Coalgebraic Models for Equivariant Homotopy Types

This thesis investigates and continues a line of mathematical research dating back to the 1960's. Algebraic topology studies the algebraic invariants associated with topological spaces, and Quillen [Q69] was the first to describe an algebraic invariant that allows one to completely recover the space up rational homotopy equivalence. In the nineties, Goerss [G95] showed that the simplicial coalgebra of $\F$-chains associated to a simplicial set could recover the space up to $\F$-homology equivalence, for $\F$ an algebraically closed field. More recently, Raptis and Rivera [RR22] introduced another notion of equivalence between simplicial sets which remembers information about the classical invariant known as the fundamental group and showed that the simplicial coalgebra of $\F$-chains could be used to reconstruct the simplicial set. The work in this thesis explores what occurs equivariantly, i.e., when the action of a group, $G$, is present on the simplicial set, and promotes the results of Raptis and Rivera [RR22] to the equivariant setting.

The first chapter provides some motivation and historical context for this thesis work. Next, in Chapter 2, we will review some nonequivariant definitions and notions, most of which can be found in Raptis and Rivera's article, [RR22]. In the third chapter, we will introduce the relevant equivariant notions and model structures that will be used in the fourth chapter. Chapter 4 is dedicated to the bulk of the equivariant homotopical results. Finally, in the last chapter we provide some partial results towards describing a notion of equivariant Koszul Duality motivating future research directions.