Category of vector spaces

In mathematics, especially category theory, the category K-Vect (some authors use Vect_K) has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms. If K is the field of real numbers, then the category is also known as Vec.

Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod, the category of left R-modules. K-Vect is an important example of an abelian category.

Much of linear algebra concerns the description of K-Vect. For example, the dimension theorem for vector spaces says that the isomorphism classes in K-Vect correspond exactly to the cardinal numbers, and that K-Vect is equivalent to the subcategory of K-Vect which has as its objects the free vector spaces Kⁿ, where n is any cardinal number.

There is a forgetful functor from K-Vect to Ab, the category of abelian groups, which takes each vector space to its additive group. This can be composed with forgetful functors from Ab to yield other forgetful functors, most importantly one to Set.

K-Vect is a monoidal category with K (as a one-dimensional vector space over K) as the identity and the tensor product as the monoidal product.

See also

• Category of graded vector spaces

References

1. Mac Lane, Saunders (September 1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (second ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001.