Animated Condensed Sets and Their Homotopy Groups

Abstract: The theory of condensed mathematics by Dustin Clausen and Peter Scholze claims that topological spaces should be replaced by the definition of condensed sets. The main purpose of this paper is to investigate in which way the theory of homotopy groups on topological spaces can be extended to the theory of condensed sets. We show that there exist functors from the category of condensed sets to the category of pro-groups, s.t. restriction to CW-spaces coincides with the ordinary notion of homotopy groups on topological spaces. These functors can be extended to the $\infty$-categories of condensed anima and pro-anima. On our way to these results we will prove that the compact projective objects in the category of condensed sets are given by the extremally disconnected profinite sets.