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EspañolCATEGORY OF VECTOR SPACES
(Redirected from K-Vect)
In mathematics, especially category theory, the category 'K-Vect' 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 ''K''''n'', 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.
★ Category of graded vector spaces
In mathematics, especially category theory, the category 'K-Vect' 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 ''K''''n'', 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.
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See also
★ Category of graded vector spaces
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