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 viewpoints 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.
Grégoire Marc was born on the 5th of September 1998 in Étampes, France. After finishing high school in Saint-Girons, he started in 2016 the bachelor parcours spécial at Université Paul-Sabatier in Toulouse. In 2019, he completed his first year of a Master’s degree in mathematics in Marseille. In 2020, he received the Lebesgue Master Scholarship and completed his second year of the Master’s program at Université Rennes 1. He wrote his master’s thesis, entitled “Problèmes de modules formels”, under the supervision of Sinan Yalin in Angers. Since April 2022, he has been a PhD candidate at the mathematics department of Radboud University, under the supervision of Magdalena Kędziorek.