Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories

TL;DR

This paper explores the categorical structures underpinning quantum mechanics, focusing on strongly compact closed categories, scalar notions, and free constructions, with emphasis on geometric and combinatorial methods.

Contribution

It introduces a detailed analysis of free constructions for strongly compact closed categories and their scalar structures, advancing categorical quantum mechanics theory.

Findings

Strongly compact closed categories support a scalar notion for quantum theory.

Extended free constructions for complex structured categories are developed.

Variations with prescribed scalar monoids are discussed.

Abstract

We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category. We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be "glued in" to the free construction.