Towards a unification of equivariant homotopical algebra and parametrized higher algebra

In this dissertation, we compare two approaches for equivariant algebra in homotopy theory. A central principle of homotopy theory is the Grothendieck hypothesis, which states that topological spaces should be interpreted, up to homotopy, as ∞-groupoids.This links two view points on classical homotopy theory namely, topology and higher category theory. In equivariant  homotopy theory, a similar story can be told with Elmendorf’s theorem which provides a description of equivariant spaces in terms of G-∞-groupoids. In particular, equivariant homotopy theory can be studied through the lens of equivariant higher category theory. 

A natural question now arises:  “Given a mathematical notion with some topological and equivariant flavor, can it be described, up to homotopy, within the framework of equivariant higher categories?” 

In this thesis, we study this question in the case of algebraic objects. We provide a full answer for commutative algebras and we develop ideas and constructions toward a complete answer to more general algebraic objects.