Category of vector spaces (${\mathbf{Vect}}_k$)

For a fixed field $k$, the category of vector spaces ${\mathbf{Vect}}_k$ has $k$-vector spaces as objects and $k$-linear maps as morphisms. Composition is ordinary composition of linear maps, and each vector space has its identity map. The full subcategory of finite-dimensional vector spaces is often denoted ${\mathbf{FinVect}}_k$; allowing arbitrary dimensions gives the larger category.

After choosing bases, morphisms between finite-dimensional objects are matrices, and composition becomes ordered matrix multiplication. For example, over $\mathbb{Q}$, putThen $BA=\left(\begin{array}{cc}2& 2\\ 0& 1\end{array}\right)$ while $AB=\left(\begin{array}{cc}2& 1\\ 0& 1\end{array}\right)$. The order of arrows in a composite matters, although the category itself is defined without choosing bases.

The tensor product over $k$ gives a symmetric monoidal category with unit object $k$. Its dualizable objects are exactly the finite-dimensional vector spaces.^[ref] This tensor product differs from the categorical product: the product of two copies of $k$ is $k\oplus k$, of dimension two, while $k{\otimes }_{k}k$ has dimension one. The zero vector space is both initial and terminal.