data as a bilinear map $A \times B \to C$ .

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: $R \otimes_{R} N ≅ N$ and $M \otimes_{R} R ≅ M$ ).

[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 $A \otimes_{R} B$ are in bijection with balanced maps out of $A \times B$ (Universal property of the tensor product for balanced maps into abelian groups).

Proof

technique · direct

1.1givenL1

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 $A b$ .

1.2L3L4

The associativity isomorphism $((A \otimes_{Z} B) \otimes_{Z} C) ≅ A \otimes_{Z} (B \otimes_{Z} C)$ and the unit isomorphisms $Z \otimes_{Z} A ≅ A$ and $A \otimes_{Z} Z ≅ A$ are exactly the maps supplied by [L3] and [L4] for the commutative ring $Z$ .

2.1step 1.1step 1.2L1L5

By [L5], homomorphisms $A \otimes_{Z} B \to C$ are the same as balanced maps $A \times B \to C$ . By step 1.1 and [L1], balanced over $Z$ is exactly bilinear for abelian groups, so the same universal property identifies homomorphisms $A \otimes_{Z} B \to C$ with bilinear maps $A \times B \to C$ . Together with step 1.2 this is the monoidal structure on $A b$ .

3.1step 1.1step 1.2step 2.1L2L3L4L5∎

Therefore $A b$ is monoidal under $\otimes_{Z}$ with unit object $Z$ .

Depends on

Used by

Dependency tree · two levels

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Sources

The Stacks Project, Section 10.12: Tensor products (standard reference, not scraped)