# nLab tensor product of abelian groups

Contents

group theory

### Cohomology and Extensions

#### Monoidal categories

monoidal categories

# Contents

## 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).

The original definition is due to Hassler Whitney:

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

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)