Maurer–Cartan Equation and Deligne Groupoids

Abstract

In 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.