Normed symmetric monoidal categories

TL;DR

This paper introduces normed symmetric monoidal categories as models for $N_ abla$ spaces, connecting equivariant symmetric monoidal categories and extending Mac Lane's coherence theorem.

Contribution

It develops the concept of NSMCs, provides an operadic interpretation of coherence, and links these structures to $N_ abla$ spaces and existing equivariant categories.

Findings

NSMCs serve as models for $N_ abla$ spaces.

The coherence theorem ensures classifying spaces of NSMCs are $N_ abla$ spaces.

Extended coherence results for NSMCs with strict relations.

Abstract

We introduce categorical models of $N_\infty$ spaces, which we call normed symmetric monoidal categories (NSMCs). These are ordinary symmetric monoidal categories equipped with compatible families of norm maps, and when specialized to a particular class of examples, they reveal a connection between the equivariant symmetric monoidal categories of Guillou-May-Merling-Osorno and those of Hill-Hopkins. We also give an operadic interpretation of the Mac Lane coherence theorem and generalize it to include NSMCs. Among other things, this theorem ensures that the classifying space of a NSMC is a $N_\infty$ space. We conclude by extending our coherence theorem to include NSMCs with strict relations.