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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.
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:
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.
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.
By definition of the free construction and the quotient there is a canonical function of the underlying sets
(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.
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:
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 $\mathbb{Z}$.
To see that $\mathbb{Z}$ is the unit object, consider for any abelian group $A$ the map
which sends for $n \in \mathbb{N} \subset \mathbb{Z}$
Due to the quotient relation defining the tensor product, the element on the left is also equal to
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.
The tensor product of abelian groups distributes over the direct sum of abelian groups
Let $(A, \cdot)$ be a monoid in $(Ab, \otimes)$. The fact that the multiplication
is bilinear means by the above that for all $a_1, a_2, b \in A$ we have
and
This is precisely the distributivity law of the ring.
For $n \in \mathbb{N}$ positive we write $\mathbb{Z}_n$ for the cyclic group of order $n$, as usual.
For $a,b \in \mathbb{N}$ and positive, we have
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:
An exposition (in the case of vector spaces) is in
and, in the further generality of the tensor product of modules, in
Last revised on August 22, 2020 at 13:05:04. See the history of this page for a list of all contributions to it.