A Hopf algebra model for Dwyer's tame spaces

In this thesis, we give a modern treatment of Dwyer's tame homotopy theory using the language of $\infty$-categories.We introduce the notion of tame spectra and show it has a concrete algebraic description.We then carry out a study of $\infty$-operads and define tame spectral Lie algebras and tame spectral Hopf algebras. Finally, we prove that the homotopy theory of tame spectral Hopf algebras is equivalent to that of tame spaces. To recover Dwyer's Lie algebra model for tame spaces, we use Koszul duality to construct a universal enveloping algebra functor, and show it is an equivalence from the $\infty$-category of tame spectral Lie algebras to the $\infty$-category of tame spectral Hopf algebras.