Normed symmetric monoidal categories
Jonathan Rubin

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 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 space. We conclude by extending our coherence theorem to include NSMCs with strict relations.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Intracerebral and Subarachnoid Hemorrhage Research
