ncts

Deformation theory aims to classify infinitismal deformations of mathematical objects. For many geometric objects, this comes out to understanding the Lie algebra structure on their tangent spaces. A key technique in proving these arguments is Koszul duality, which gives the cotangent complex of an algebra the extra structure of a Lie coalgebra, then you take a linear dual to get the Lie algebra structure on the tangent space. My work currently aims to understand the role of coalgebras in deformation theory. We develop a theory of a tangent complex and a coalgebraic version of formal moduli problems, and relate them to the algebraic theories. On the way we also need to understand the problem of rectification of coalgebras: relating the model structure of coalgebras with the infinity categorical definition of coalgebras, which unlike with algebras is not true in many cases.