Rational motivic path spaces and Kim's relative unipotent section conjecture

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

Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories

Periodizable motivic ring spectra

We show that the cellular objects in the module category over a motivic E infinity ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is strongly periodizable we obtain topological incarnations of motivic Landweber spectra. Under some categorical assumptions the unit object of the model category for triangulated motives is as well strongly periodizable giving motivic cochains whose module category models integral triangulated categories of Tate motives.Comment: 15 page