Categorifying non-commutative deformation theory

I will discuss a categorification of the non-commutative deformation theory of n objects in an abelian category. A suitable abelian category plays the role of the non-commutative base for a deformation in this approach. A motivation coming from a categorical description of flops will be outlined. I will give sufficient conditions for the pro-representability of the deformation functor. I will also construct a pro-representing hull for the deformation functor in general situation and discuss how to recover the functor from the hull.