Free higher groups in homotopy type theory

Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the 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). Equivalently, we can take the loop space of the suspension of A + 1. 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 [20, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ∥F(A)∥1 is a set

Free higher groups in homotopy type theory

Given a type A in homotopy type theory (HoTT), we can define the free ∞-group on A as the 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). Equivalently, we can take the loop space of the suspension of A + 1. 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 [20, Ex. 8.2]. We show an approximation to the question, namely that the fundamental groups of F(A) are trivial, i.e. that ∥F(A)∥1 is a set

On the Nielsen-Schreier Theorem in Homotopy Type Theory

We give a formulation of the Nielsen-Schreier theorem (subgroups of free groups are free) in homotopy type theory using the presentation of groups as pointed connected 1-truncated types. We show the special case of finite index subgroups holds constructively and the full theorem follows from the axiom of choice. We give an example of a boolean infinity topos where our formulation of the theorem does not hold and show a stronger "untruncated" version of the theorem is provably false in homotopy type theory

Path spaces of higher inductive types in homotopy type theory

The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions. To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman '13), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman '16). We provide a formalization of the main technical results in the proof assistant Lean.Comment: v1: 23 pages; v2: 24 pages, small reformulations and reorganization

A Rewriting Coherence Theorem with Applications in Homotopy Type Theory

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution "Coherence via Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the language of higher-dimensional rewriting.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2001.0765

A rewriting coherence theorem with applications in homotopy type theory

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higher-dimensional rewriting

On the Nielsen-Schreier Theorem in Homotopy Type Theory

We give a formulation of the Nielsen-Schreier theorem (subgroups of free groups are free) in homotopy type theory using the presentation of groups as pointed connected 1-truncated types. We show the special case of finite index subgroups holds constructively and the full theorem follows from the axiom of choice. We give an example of a boolean infinity topos where our formulation of the theorem does not hold and show a stronger "untruncated" version of the theorem is provably false in homotopy type theory