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Complete W*-categories

André Henriques, Nivedita, David Penneys

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 $d \in D$.

Complete W*-categories

Abstract

We study -categories, and explain the ways in which complete -categories behave like categorified Hilbert spaces. Every -category admits a canonical categorified inner product . Moreover, if and are complete -categories there is an antilinear equivalence characterised by , for and .

Paper Structure

This paper contains 10 sections, 61 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Table of analogies
  3. Summary of notations and definitions:
  4. Acknowledgements
  5. Hilbert spaces and von Neumann algebras
  6. W*-categories: basic properties and operations
  7. The Hilb-valued inner product
  8. Positive cones
  9. Small W*-categories
  10. W*-tensor categories

Key Result

Theorem 1

There is an equivalence of bi-involutive bicategories $\mathrm{vN2}\stackrel{\simeq}\to\mathrm{W^*Cat}$ between the bicategory of von Neumann algebras, bimodules, and intertwiners, and the bicategory of complete ${\rm W}^*$-categories admitting a set of generators, functors, and natural transformati

Theorems & Definitions (144)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • Corollary 2.7
  • Lemma 2.8: MR0407615
  • ...and 134 more