Why do physicists care about the triangulated/derived categories? I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror symmetry and stability of D-branes.

Also I would like to know some problems in algebraic geometry which we hope to solve using triangulated categories, e.g. the derived category of coherent sheaves on an algebraic variety is believed to capture the geometry of the variety.

I am not sure whether to make it community wiki, feel free to do so. Thanks.