For a fixed real parameter 0 < p <= 1, a p-liquid real vector space is a condensed R-vector space carrying a canonical action of the spaces of measures M_<p(S) on profinite sets S. The point of the construction is that the resulting category is abelian and stable under kernels, cokernels and extensions - exactly the property that ordinary topological vector spaces fail - and that the M_<p(S) are compact projective generators, so homological algebra over the reals becomes workable. This is what makes it possible to do analytic geometry over R in the same language as the non-archimedean case.
Lecture notes for a course taught by Peter Scholze at the University of Bonn in the winter term 2019/20, on joint work with Dustin Clausen; the posted version is lightly updated as of May 2026. A con…