This is work in progress with Pranav Pandit. I will explain how to use Lurie's theory of formal moduli problems and their approximations in order to obtain a better understanding of the deformation theory of a dg-category up to Morita equivalence. In the case of a smooth proper dg-category, this leads to the compact generation of any formal deformation.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3946