« Seminar on homological algebra »
A Maurer-Cartan (MC) element in a differential graded (dg) algebra A is an odd element x satisfying the equation dx+x2=0. The group of invertible elements of A acts on MC element by gauge transformations: g(x):=gxg-1-dgg-1. MC elements are an abstraction of the notion of a flat connection and are fundamental in many problems of homological algebra, deformation theory, differential geometry etc.
There is a notion of a (Sullivan) homotopy of MC elements: two such are homotopic if they could be extended to a family over the de Rham algebra on the interval R[x,dx]. A fundamental result (over 40 years old) due to Schlessinger and Stasheff (SS) states that (under certain assumptions) two MC elements are gauge equivalent if an only if they are homotopic.
There is also another notion of homotopy of MC elements, based on the singular cochain complex of the interval, and a corresponding SS type theorem.