Negligible morphisms in the Temperley–Lieb category

Work over $\mathbb{C}$ with a fixed loop parameter $\delta$. Let ${\mathcal{T}}_{\delta }$ be the strict Temperley–Lieb diagram category of Temperley–Lieb planar diagram data and endomorphism modules: its objects are $n\in \mathbb{N}$, and its morphism space ${\mathrm{Hom}}_{{\mathcal{T}}_{\delta }}(m,n)$ is the complex span of the noncrossing matching diagrams with $m$ bottom and $n$ top boundary points, composed by stacking and loop removal.

For a basis endomorphism $D:n\to n$, let $c(D)$ be the number of components after joining each top point to the corresponding bottom point outside the rectangle. Define the unnormalized closure trace by ${\mathrm{Tr}}_n(D)={\delta }^{c(D)},$ and extend linearly. This definition is valid for every $\delta \in \mathbb{C}$, including $\delta =0$.

For $m,n\in \mathbb{N}$, the subspace of negligible morphisms is ${\mathcal{N}}_{\delta }(m,n)=\{f\in {\mathrm{Hom}}_{{\mathcal{T}}_{\delta }}(m,n):{\mathrm{Tr}}_m(g\circ f)=0\text{ for every }g\in {\mathrm{Hom}}_{{\mathcal{T}}_{\delta }}(n,m)\}\mathrm{.}$ This definition specifies subspaces only. Their closure under composition and tensor product, and hence the quotient category, are supplied separately.