Relative tensor products of bimodules over C times C

Finite-dimensional complex vector spaces form a finite semisimple category, their tensor unit $\mathbb{C}$ is simple, and vector-space duals give rigidity; thus they satisfy the fusion conditions in [F2]. The flip $v\otimes w\mapsto w\otimes v$ is natural and invertible, and on triples both hexagon composites perform the same adjacent transpositions to reverse the order past one tensor factor; hence it is a braiding as in [F3]. The element $E=e_1\otimes e_1+e_2\otimes e_2$ satisfies $m_R(E)=e_1+e_2=1_R$ and $rE=Er$ for every $r\in R$, so it is a separability idempotent. On $M_{ij}\otimes M_{kl}\cong \mathbb{C}$, the projector ${\sum }_{s=1}^2(r_{e_s}\otimes l_{e_s})$ acts by the scalar ${\sum }_s{\delta }_{j,s}{\delta }_{k,s}$; this is $1$ when $j=k$ and $0$ when $j\mathrm{\ne }k$. By the image description, $M_{ij}{\otimes }_R{M}_{kl}\cong \{\begin{array}{ll}{M}_{il},& j=k,\\ 0,& j\mathrm{\ne }k\mathrm{.}\end{array}$

Applying the rule twice gives $(M_{ij}{\otimes }_R{M}_{kl}){\otimes }_R{M}_{mn}\cong {\delta }_{j,k}{\delta }_{l,m}{M}_{in}\cong M_{ij}{\otimes }_R(M_{kl}{\otimes }_R{M}_{mn}),$ where ${\delta }_{u,v}$ is $1$ if $u=v$ and $0$ otherwise. The regular bimodule decomposes as $R=M_{11}\oplus M_{22}$ and acts as a unit on every $M_{ij}$. These explicit products verify the associativity and unit rules in this finite split-semisimple example.