= Category of vector spaces
{title2=$\mathbf{Vect}_k$}
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 $\mathbb Q$, put
$$
A=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad B=\begin{pmatrix}2&0\\0&1\end{pmatrix}.
$$
Then $BA=\left(\begin{smallmatrix}2&2\\0&1\end{smallmatrix}\right)$ while $AB=\left(\begin{smallmatrix}2&1\\0&1\end{smallmatrix}\right)$. 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.<On the Classification of Topological Field Theories by Jacob Lurie>[Example 1.1.4 and Example 2.3.4]{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.
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