Kerodon

Example 2.1.4.5. Let $k$ be a field and let $\operatorname{Vect}_ k$ denote the category of vector spaces over $k$, endowed with the monoidal structure of Example 2.1.3.1. The construction of this monoidal structure involved certain choices: for every pair of vector spaces $U,V \in \operatorname{Vect}_{k}$, we selected a universal $k$-bilinear map $b_{U,V}: U \times V \rightarrow U \otimes _{k} V$. The collection of functions $b = \{ b_{U,V} \} _{U,V \in \operatorname{Vect}_{k} }$ is then a nonunital lax monoidal structure on the forgetful functor $\operatorname{Vect}_{k} \rightarrow \operatorname{Set}$ (where we equip $\operatorname{Set}$ with the monoidal structure given by cartesian products; see Example 2.1.3.2). Note that the tensor product functor $\otimes _{k}: \operatorname{Vect}_ k \times \operatorname{Vect}_ k \rightarrow \operatorname{Vect}_ k$ is characterized by the requirement that it is given on objects by $(U,V) \mapsto U \otimes _{k} V$ and satisfies condition $(a)$ of Definition 2.1.4.3, and the associativity constraint on $\operatorname{Vect}_ k$ is characterized by the requirement that it satisfies condition $(b)$ of Definition 2.1.4.3. Note that $b$ is not a nonunital monoidal structure: the bilinear maps $b_{U,V}: U \times V \rightarrow U \otimes _{k} V$ are never bijective, except in the trivial case where $U \simeq 0 \simeq V$.