The Pro-\'Etale Homotopy Type

In this paper we define the pro-\'etale homotopy type of a scheme and prove some of its expected properties. Our definition is similar to the definition of the \'etale homotopy type by Michael Artin and Barry Mazur. We prove that for a qcqs scheme the pro-\'etale homotopy type is profinite, determined by a single split affine weakly contractible hypercovering and computes the cohomology of a certain class of sheaves. We show that the pro-\'etale homotopy type of a w-contractible scheme is trivial and compute the pro-\'etale homotopy type of the real numbers. Moreover, we prove that a suitable version of $\pi_0$ composed with the pro-\'etale homotopy type gives back the space of components of the base scheme. We make some progress towards describing the pro-\'etale homotopy type of arbitrary fields. Lastly, we give a refined definition of the pro-\'etale homotopy type using the theory by Ilan Barnea and Tomer M. Schlank and the theory of condensed sets by Dustin Clausen and Peter Scholze. This allows us to define pro-\'etale homotopy groups associated to pointed qcqs schemes.