We initiate a study of path spaces in the nascent context of "motivic dga's",
under development in doctoral work by Gabriella Guzman. This enables us to
reconstruct the unipotent fundamental group of a pointed scheme from the
associated augmented motivic dga, and provides us with a factorization of Kim's
relative unipotent section conjecture into several smaller conjectures with a
homotopical flavor. Based on a conversation with Joseph Ayoub, we prove that
the path spaces of the punctured projective line over a number field are
concentrated in degree zero with respect to Levine's t-structure for mixed Tate
motives. This constitutes a step in the direction of Kim's conjecture.Comment: Minor corrections, details added, and major improvements to
exposition throughout. 52 page
