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Relative and lax volutive categories

Tim Lüders

Abstract

In this paper we introduce the notion of a relative volutive (higher) category, specializing to the notion of a lax volutive (higher) category. Our primary motivation to study these objects is the following: while any rigid symmetric monoidal category admits a volutive structure, any closed symmetric monoidal category admits a lax volutive structure. We develop some of the basic theory of relative volutive categories and provide several equivalent formulations of lax volutive categories. We then study examples of interest, including categories of complete bornological vector spaces and modules over star-rings. We will also separately discuss unbounded operators between Hilbert spaces and Morita 2-categories, the latter of which in the context of fully closed symmetric monoidal 2-categories.

Relative and lax volutive categories

Abstract

In this paper we introduce the notion of a relative volutive (higher) category, specializing to the notion of a lax volutive (higher) category. Our primary motivation to study these objects is the following: while any rigid symmetric monoidal category admits a volutive structure, any closed symmetric monoidal category admits a lax volutive structure. We develop some of the basic theory of relative volutive categories and provide several equivalent formulations of lax volutive categories. We then study examples of interest, including categories of complete bornological vector spaces and modules over star-rings. We will also separately discuss unbounded operators between Hilbert spaces and Morita 2-categories, the latter of which in the context of fully closed symmetric monoidal 2-categories.
Paper Structure (30 sections, 64 theorems, 153 equations)

This paper contains 30 sections, 64 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $\operatorname{O(1)}$-volutive and dagger categories
  4. $\operatorname{O(1)}$-volutive 2-categories and dagger categories
  5. The 2-categories of $\operatorname{O(1)}$-volutive and dagger categories
  6. Relative $\operatorname{O(1)}$-volutive categories
  7. Relative $\operatorname{O(1)}$-volutive categories
  8. Lax $\operatorname{O(1)}$-volutive categories
  9. Lax $\operatorname{O(1)}$-volutive structures are relative
  10. Relative $\operatorname{SO(2)}$-volutive 2-categories
  11. Adjoints in 2- and 3-categories
  12. $\operatorname{SO(2)}$-volutive 2-categories
  13. Relative $\operatorname{SO(2)}$-volutive 2-categories
  14. Lax $\operatorname{SO(2)}$-volutive 2-categories
  15. Lax hermitian fixed points
  16. ...and 15 more sections

Key Result

Theorem 1

Any closed symmetric monoidal category $\mathcal{C}$ admits a lax $\operatorname{O(1)}$-volutive structure.

Theorems & Definitions (297)

  • Theorem 1
  • Example 1
  • Theorem 2
  • Definition 1
  • Theorem 3
  • Definition 2.1.1
  • Example 2.1.2
  • Proposition 2.1.3
  • Remark 2.1.4
  • Remark 2.1.5
  • ...and 287 more