Negligible morphisms for a pivotal trace

Let $\mathcal{C}$ be an additive $\mathbb{C}$-linear rigid monoidal category with ${\mathrm{End}}_{\mathcal{C}}(\mathbf{1})=\mathbb{C}$, and fix a pivotal structure $j_X:X\to X^{\vee \vee }$ as in Pivotal structure. For $f:X\to Y$, call $f$ negligible if

${\mathrm{Tr}}_L(j_X\circ g\circ f)=0\text{for every }g:Y\to X,$

where the left trace is The categorical trace of a morphism into the double dual. The negligible morphisms form the intersection of the kernels of these linear functionals; this definition alone does not assert tensor-ideal closure.

For the complex Temperley–Lieb diagram category, use its cup/cap pivotal structure. The resulting trace is exactly the unnormalized planar closure of Unnormalized diagram closure and partial trace. The same definition applies to the complex type-1 tilting category of The specified rank-one divided-power tilting category when its stated ribbon pivotal structure is used.