Animated Condensed Sets and Their Homotopy Groups

TL;DR

This paper explores extending homotopy group concepts from topological spaces to condensed sets, establishing functors to pro-groups and extending to $mbda$-categories, with a focus on extremally disconnected profinite sets.

Contribution

It introduces functors from condensed sets to pro-groups that align with classical homotopy groups on CW-spaces and extends these to condensed anima and pro-anima categories.

Findings

Existence of functors from condensed sets to pro-groups matching classical homotopy groups on CW-spaces.

Extension of these functors to $mbda$-categories of condensed anima and pro-anima.

Identification of extremally disconnected profinite sets as compact projective objects in condensed sets.

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.