Normed symmetric monoidal categories

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.

37 pages, significant revision. Several portions have been reorganized, a new section on NSMCs with strict relations has been added, and an old section on change of norms has been removed

Cited by in corpus (1)

• Combinatorial $N_{\infty }$ operads