A non-invertible direct sum in Vec_Z2

Every graded vector space is the direct sum of its two finite-dimensional homogeneous components, each of which is a finite direct sum of one-dimensional graded lines. Thus the category is semisimple with exactly two simple isomorphism classes, represented by ${\delta }_0$ and ${\delta }_1$; its Hom spaces are finite-dimensional and its kernels and cokernels inherit the grading. The displayed tensor product is bilinear and has unit ${\delta }_0$. Its associator is the ordinary vector-space rebracketing map, which preserves degree; the two pentagon composites send every pure tensor to the same rebracketed tensor, and the unit triangle follows from the canonical identifications with $\mathbb{C}\otimes W$ and $W\otimes \mathbb{C}$. The unit ${\delta }_0$ is simple.

The four dual maps displayed in the Example preserve degree: each evaluation pairs only a component of degree $a$ with its dual component of degree $-a$, and each coevaluation is a sum of tensors of total degree $0$. With the vector-space associator and unit identifications, the two left snake composites act by $w\mapsto {\sum }_{a,i}{e}^{a,i}(w)e_{a,i}=w$ and $\varphi \mapsto {\sum }_{a,i}\varphi (e_{a,i})e^{a,i}=\varphi$. The two right snake composites act by the same contractions, with evaluation and coevaluation in the reversed map orientations: $w\mapsto {\sum }_{a,i}{e}^{a,i}(w)e_{a,i}=w$ and $\varphi \mapsto {\sum }_{a,i}\varphi (e_{a,i})e^{a,i}=\varphi$. Thus every object has both left and right duals, and the category has the fusion properties in [F1].

Distributing the tensor product over direct sums gives $({\delta }_0\oplus {\delta }_1)\otimes ({\delta }_0\oplus {\delta }_1)\cong ({\delta }_0\otimes {\delta }_0)\oplus ({\delta }_0\otimes {\delta }_1)\oplus ({\delta }_1\otimes {\delta }_0)\oplus ({\delta }_1\otimes {\delta }_1)\cong 2{\delta }_0\oplus 2{\delta }_1,$ since ${\delta }_a\otimes {\delta }_b\cong {\delta }_{a+b}$.

Any object $X$ has the form ${\delta }_0^{\oplus a}\oplus {\delta }_1^{\oplus b}$ for nonnegative integers $a,b$. Hence $D\otimes X\cong (a+b){\delta }_0\oplus (a+b){\delta }_1\mathrm{.}$ If $a+b=0$ this is the zero object; if $a+b>0$ it has a nonzero grade-$1$ summand. In neither case is it isomorphic to the unit ${\delta }_0$. Therefore $D$ has no right tensor inverse and is non-invertible by [F1], while ${\delta }_0$ and ${\delta }_1$ each have themselves as two-sided inverses.