Compact closed categories and $\Gamma$-categories (with an appendix by Andr\'e Joyal)

TL;DR

This paper develops new model category structures to study compact closed categories within homotopical algebra, showing that fibrant objects correspond to compact closed categories and establishing a fibrant replacement for free symmetric monoidal categories.

Contribution

It introduces two localized model category structures that characterize compact closed categories as fibrant objects, linking homotopical algebra with categorical structures.

Findings

Constructed two new model categories for symmetric monoidal categories.

Identified fibrant objects as compact closed categories.

Established fibrant replacement of free symmetric monoidal categories.

Abstract

In this paper we study compact closed categories within the context of homotopical algebra. We construct two new model category structures by localizing two (Quillen equivalent) model categories of symmetric monoidal categories with the objective of establishing the free compact closed category on one generator as a fibrant replacement of the free symmetric monoidal category on one generator, in our localized model categories. We go on to show that the fibrant objects in our model categories are compact closed categories.