Equivariant higher algebra

Abstract

In this thesis, we set out foundations for a homotopy-coherently parameterized algebra suited towards equivariant homotopy theory and algebraic topology. In particular, the first chapter after the introduction introduces parameterized (∞, 2)-category theory and a theory of parameterized algebraic patterns lifting the theory of Chu-Haugseng; intuivitvely, for base of parameterization the orbit ∞-category O_G, a G-(∞, 2)-category is an ∞-category enriched in ∞-categories enriched in G-spaces (i.e. enriched in G-∞-categories) and O_G-parameterized algebraic patterns are G-∞-categorical objects which functorially present G-(∞, 2)-categories of generalized G-operads and G-symmetric monoidal ∞-categories via the fibrational perspective. We present a Morita theory for such parameterized patterns lifting the approximation theory of Lurie (and Barkan, Barkan-Haugseng-Steinebrunner) and apply it to identify many models for parameterized operads, compatibly with the corresponding theories of parameterized symmetric monoidal categories (considered as normed algebras in ∞-categories). In the second chapter, we apply this to develop some rudiments of equivariant higher algebra. In particular, we center the roles of (co)cartesian G-symmetric monoidal structures, parameterized Segal-object models for algebras, Boardman-Vogt tensor products/homotopy-coherent interchange, and parameterized distributivity. At the end of the day, we use these to understand a parameterized version of Lurie, Gepner-Groth-Nikolaus, Harpaz, and Carmeli-Schlank-Yanovski’s theory of modes, giving existence and uniqueness of a presentable G-symmetric monoidal structure on G-spectra when G is a compact Lie group (with unit Σ^∞S^0) incorporating finite-index Hill-Hopkins-Ravanel norms. In the final chapter, we use these developments to perform two important arguments regarding interchange: we prove a parameterized version of Schlank-Yanovski’s ∞-categorical Eckmann-Hilton arguments (realizing a mild generalization of Blumberg-Hill’s N_∞-operads precisely as the G-operads arising from infinitely iterated Eckmann-Hilton arguments), and we lift Dunn-Lurie additivity to little disk G-operads for finite G, as well as versions with tangential structure specified by an arbitrary map of G-spaces B → B_GO(n).