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We give a general treatment of deformation theory from the point of view ofhomotopical algebra following Hinich, Manetti and Pridham. In particular, weshow that any deformation functor in characteristic zero is controlled by acertain differential graded Lie algebra defined up to homotopy, and alsoformulate a noncommutative analogue of this result valid in any characteristic.

Review of deformation theory II: a homotopical approach