Non-abelian motives

Non-abelian Hodge theory, developed by Corlette, Simpson and others, equips representation spaces of the fundamental group of a complex variety with structures analogous to those on its cohomology. Taking that analogy further suggests that every cohomological structure should admit a counterpart in the non-abelian setting. Daniel Litt presents work toward this programme, related conjectures, and new structures whose analogues in the abelian setting remain less clear.