nLab tensor product of abelian groups



Group Theory

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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

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,,A nBA_1, \cdots, A_n \to B.

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

Equivalently this means explicitly:


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

  • (a 1,b)+(a 2,b)(a 1+a 2,b)(a_1,b)+(a_2,b)\sim (a_1+a_2,b)

  • (a,b 1)+(a,b 2)(a,b 1+b 2)(a,b_1)+(a,b_2)\sim (a,b_1+b_2)

for all a,a 1,a 2Aa, a_1, a_2 \in A and b,b 1,b 2Bb, b_1, b_2 \in B.

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


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


The 0-ary relations (0,b)0(0,b)\sim 0 and (a,0)0(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

p A,B:U(A)×U(B)U(AB) p_{A,B} \;\colon\; U(A) \times U(B) \overset{}{\longrightarrow} U(A \otimes B)

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

On elements this sends (a,b)(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×BCf : A \times B \to C is a bilinear function precisely if it factors by the morphism of through a group homomorphism ϕ:ABC\phi : A \otimes B \to C out of the tensor product:

f:A×BABϕC. 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 AA and BB can be expressed as a quotient inductive type ABA \otimes B, generated by a function f:A×BABf:A \times B \to A \otimes B, terms, functions, and dependent identifications guaranteeing that ABA \otimes B is an abelian group

  • 0:AB0:A \otimes B
  • +:(AB)×(AB)(AB)+:(A \otimes B) \times (A \otimes B) \to (A \otimes B)
  • :(AB)(AB)-:(A \otimes B) \to (A \otimes B)
  • assoc(a,b,c):a+(b+c)= AB(a+b)+c\mathrm{assoc}(a, b, c):a + (b + c) =_{A \otimes B} (a + b) + c for all a,b,c:ABa, b, c:A \otimes B
  • lunit(a):0+a= ABa\mathrm{lunit}(a):0 + a =_{A \otimes B} a for all a:ABa:A \otimes B
  • runit(a):a+0= ABa\mathrm{runit}(a):a + 0 =_{A \otimes B} a for all a:ABa:A \otimes B
  • linv(a):a+a= AB0\mathrm{linv}(a):-a + a =_{A \otimes B} 0 for all a:ABa:A \otimes B
  • rinv(a):a+(a)= AB0\mathrm{rinv}(a):a + (-a) =_{A \otimes B} 0 for all a:ABa:A \otimes B
  • settrunc(a,b):isProp(a= ABb)\mathrm{settrunc}(a, b):\mathrm{isProp}(a =_{A \otimes B} b) for all a,b:ABa, b:A \otimes B

and dependent identifications stating that ff is a bilinear map

  • ldist(a 1,a 2,b):f(a 1,b)+f(a 2,b)= ABf(a 1+a 2,b)\mathrm{ldist}(a_1, a_2, b):f(a_1,b)+f(a_2,b) =_{A \otimes B} f(a_1+a_2,b)

  • rdist(a,b 1,b 2):f(a,b 1)+f(a,b 2)= ABf(a,b 1+b 2)\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:Aa, a_1, a_2:A and b,b 1,b 2:Bb, b_1, b_2:B.


Symmetric monoidal category structure


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

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


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

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

which sends for nn \in \mathbb{N} \subset \mathbb{Z}

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

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

Showing that σ A,B\sigma_{A, B} is natural in A,BA, B is trivial, so σ\sigma is a braiding. σ 2\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

A sSB s sS(AB s). A \otimes \oplus_{s \in S} B_s \simeq \oplus_{s \in S} ( A \otimes B_s ) \,.



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


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

:AAA \cdot : A \otimes A \to A

is bilinear means by the above that for all a 1,a 2,bAa_1, a_2, b \in A we have

(a 1+a 2)b=a 1b+a 2b (a_1 + a_2) \cdot b = a_1 \cdot b + a_2 \cdot b


b(a 1+a 2)=ba 1+ba 2. b \cdot (a_1 + a_2) = b \cdot a_1 + b \cdot a_2 \,.

This is precisely the distributivity law of the ring.


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


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

a b (a,b), \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:

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)

Last revised on December 23, 2023 at 18:11:09. See the history of this page for a list of all contributions to it.