Shape theory for condensed anima

TL;DR

This paper explores the concept of shape in condensed anima, connecting it to classical topological notions and extending cohomology comparison results, while also investigating condensed homotopy groups and related topological group structures.

Contribution

It introduces new perspectives on shape theory for condensed anima, recovering classical notions and extending cohomology comparison results, along with describing condensed homotopy groups and topological groups.

Findings

Shape for condensed anima recovers classical shape notions for certain spaces.

Extends comparison results on sheaf and condensed cohomology.

Provides a description of condensed homotopy groups and related topological groups.

Abstract

We give different perspectives on the notion of shape for condensed anima. We prove that it recovers more classical notions of shape for topological spaces in the cases of all paracompact compactly generated spaces and all locally contractible spaces. These recovering statements imply and extend comparison results on sheaf and condensed cohomology by Clausen-Scholze and Haine. Another homotopy-theoretical direction for condensed anima are their condensed homotopy groups. Connected to this, we give a description of the underlying topological group functor on condensed groups via quasi-topological groups.