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 *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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—Vladimir Ilyich Lenin (1870–1924)