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In mathematics, the category K-Vect has all vector spaces over a fixed field K as objects and linear transformations as morphisms. 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 subcategoryA subcategory in Wikipedia is a category that depends on another category. See Categorization In mathematics, a subcategory of a category C consists of subsets of the morphisms and of the objects of C such that the subset X of morphisms is closed under co of K-Vect which has as its objects the free vector spaces Kn, where n is any cardinal number.
There is a forgetful functorA forgetful functor is a type of functor in mathematics. The nomenclature is suggestive of such a functor's behaviour: given some algebraic object as input, some or all of the object's structure is 'forgotten' in the output. For an algebraic structure of from K-Vect to Ab, the category of abelian groupsIn mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group, 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 SetCategory theory In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics. The category is usually denoted simply as Set ..
K-Vect is a monoidal categoryIn mathematics, a strict monoidal category is a category with a product operation × on objects that has properties analogous to those of the tensor product. The product is assumed to be associative, and have an left and right identity, I. The correspondin with K (as a one dimensional vector space over K) as the identity and the tensor productAbstract algebra Algebra In mathematics, the tensor product denoted by , may be applied in different contexts to vectors, matrices, tensors and vector spaces. In each case the significance of the symbol is the same: the most general bilinear operation. as the monoidal product.
See also: K-Z_2Vect
Category theory
Linear algebra
