Big fundamental groups: generalizing homotopy and big homotopy

TL;DR

This paper extends big homotopy theory by identifying a canonical cardinal for each space that captures all big loops and homotopies, generalizing the original concept introduced by Cannon and Conner.

Contribution

It introduces a canonical cardinal for each space that suffices to detect all big loops and homotopies, broadening the scope of big homotopy theory.

Findings

Existence of a canonical cardinal for each space

Detection of all big loops and homotopies using this cardinal

Generalization of big homotopy theory concepts

Abstract

The concept of big homotopy theory was introduced by J. Cannon and G. Conner using big intervals of arbitrarily large cardinality to detect big loops. We find, for each space, a canonical cardinal that is sufficient to detect all big loops and all big homotopies in the space.