Fusion 2-category

A fusion 2-category is a finite semisimple linear 2-category with a weak monoidal structure, a simple monoidal unit, and left and right duals for every object. For a right dual $A^{\mathrm{\#}}$ of $A$, the duality data include 1-morphisms $e_A:A\otimes A^{\mathrm{\#}}\to I,i_A:I\to A^{\mathrm{\#}}\otimes A,$ and invertible cusp 2-morphisms from $(e_A\otimes A)\circ (A\otimes i_A)$ to $1_A$ and from $(A^{\mathrm{\#}}\otimes e_A)\circ (i_A\otimes A^{\mathrm{\#}})$ to $1_{A^{\mathrm{\#}}}$, with the associator and unitors inserted at the indicated sources and targets.

For a left dual ${}^{\mathrm{\#}}A$ of $A$, the data separately include 1-morphisms $e_A^{\mathrm{\ell }}:{}^{\mathrm{\#}}A\otimes A\to I,i_A^{\mathrm{\ell }}:I\to A\otimes {}^{\mathrm{\#}}A,$ and invertible cusp 2-morphisms from $(A\otimes e_A^{\mathrm{\ell }})\circ (i_A^{\mathrm{\ell }}\otimes A)$ to $1_A$ and from $(e_A^{\mathrm{\ell }}\otimes {}^{\mathrm{\#}}A)\circ ({}^{\mathrm{\#}}A\otimes i_A^{\mathrm{\ell }})$ to $1_{{}^{\mathrm{\#}}A}$. The associator and unitors are inserted at these sources and targets as well. No identification of the chosen left and right duals is required.

Here “simple unit” means that $1_I$ is a simple 1-morphism. A pivotal or spherical enhancement is extra structure: it requires its own duality coherence and trace axioms. Neither such enhancement nor a spherical trace is included in “fusion 2-category.”