The ribbon evaluation of an $X$-colored closed braid

Let $\mathcal{C}$ be a ribbon category with chosen left duals, twist $\theta$ (Twist and ribbon structure) and Drinfeld morphism $u_X\text{\hspace{0.17em}⁣}:X\to X^{\vee \vee }$ of A braided rigid category has a Drinfeld morphism. Put

$j_X:=u_X{\theta }_X\text{\hspace{0.17em}⁣}:X⟶X^{\vee \vee }\mathrm{.}$

Then $j$ is a natural isomorphism: it is the composite of the natural isomorphism $u$ with the natural automorphism $\theta$ of the identity. It is monoidal up to the monoidal comparison $d_{X,Y}\text{\hspace{0.17em}⁣}:X^{\vee \vee }\otimes Y^{\vee \vee }\to (X\otimes Y{)}^{\vee \vee }$ of the double-dual functor,

$j_{X\otimes Y}=d_{X,Y}\circ (j_X\otimes j_Y),$

which is precisely the statement that $u\theta$ is the pivotal comparison induced by the ribbon structure; the identity follows from the tensor relation $d_{X,Y}\circ (u_X\otimes u_Y)=u_{X\otimes Y}\circ c_{Y,X}\circ c_{X,Y}$ for the Drinfeld morphism together with the twist axiom ${\theta }_{X\otimes Y}=({\theta }_X\otimes {\theta }_Y)\circ c_{Y,X}\circ c_{X,Y}$ and the naturality of $\theta$. Iterating the comparison identifies $j_{X^{\otimes n}}$ with the corresponding composite of the $j_X$ and the coherence isomorphisms of the tensor power.

Let $n\ge 1$ and $\beta \in B_n$ (The braid group by Artin presentation), and let ${\rho }_n\text{\hspace{0.17em}⁣}:B_n\to {\mathrm{Aut}}_{\mathcal{C}}(X^{\otimes n})$ be the canonical braid action (An object of a braided category carries canonical braid actions). The ribbon evaluation of the $X$-colored closed braid is the value

$t_n(\beta ):={\mathrm{Tr}}_L\text{\hspace{0.17em}⁣}(j_{X^{\otimes n}}\circ {\rho }_n(\beta ))\in {\mathrm{End}}_{\mathcal{C}}(\mathbf{1}),$

in the sense of The categorical trace of a morphism into the double dual: the composite $j_{X^{\otimes n}}\circ {\rho }_n(\beta )$ is a morphism $X^{\otimes n}\to (X^{\otimes n}{)}^{\vee \vee }$, which is exactly the input type of the left categorical trace. When ${\mathrm{End}}_{\mathcal{C}}(\mathbf{1})=k$ the value $t_n(\beta )$ is a scalar. For $n=1$ the group $B_1$ is trivial and $t_1(e)={\mathrm{Tr}}_L(j_X)$, the left dimension of $X$.

For $n=0$ put $t_0(e)=1_{\mathbf{1}}$, the evaluation of the empty closed tangle. This is a separate convention; no generator or dual pairing is needed for the empty diagram.

The identification of $t_n(\beta )$ with the evaluation $F_X({\widehat{\beta }}^{\mathrm{f}\mathrm{r}})$ of the blackboard-framed closure of $\beta$ under the functor of A ribbon object defines a unique framed-tangle evaluation functor is the content of the closure-comparison lemma of this page; it must be proved before the trace is used as a link evaluation, and it is not assumed here. The functor $F_X$ is constructed under countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), while the definition of $t_n$ above is choice-free and uses neither that functor nor that principle; only the comparison, not the trace, carries the choice cost.