Historically, differential graded Lie algebras (DGLA) and the Maurer–Cartan equation da+[a,a]/2 = 0 have played a central role in many areas, from deformation theory of various algebraic structures to models in rational homotopy theory and differential geometry. We will discuss the abstract problem of associating a DGLA to a cell complex in a functorial and meaningful way, along with reconstruction of the original geometry (topology) and determination of explicit formulae. The model embeds the infinity cocommutative coalgebra structure on the chains of the cell complex and thus its subdivision maps are expected to give correction maps between different scales relevant to the problem of finding good discrete computational models.

Based on joint work with Dennis Sullivan, Nir Gadish and Itay Griniasty.