Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
For and two abelian groups, their tensor product is a new abelian group such that a group homomorphism is equivalently a bilinear map out of and .
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 .
The tensor product in this multicategory is the tensor product of abelian groups.
Equivalently this means explicitly:
For two abelian groups, their tensor product of abelian groups is the abelian group which is the quotient of the free abelian group on the product of their underlying sets by the relations
for all and .
In words: it is the group whose elements are presented by pairs of elements in and 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 and 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 is the forgetful functor).
On elements this sends 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 is a bilinear function precisely if it factors by the morphism of through a group homomorphism out of the tensor product:
In dependent type theory, the tensor product of two abelian groups and can be expressed as a quotient inductive type , generated by a function , terms, functions, and dependent identifications guaranteeing that is an abelian group
and dependent identifications stating that is a bilinear map
for all and .
Equipped with the tensor product of def. and the exchange map generated by , Ab becomes a symmetric monoidal category.
The unit object in is the additive group of integers .
To see that is the unit object, consider for any abelian group the map
which sends for
Due to the quotient relation defining the tensor product, the element on the left is also equal to
This shows that is in fact an isomorphism.
Showing that is natural in is trivial, so is a braiding. 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 be a monoid in . The fact that the multiplication
is bilinear means by the above that for all we have
and
This is precisely the distributivity law of the ring.
For positive we write for the cyclic group of order , as usual.
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 December 23, 2023 at 18:11:09. See the history of this page for a list of all contributions to it.