Normed symmetric monoidal categories

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 an NSMC is an \(N_\infty \) space. We conclude by extending our coherence theorem to include NSMCs with strict relations.