Complete W*-categories

TL;DR

This paper explores the structure of complete W*-categories, demonstrating their similarities to categorified Hilbert spaces and establishing a canonical inner product and duality functor between functor categories.

Contribution

It introduces a canonical categorified inner product for W*-categories and characterizes a duality equivalence between functor categories of complete W*-categories.

Findings

Complete W*-categories behave like categorified Hilbert spaces.

Existence of a canonical categorified inner product.

Duality functor between functor categories is characterized by inner product.

Abstract

We study $\mathrm{W}^*$-categories, and explain the ways in which complete $\mathrm{W}^*$-categories behave like categorified Hilbert spaces. Every $\mathrm{W}^*$-category $C$ admits a canonical categorified inner product $\langle\,\,,\,\rangle_{\mathrm{Hilb}}\,:\,\overline C\times C\,\to\, \mathrm{Hilb}$. Moreover, if $C$ and $D$ are complete $\mathrm{W}^*$-categories there is an antilinear equivalence $$\dagger:\mathrm{Func}(C,D) \leftrightarrow \mathrm{Func}(D,C)$$ characterised by $\langle c,F^\dagger(d)\rangle_{\mathrm{Hilb}} \simeq \langle F(c),d\rangle_{\mathrm{Hilb}}$, for $c\in C$ and $d \in D$.