Operads and Maurer–Cartan spaces

Overview & Synopsis

This thesis is inscribed in the topics of operad theory and homotopical algebra. Suppose we are given a type of algebras, a type of coalgebras, and a relationship between those types of algebraic structures (encoded by an operad, a cooperad, and a twisting morphism respectively). Then, it is possible to endow the space of linear maps from a coalgebra C and an algebra A with a natural structure of Lie algebra up to homotopy. We call the resulting homotopy Lie algebra the convolution algebra of A and C. In this thesis, we study the theory of convolution algebras and their compatibility with the tools of homotopical algebra : infinity morphisms and the homotopy transfer theorem. After doing that, we apply this theory to various domains, such as derived deformation theory and rational homotopy theory. In the first case, we use the tools we developed to construct an universal Lie algebra representing the space of Maurer-Cartan elements, a fundamental object of deformation theory. In the second case, we generalize a result of Berglund on rational models for mapping spaces between pointed topological spaces.