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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.

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I would make this community wiki, if I was you. (I don't have the power to do so) – David Roberts Dec 30 '10 at 6:54
Have you checked out Eric Sharpe's paper "Derived categories and stacks in physics" ? – j.c. Dec 30 '10 at 8:17
@ Gjergji - I looked at the questions, what I am really interested in are the problems which we hope to approach via triangulated categories. Like the homological mirror symmetry is relation between derived categories, D-branes in string theory etc. – J Verma Dec 30 '10 at 9:32
see the answer at – Arnold Neumaier Sep 11 '14 at 7:39

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