The free compact closure of a symmetric monoidal category

TL;DR

This paper presents a method to construct a compact closed category from any symmetric monoidal category by freely adding adjoints, using string diagrams, and explores the relationship with the Int construction.

Contribution

It introduces a new free construction of compact closed categories from symmetric monoidal categories, extending the Int construction, and analyzes the embedding properties.

Findings

The embedding into the completion is faithful but not full.

The construction uses string diagrams annotated by original category elements.

The approach generalizes the free traced monoidal category via the Int construction.

Abstract

We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original category. The symmetric monoidal category embeds via a faithful monoidal functor into its completion, but in contrast to the non-symmetric case, this embedding is not full. Our construction factors through the Int construction, which yields another free construction: the free traced monoidal category on a symmetric monoidal category.