Review of deformation theory II: a homotopical approach

Abstract

We give a general treatment of deformation theory from the point of view of homotopical algebra following Hinich, Manetti and Pridham. In particular, we show that any deformation functor in characteristic zero is controlled by a certain differential graded Lie algebra defined up to homotopy, and also formulate a noncommutative analogue of this result valid in any characteristic. Contents1. Introduction 2. Closed model categories 3. Brown representability theorem for compactly generated model categories 4. MC elements and MC moduli sets 4.1. MC moduli in dg Lie algebras 4.2. MC moduli in dg algebras 5. Koszul duality 5.1. DG coalgebras and pseudocompact dg algebras 5.2. Quillen equivalence between DGLA and pcCDGA op loc 5.3. Quillen equivalence between DGA/k and pcDGA op loc 5.4. Relationship between two types of Koszul duality 6. Main theorems 6.1. MC elements and the deformation functor based on a dg Lie algebra 6.2. Finding a dg Lie algebra associated with a deformation functor 6.3. Associative deformation theory 6.4. Finding a dg algebra associated with a deformation functor 6.5. Comparing commutative and associative deformations References