Relative and lax volutive categories

TL;DR

This paper introduces the concepts of relative and lax volutive (higher) categories, exploring their theoretical foundations and examples, especially in the context of symmetric monoidal categories and operator theory.

Contribution

It defines and develops the theory of relative and lax volutive categories, providing equivalent formulations and exploring key examples in functional analysis and higher category theory.

Findings

Established basic theory of relative volutive categories

Provided equivalent formulations of lax volutive categories

Analyzed examples including bornological vector spaces and operator 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.