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

## TL;DR

This paper explores motivic path spaces and their relation to Kim's conjecture, providing new insights into the structure of unipotent fundamental groups within the motivic framework.

## Contribution

It introduces the concept of motivic dga's for studying path spaces and decomposes Kim's conjecture into smaller, homotopically flavored conjectures.

## Key findings

- Path spaces of punctured projective line over a number field are concentrated in degree zero.
- Reconstruction of unipotent fundamental groups from motivic dga's.
- A step towards Kim's relative unipotent section conjecture.

## Abstract

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.