Abstracts Mathematics

Equivariant Stable Homotopy Theory for Diagrams

by Hannah Christine Housden

We begin with the observation that a group G is just a category with one object where every morphism is an isomorphism and that a G-space is just a functor out of G. The rest of the dissertation re-develops equivariant stable homotopy theory by replacing G with a (usually finite) category, D.The first chapter considers the unstable case. Our main tool is that of an orbit, a generalization of subgroups of the form G/H. We show that several notions often framed in terms of subgroups H ≤ G can be rephrased purely in terms of orbits.The second chapter explores the homotopical structure and homotopy invariants of D- spaces. Its final section gives a generalization of “Elmendorf’s Theorem,” which states that the category of D-spaces is Quillen equivalent to the category of functors out of the orbit category of D.The third chapter considers the stable case. We build equivariant spectra as D-spectral Mackey functors, originally introduced in the group case by Guillou and May. We then construct examples including suspension spectra and Eilenberg-MacLane spectra. The penul- timate section gives a generalization of geometric fixed points, and the final section gives specific computations of Mackey functors over the two-object category J.