Complete W*-categories

We study ${\mathrm{W}}^{\ast }$-categories, and explain the ways in which complete ${\mathrm{W}}^{\ast }$-categories behave like categorified Hilbert spaces. Every ${\mathrm{W}}^{\ast }$-category $C$ admits a canonical categorified inner product $⟨,{⟩}_{\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}:\bar{C}\times C\to \mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}$. Moreover, if $C$ and $D$ are complete ${\mathrm{W}}^{\ast }$-categories there is an antilinear equivalence $$†:\mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}(C,D)\leftrightarrow \mathrm{F}\mathrm{u}\mathrm{n}\mathrm{c}(D,C)$$ characterised by $⟨c,F^{†}(d){⟩}_{\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}\simeq ⟨F(c),d{⟩}_{\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}$, for $c\in C$ and $d\in D$.