Complete W*-categories

Abstract

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 $⟨,{⟩}_{\mathrm{Hilb}}:\overline{C}\times C\to \mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete ${\mathrm{W}}^{\ast }$-categories there is an antilinear equivalence $$†:\mathrm{Func}(C,D)\leftrightarrow \mathrm{Func}(D,C)$$ characterised by $⟨c,F^{†}(d){⟩}_{\mathrm{Hilb}}\simeq ⟨F(c),d{⟩}_{\mathrm{Hilb}}$, for $c\in C$ and $d\in D$.

41 pages; 1 very long table of analogies between Hilbert spaces and complete W*-categories