nLab tensor product of abelian groups

Idea

For $A$ and $B$ two abelian groups, their tensor product $A \otimes B$ is a new abelian group such that a group homomorphism $A \otimes B \to C$ is equivalently a bilinear map out of $A$ and $B$.

Tensor products of abelian groups were defined by Hassler Whitney in 1938.

Definition

Definition

Let Ab be the collection of abelian groups, regarded as a multicategory whose multimorphisms are the multilinear maps $A_1, \cdots, A_n \to B$.

The tensor product $A, B \mapsto A \otimes B$ in this multicategory is the tensor product of abelian groups.

Equivalently this means explicitly:

Definition

For $A, B$ two abelian groups, their tensor product of abelian groups is the abelian group $A \otimes B$ which is the quotient of the free abelian group on the product of their underlying sets $A \times B$ by the relations

• $(a_1,b)+(a_2,b)\sim (a_1+a_2,b)$

• $(a,b_1)+(a,b_2)\sim (a,b_1+b_2)$

for all $a, a_1, a_2 \in A$ and $b, b_1, b_2 \in B$.

In words: it is the group whose elements are presented by pairs of elements in $A$ and $B$ and such that the group operation for one argument fixed is that of the other group in the other argument.

Remark

If one generalises to abelian semigroups, this definition of the tensor product also defines the tensor product of abelian semigroups.

Remark

The 0-ary relations $(0,b)\sim 0$ and $(a,0)\sim 0$ follow automatically; one needs them explicitly only if one generalises to abelian monoids.

Remark

By definition of the free construction and the quotient there is a canonical function of the underlying sets

$$p_{A,B} \;\colon\; U(A) \times U(B) \overset{}{\longrightarrow} U(A \otimes B)$$

(where $U \colon Ab \to Set$ is the forgetful functor).

On elements this sends $(a,b)$ to the equivalence class that it represents under the above equivalence relations.

The following relates the tensor product to bilinear functions. It is a definition or a proposition dependening on whether one takes the notion of bilinear function to be defined before or after that of tensor product of abelian groups.

Definition/Proposition

A function of underlying sets $f : A \times B \to C$ is a bilinear function precisely if it factors by the morphism of through a group homomorphism $\phi : A \otimes B \to C$ out of the tensor product:

$$f : A \times B \stackrel{\otimes}{\to} A \otimes B \stackrel{\phi}{\to} C \,.$$

As a quotient inductive type

In dependent type theory, the tensor product of two abelian groups $A$ and $B$ can be expressed as a quotient inductive type $A \otimes B$, generated by a function $f:A \times B \to A \otimes B$, terms, functions, and dependent identifications guaranteeing that $A \otimes B$ is an abelian group

• $0:A \otimes B$
• $+:(A \otimes B) \times (A \otimes B) \to (A \otimes B)$
• $-:(A \otimes B) \to (A \otimes B)$
• $\mathrm{assoc}(a, b, c):a + (b + c) =_{A \otimes B} (a + b) + c$ for all $a, b, c:A \otimes B$
• $\mathrm{lunit}(a):0 + a =_{A \otimes B} a$ for all $a:A \otimes B$
• $\mathrm{runit}(a):a + 0 =_{A \otimes B} a$ for all $a:A \otimes B$
• $\mathrm{linv}(a):-a + a =_{A \otimes B} 0$ for all $a:A \otimes B$
• $\mathrm{rinv}(a):a + (-a) =_{A \otimes B} 0$ for all $a:A \otimes B$
• $\mathrm{settrunc}(a, b):\mathrm{isProp}(a =_{A \otimes B} b)$ for all $a, b:A \otimes B$

and dependent identifications stating that $f$ is a bilinear map

• $\mathrm{ldist}(a_1, a_2, b):f(a_1,b)+f(a_2,b) =_{A \otimes B} f(a_1+a_2,b)$

• $\mathrm{rdist}(a, b_1, b_2):f(a, b_1)+f(a, b_2) =_{A \otimes B} f(a,b_1+b_2)$

for all $a, a_1, a_2:A$ and $b, b_1, b_2:B$.

Properties

Symmetric monoidal category structure

Proposition

Equipped with the tensor product $\otimes$ of def. and the exchange map $\sigma_{A, B}: A\otimes B \to B \otimes A$ generated by $\sigma_{A, B}(a, b) = (b, a)$, Ab becomes a symmetric monoidal category.

The unit object in $(Ab, \otimes)$ is the additive group of integers $\mathbb{Z}$.

Proof

To see that $\mathbb{Z}$ is the unit object, consider for any abelian group $A$ the map

$$A \otimes \mathbb{Z} \to A$$

which sends for $n \in \mathbb{N} \subset \mathbb{Z}$

$$(a, n) \mapsto n \cdot a \coloneqq \underbrace{a + a + \cdots + a}_{n\;summands} \,.$$

Due to the quotient relation defining the tensor product, the element on the left is also equal to

$$(a, n) = (a, \underbrace{1 + 1 \cdots + 1}_{n\; summands}) = \underbrace{ (a,1) + (a,1) + \cdots + (a,1) }_{n\; summands} \,.$$

This shows that $A \otimes \mathbb{Z} \to A$ is in fact an isomorphism.

Showing that $\sigma_{A, B}$ is natural in $A, B$ is trivial, so $\sigma$ is a braiding. $\sigma^2$ is identity, so it gives Ab a symmetric monoidal structure.

Proposition

The tensor product of abelian groups distributes over the direct sum of abelian groups

$$A \otimes \oplus_{s \in S} B_s \simeq \oplus_{s \in S} ( A \otimes B_s ) \,.$$

Monoids

Proposition

A monoid in $(Ab, \otimes)$ is equivalently a ring.

Proof

Let $(A, \cdot)$ be a monoid in $(Ab, \otimes)$. The fact that the multiplication

$$\cdot : A \otimes A \to A$$

is bilinear means by the above that for all $a_1, a_2, b \in A$ we have

$$(a_1 + a_2) \cdot b = a_1 \cdot b + a_2 \cdot b$$

and

$$b \cdot (a_1 + a_2) = b \cdot a_1 + b \cdot a_2 \,.$$

This is precisely the distributivity law of the ring.

Examples

For $n \in \mathbb{N}$ positive we write $\mathbb{Z}_n$ for the cyclic group of order $n$, as usual.

Example

For $a,b \in \mathbb{N}$ and positive, we have

$$\mathbb{Z}_a \otimes \mathbb{Z}_b \simeq \mathbb{Z}_{(a,b)} \,,$$

where $(-,-)$ denotes the greatest common divisor.

A proof is spelled out for instance as (Conrad, theorem 4.1).

Related concepts

• tensor product of commutative monoids

• tensor product of chain complexes

• tensor product of modules

• tensor product of algebras over a commutative monad

References

The original definition is due to Hassler Whitney:

• Hassler Whitney, Tensor products of Abelian groups, Duke Mathematical Journal 4 3 (1938) 495-528

  [doi:10.1215/S0012-7094-38-00442-9]

An exposition (in the case of vector spaces) is in

• Tim Gowers, How to lose your fear of tensor products, (web)

and, in the further generality of the tensor product of modules, in

• Keith Conrad, Tensor products (pdf)