Abstract

After recalling Lurie's axiomatic framework for deformation theory, I will formulate the deformation theory of k-linear ∞-categories within this framework. Following joint work with Blanc and Katzarkov, I will describe the relationship between formal deformations of a category and solutions to the Maurer-Cartan equation in a certain differential-graded Lie algebra associated with the category. Simultaneous deformations of categories and objects in them will play an important role. Finally, I will discuss generalizations of these ideas to deformations of E_n-monoidal categories and applications to topological field theories. This is based on joint work in progress with Bhanu Kiran.