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.
Highlights
An important feature of Martin-Löf type theory (MLTT) is the identity type which makes it possible to express equality inside type theory
Homotopy type theory (HoTT) embraces the fact that x = y may come with interesting structure
We define F(A) to be the higher inductive type (HIT) [22, Chp 6] which as constructors has a neutral element unit : F(A) and a multiplication operation cons : A → F(A) → F(A), together with conditions ensuring that each cons(a) : F(A) → F(A) is an equivalence
Summary
An important feature of Martin-Löf type theory (MLTT) is the identity type which makes it possible to express equality inside type theory. We define F(A) to be the higher inductive type (HIT) [22, Chp 6] which as constructors has a neutral element unit : F(A) and a multiplication operation cons : A → F(A) → F(A), together with conditions ensuring that each cons(a) : F(A) → F(A) is an equivalence. This definition encodes the a priori infinite tower of coherence condition suitably and it will turn out that it is equivalent to the loop space of Σ(A+ 1).
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