Categorical resolutions of singularities

A categorical resolution of singularities of an algebraic variety $Y$ is a triangulated category $T$ with an adjoint pair of functors between $D(Y)$ (the derived category of quasicoherent sheaves on $Y$) and $T$, such that the composition is the identity endofunctor of $D(Y)$. If $X \to Y$ is a usual resolution, the derived category $D(X)$ with pullback and pushforward functors is a categorical resolution only if $Y$ has rational singularities. However, I will explain that even if $Y$ has nonrational singularities, still one can construct a categorical resolution of $D(Y)$ by gluing derived categories of appropriate smooth algebraic varieties. This is a work in progress, joint with Valery Lunts.