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Abstract

Let G be a finite, discrete group. This thesis studies equivariant symmetric monoidal G-categories and the operads that parametrize them. We devise explicit tools for working with these objects, and then we use them to tackle two conjectures of Blumberg and Hill and a presentation problem of Guillou-May-Merling-Osorno, with varying degrees of success. ,The first half of this thesis introduces normed symmetric monoidal categories, and develops their basic theory. These are direct generalizations of the classical structures, and they are presented by generators and isomorphism relations. We explain how to construct an operad action from these generators via an equivariant version of the Kelly-Mac Lane coherence theorem, and then we study the resulting operads in their own right. We show that the operads for normed symmetric monoidal categories are precisely the cell complexes in a certain model structure, and that they are cofibrant replacements for the commutativity operad in a family of other model structures. Our work resolves a conjecture of Blumberg and Hill on the classification of N-infinity operads in the affirmative. Finally, we prove a number of homotopy invariance results for the structures under consideration. We show that weak equivalences between certain categorical N-infinity operads induce equivalences on the level of algebras, and that pseudoalgebras over such operads are strict algebras over larger, equivalent operads. We deduce that the symmetric monoidal G-categories of Guillou-May-Merling-Osorno are equivalent to E-infinity normed symmetric monoidal categories.,The second half of this thesis studies a number of examples. We explain how to construct normed symmetric monoidal structures by twisting a given operation over a diagram, and we examine a shared link between the symmetric monoidal G-categories of Guillou-May-Merling-Osorno and the G-symmetric monoidal categories of Hill and Hopkins. We give functorial constructions of N-infinity operads, and we examine how the lattice of indexing systems is reflected on the level of operads. We prove a combinatorial analogue to a conjecture of Blumberg and Hill on the Boardman-Vogt tensor product of N-infinity operads, and while our work does not solve their original problem, it does imply a space-level interchange result.

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