Free Higher Groups in Homotopy Type Theory

TL;DR

This paper explores the properties of free higher groups in homotopy type theory, specifically examining whether the free group constructed from a set A retains set-like properties, with results on its fundamental groups.

Contribution

It provides an approximation to an open problem by showing that the 1-truncation of the free higher group F(A) has trivial fundamental groups.

Findings

Fundamental groups of F(A) are trivial.

The 1-truncation of F(A) is a set.

Addresses an open problem in HoTT regarding the nature of free higher groups.

Abstract

Given a type A in homotopy type theory (HoTT), we can define the free infinity-group on A as the loop space of the suspension of A+1. Equivalently, this free higher group can be defined as a higher inductive type F(A) with constructors unit : F(A), cons : A -> F(A) -> F(A), and conditions saying that every cons(a) is an auto-equivalence on F(A). Assuming that A is a set (i.e. satisfies the principle of unique identity proofs), we are interested in the question whether F(A) is a set as well, which is very much related to an open problem in the HoTT book. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that the 1-truncation of F(A) is a set.