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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.

A fully dualizable object in a symmetric monoidal $(\infty ,n)$-category has a categorical dual together with adjoints for the evaluation and coevaluation data, recursively through the required higher levels. The meaning depends on $n$. For $n=1$, full dualizability is just ordinary dualizability. For $n=2$, the evaluation morphism and coevaluation morphism must have both left and right adjoints. This is a finiteness condition expressed through composition and the monoidal tensor product.^[ref]

In the ordinary symmetric monoidal category of vector spaces over $k$, a finite-dimensional vector space $V$ has dual $V^{\ast }$, evaluation $v\otimes \lambda \mapsto \lambda (v)$ and coevaluationThe dual-basis identities make the two resulting zigzag composites the identity. Conversely, any algebraic tensor used as a coevaluation is a finite sum, so these identities force the whole space to be spanned by finitely many vectors. Infinite-dimensional vector spaces have algebraic duals but are not dualizable in this monoidal category.^[ref]

The cobordism hypothesis identifies fully dualizable objects as the possible values on a point of fully extended framed topological quantum field theories. The higher adjoints supply the compatibility needed when bordisms are cut and glued in multiple dimensions.