Can Yaylali (TU Darmstadt): A^1-homotopy theory of rigid analytic spaces
In this talk, I will report about work with Christian Dahlhausen (Heidelberg) on A^1-homotopy theory of rigid analytic spaces. The B^1-homotopy category has already been defined and studied by Ayoub and a full six-functor formalism was established by Ayoub-Gallauer-Vezzani. One drawback of the B^1-invariant theory is that analytic K-theory for rigid analytic spaces (as defined and studied by Kerz-Saito-Tamme) is not representable since it is not B^1-invariant. Thus we aim for an A^1-invariant version with coefficients in any presentable category. For the stable theory, we can prove the existence of a partial six-functor formalism for analytifications of schemes and algebraic morphisms between them by using the results of Ayoub’s thesis. Furthermore, using coefficients in condensed spectra, we can represent analytic K-theory as the P^1-ring spectrum Z x BGL. If time permits I will also highlight some of the remaining questions.

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