# Homotopy types and geometries below Spec Z

@article{Manin2018HomotopyTA, title={Homotopy types and geometries below Spec Z}, author={Yu. I. Manin and Matilde Marcolli}, journal={arXiv: Algebraic Geometry}, year={2018} }

After the first heuristic ideas about `the field of one element' F_1 and `geometry in characteristics 1' (J.~Tits, C.~Deninger, M.~Kapranov, A.~Smirnov et al.), there were developed several general approaches to the construction of `geometries below Spec Z'. Homotopy theory and the `the brave new algebra' were taking more and more important places in these developments, systematically explored by B.~Toen and M.~Vaquie, among others.
This article contains a brief survey and some new results on…

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