This PhD thesis consists of four papers treating topics in rational homotopy theory.

In Paper I, we establish two formality conditions in characteristic zero. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg associative algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg associative algebra. We present some consequences of these theorems in rational homotopy theory.

In Paper II, which is coauthored with Alexander Berglund, we construct a dg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a subspace, so called relative homotopy automorphisms.

In Paper III, which is coautohored with Hadrien Espic, we prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel.

In Paper IV, we study rational homological stability for the classifying space of the monoid of homotopy automorphisms of iterated connected sums of complex projective 3-spaces.

This PhD thesis consists in a collection of three papers on Koszul duality of categories and on an analogue of the Sullivan-Wilkerson theorem for relative CW-complexes.

In Paper I, we define a general notion of Koszul dual in the context of a monoidal biclosed model category. We apply it to a category of enriched graphs to define the Koszul dual of an augmented enriched category C. We mostly study the case of categories enriched over a stable model category. We establish the expected adjunctions between categories of modules over C and modules over its Koszul dual K(C), and investigate the question of when the map from C to its double dual K(K(C)) is an equivalence. We also, importantly, show that Koszul duality of operads can be understood as a special case of Koszul duality of categories.

In Paper II, we investigate further this notion of Koszul duality in the case of categories enriched over a category of chain complexes. In this setting, there is a natural cocategory structure on the bar construction on a category C. We show that the dual of this bar cocategory is equivalent to our definition of the Koszul dual of C.

In Paper III, coauthored with Bashar Saleh, we prove more general versions of two important consequences of the Sullivan-Wilkerson theorem. Namely, we show that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented, and that its associated rationalization map has finite kernel.