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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2017
- DOI:
- 10.48550/arXiv.1708.04777
- arXiv:
- arXiv:1708.04777
- Bibcode:
- 2017arXiv170804777R
- Keywords:
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- Mathematics - Algebraic Topology;
- Mathematics - Category Theory;
- 55P91 (Primary) 18D10 (Secondary)
- E-Print:
- 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