Categorical resolutions and categorical absorptions of singularities

A resolution of singularities of a singular algebraic variety is a proper morphism from a smooth algebraic variety which induces an isomorphism of dense open subsets. An old but extremely important theorem of Hironaka proves that, over a field of characteristic zero, any algebraic variety admits a resolution.

Similarly, an absorption of singularities of a singular algebraic variety is a proper morphism to a smooth algebraic variety which induces an isomorphism of dense open subsets. In contrast to resolution, existence of absorption is a very rare phenomenon.

I will talk about a categorical version of resolution and absorption of singularities, give some examples, and try to demonstrate how useful these notions are.