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\begin{document}

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\section*{analytic motives}

Analytic motives give a common generalization of classical homotopy theory (Ayoub's analytic motives over C), algebraic motivic homotopy theory (strict analytic motives over Z with the trivial non-archimedean norm), and Ayoub's non-archimedean motives.

The basic idea is to extend motivic homotopy theory to a general base Banach ring, by replacing the affine line by the overconvergent unit disc.

Analytic stable homotopy theory gives a natural setting to define homotopy invariant cohomologies, that are particularly interesting when one works over an extension of the field of rational numbers. However, they seem to be less well adapted to the study of torsion phenomena in de Rham type cohomology theories over an integral base, because homotopy invariance is not available in this situation, and imposing it artificially kills the p-torsion information in caracteristic p.

References:

Joseph Ayoub: Betti realization of motives and motives for rigid analytic varieties.

[[Frederic Paugam]]: Overconvergent global analytic geometry (in preparation).



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