Morphisms of Rational Motivic Homotopy Types

Ishai Dan-Cohen*, Tomer Schlank

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We investigate several interrelated foundational questions pertaining to the study of motivic dga’s of Dan-Cohen and Schlank (Rational motivic path spaces and Kim’s relative unipotent section conjecture. arXiv:1703.10776) and Iwanari (Motivic rational homotopy type. arXiv:1707.04070). In particular, we note that morphisms of motivic dga’s can reasonably be thought of as a nonabelian analog of motivic cohomology. Just as abelian motivic cohomology is a homotopy group of a spectrum coming from K-theory, the space of morphisms of motivic dga’s is a certain limit of such spectra; we give an explicit formula for this limit—a possible first step towards explicit computations or dimension bounds. We also consider commutative comonoids in Chow motives, which we call “motivic Chow coalgebras”. We discuss the relationship between motivic Chow coalgebras and motivic dga’s of smooth proper schemes. As a small first application of our results, we show that among schemes which are finite étale over a number field, morphisms of associated motivic dga’s are no different than morphisms of schemes. This may be regarded as a small consequence of a plausible generalization of Kim’s relative unipotent section conjecture, hence as an ounce of evidence for the latter.

Original languageAmerican English
Pages (from-to)311-347
Number of pages37
JournalApplied Categorical Structures
Issue number2
StatePublished - Apr 2021

Bibliographical note

Publisher Copyright:
© 2020, Springer Nature B.V.


  • Algebraic K-theory
  • Infinity categories
  • Motives
  • Rational homotopy


Dive into the research topics of 'Morphisms of Rational Motivic Homotopy Types'. Together they form a unique fingerprint.

Cite this