Unitary fusion category

Let $\mathcal{C}$ be a fusion category over $\mathbb{C}$ in the sense of Fusion and multifusion categories. A dagger structure assigns to every morphism $f:X\to Y$ a morphism $f^{†}:Y\to X$, conjugate-linearly, with $(g\circ f{)}^{†}=f^{†}\circ g^{†}$ and $(f^{†}{)}^{†}=f$. A unitary fusion category is such a fusion category equipped with this dagger so that every morphism space is a finite-dimensional normed space, composition is submultiplicative, $\mathrm{\parallel }{f}^{†}\mathrm{\parallel }=\mathrm{\parallel }f\mathrm{\parallel }$, and $\mathrm{\parallel }{f}^{†}f\mathrm{\parallel }=\mathrm{\parallel }f{\mathrm{\parallel }}^2$. Thus each ${\mathrm{End}}_{\mathcal{C}}(X)$ is a finite-dimensional $C^{\ast }$-algebra. Positive endomorphisms are the elements of its usual $C^{\ast }$-positive cone (equivalently $u^{†}u$ for an endomorphism $u$); in particular, if $c$ is simple and $\mathrm{End}(c)=\mathbb{C}{\mathrm{id}}_c$, then $\lambda {\mathrm{id}}_c$ is positive exactly when $\lambda \ge 0$. For any morphism $f:X\to Y$, $f^{†}f$ is positive in $\mathrm{End}(X)$; also require $f^{†}f=0$ to imply $f=0$. Semisimple direct-sum decompositions can be chosen orthogonally: their inclusions $i_j$ satisfy $i_j^{†}{i}_k=0$ for $j\mathrm{\ne }k$, $i_j^{†}{i}_j=\mathrm{id}$, and ${\sum }_j{i}_j{i}_j^{†}=\mathrm{id}$. Tensor product preserves daggers: $(f\otimes g{)}^{†}=f^{†}\otimes g^{†}$. Its associator and unit constraints are unitary, meaning ${\alpha }^{†}={\alpha }^{-1}$, ${\lambda }^{†}={\lambda }^{-1}$, and ${\rho }^{†}={\rho }^{-1}$.

The underlying fusion category is rigid, so its evaluation and coevaluation morphisms satisfy the duality zig-zag identities. Those morphisms are not required to be unitary isomorphisms. A braiding is additional structure; when it is specified as a unitary braiding, each component satisfies $c_{X,Y}^{†}=c_{X,Y}^{-1}$. Positivity and unitarity are extra structure, not consequences of the fusion rules or of the braid relations alone.