A motivic homotopy theory without \(\mathbb {A}^{1}\)-invariance


In this paper, we continue the program initiated by Kahn–Saito–Yamazaki by constructing and studying an unstable motivic homotopy category with modulus \(\overline{\mathbf{M}}\mathcal {H}(k)\), extending the Morel–Voevodsky construction from smooth schemes over a field k to certain diagrams of schemes. We present this category as a candidate environment for studying representability problems for non \(\mathbb {A}^1\)-invariant generalized cohomology theories.

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    The paper [22] has been superseded by [21] during the revision of this manuscript. Since the latter is not yet available to the public, we have decided to keep the references to the obsolete version. To the best of our knowledge, the relevant parts will appear unchanged in [21] (but presumably with a different numbering).


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This paper is essentially based on the second chapter of my PhD thesis, written in Essen under the supervision of Marc Levine. It is a pleasure to thank him heartily for much advice, constant support and encouragement. I would also like to thank Moritz Kerz and Shuji Saito for valuable comments and many friendly conversations on these topics, as well as Lorenzo Mantovani and Mauro Porta for many remarks on a preliminary version of this paper. Finally, I’m grateful to the anonymous referee for the careful reading of the manuscript, and for suggesting many improvements in the exposition.

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Correspondence to Federico Binda.

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Partially supported by the CRC SFB 1085 “Higher Invariants” (University of Regensburg) of the Deutsche Forschungsgemeinschaft during the preparation of the manuscript.

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Binda, F. A motivic homotopy theory without \(\mathbb {A}^{1}\)-invariance. Math. Z. 295, 1475–1519 (2020). https://doi.org/10.1007/s00209-019-02399-2

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Mathematics Subject Classification

  • Primary 14F42
  • Secondary 19E15