Question
Why are pivotal categories called pivotal?
What I can think of:
- In a (right) rigid monoidal category $\mathbf C$ one draws a morphism $\varphi \in Hom_{\mathbf C}(X, Y)$ in string diagrams as follows:
Duality is represented by "switching arrows around", i.e. a morphism $\phi \in Hom_{\mathbf C}(V^*, W^*)$ is drawn as follows:
Given a morphism $\varphi \in Hom_{\mathbf C}(X, Y)$ the (strict) pivotal structure lets one "pivot" its representing string diagram (turn its arrows around):
Here we make use of the identification $X^{**}=X$. (The expression $X^*$ is not ambigous because a right rigid pivotal category is left rigid, and left and right dual objects of a given object coincide.)