Coherence for indexed symmetric monoidal categories

TL;DR

This paper extends coherence theorems to indexed symmetric monoidal categories, providing a graphical calculus to identify when operations are canonically isomorphic, which enhances understanding of their structural properties.

Contribution

It introduces a coherence theorem and a graphical calculus for indexed symmetric monoidal categories, generalizing bicategory structures and clarifying canonical isomorphisms.

Findings

Most operations are canonically isomorphic under the new coherence theorem.

The graphical calculus precisely characterizes when two operations admit a canonical isomorphism.

The results apply to important bicategories like parametrized spectra and their generalizations.

Abstract

Indexed symmetric monoidal categories are an important refinement of bicategories -- this structure underlies several familiar bicategories, including the homotopy bicategory of parametrized spectra, and its equivariant and fiberwise generalizations. In this paper, we extend existing coherence theorems to the setting of indexed symmetric monoidal categories. The most central theorem states that a large family of operations on a bicategory defined from an indexed symmetric monoidal category are all canonically isomorphic. As a part of this theorem, we introduce a rigorous graphical calculus that specifies when two such operations admit a canonical isomorphism.