Maurer–Cartan Equation and Deligne Groupoids

Abstract

AbstractIn this chapter we introduce the natural transformation \({{\text {Def}}}\) from the category of DG-Lie algebras, over a field of characteristic 0, to the category of deformation functors. Then we prove the homotopy invariance of \({{\text {Def}}}\), namely that for every quasi-isomorphism \(L\rightarrow M\), the induced natural transformation \({{\text {Def}}}_L\rightarrow {{\text {Def}}}_M\) is an isomorphism. The explicit functorial construction \(L\mapsto {{\text {Def}}}_L\) is precisely the one involved in the general philosophy that, in characteristic 0, every deformation problem is controlled by a differential graded Lie algebra. From now on, and throughout the rest of this book, every vector space is considered over a base field of characteristic 0; unless otherwise specified, by the symbol \(\otimes \) we mean the tensor product over the base field.