Example 2.1.3.1 (00CQ)

Example 2.1.3.1. Let $k$ be a field and let $\operatorname{Vect}_{k}$ denote the category of vector spaces over $k$ (where morphisms are $k$-linear maps). For every pair of vector spaces $V,W \in \operatorname{Vect}_{k}$, let us choose a vector space $V \otimes _{k} W$ and a bilinear map

\[ V \times W \rightarrow V \otimes _{k} W \quad \quad (v,w) \mapsto v \otimes w \]

which exhibits $V \otimes _{k} W$ as a tensor product of $V$ and $W$ (see Example 2.1.0.1). The construction $(V, W) \mapsto V \otimes _{k} W$ determines a functor

\[ \otimes _{k}: \operatorname{Vect}_{k} \times \operatorname{Vect}_{k} \rightarrow \operatorname{Vect}_{k}, \]

whose value on a pair of $k$-linear maps $\varphi : V \rightarrow V'$, $\psi : W \rightarrow W'$ is characterized by the identity

\[ (\varphi \otimes _{k} \psi )(v \otimes w) = \varphi (v) \otimes \psi (w). \]

For every triple of vector spaces $U, V, W \in \operatorname{Vect}_{k}$, there is a canonical isomorphism

\[ \alpha _{U,V,W}: U \otimes _ k (V \otimes _{k} W) \xrightarrow {\sim } (U \otimes _{k} V) \otimes _{k} W, \]

characterized by the identity $\alpha _{U,V,W}( u \otimes (v \otimes w) ) = (u \otimes v) \otimes w$ for $u \in U$, $v \in V$, and $w \in W$. The pair $( \otimes _{k}, \alpha ) = ( \otimes _{k}, \{ \alpha _{U,V,W} \} _{U,V,W \in \operatorname{Vect}_{k} } )$ is then a nonunital monoidal structure on the category $\operatorname{Vect}_{k}$, in the sense of Definition 2.1.1.5. We can upgrade this to a monoidal structure by taking the unit object $\mathbf{1}$ to be the field $k$ (regarded as a vector space over itself), and the unit constraint $\upsilon : \mathbf{1} \otimes _{k} \mathbf{1} \simeq \mathbf{1}$ to be the linear map corresponding to the multiplication on $k$ (so that $\upsilon (a \otimes b) = ab$).