Derived deformation functors, Koszul duality and Maurer-Cartan spaces.

We summarize the chain of comparisons (Pridham 2011 Unifying derived deformation theories. Adv. Math. 224(3), 772-826, 2010. Corrigendum 228, 2554-2556 (doi:10.48550/arXiv.0705.0344)) showing Hinich's derived Maurer-Cartan functor gives an equivalence between differential graded Lie algebras and derived Schlessinger functors on Artinian differential graded-commutative algebras. We include some motivating deformation problems and analogues for more general Koszul dual pairs of operads. This article is part of the theme issue 'Derived Lie algebras in geometry and topology'.